Kadlec plotting

kadlec-blend.c

// run this yourself at https://replit.com/@Protonk1/kadlec-testing

#include <math.h>
#include <stdint.h>
#include <stdio.h>
#include <time.h>

typedef struct {
  float input;
  float output;
  float error;
  float diff;
  int iters;
} ISRResult;

void interpolatedISR(float x, int NR, float alpha, int constraint, ISRResult *result) {
  float prev_output;
  float reference = 1 / sqrtf(x);
  // https://doi.org/10.1109/38.595279
  // Blinn makes a slight modification
  // to NR values but chooses a
  // constant purely for exponent restoration
  int BlinnM = 0x5F400000;

  // http://rrrola.wz.cz/inv_sqrt.html
  // Kadlec determined magic & NR parameters
  // via a brute force search
  int KadlecM = 0x5F1FFFF9;

  // Interpolate the magic number linearly
  int magic = BlinnM - alpha * (BlinnM - KadlecM);

  // FISR guess
  union {
    float f;
    uint32_t u;
  } y = {x};
  y.u = magic - (y.u >> 1);
  // Helps us produce a difference without
  // worrying about N - 1 for the 1st iteration
  float initial = y.f;

  // Standard newton's method approximates
  // 1/sqrt(x) (A - B x (1/sqrt(x))^2) ~ 1/sqrt(x)
  // where A = 1.5, B = 0.5.
  // Blinn chooses slightly different values
  // which meet the constraint of A = B - 1
  float A1 = 1.47f; // BlinnISR A constant
  float B1 = 0.47f; // BlinnISR B constant

  // Kadlec's method approximates
  // A/sqrt(x) (B - x (1/sqrt(x))^2) ~ 1/sqrt(x)
  float A2, B2;
  if (constraint == 1) {
    // A = 0.712 & B = 2.4045 meet the constraint
    // A(B - 1) = 1
    A2 = 0.712f; 
    B2 = 2.4045f;
  } else {
    // Original constants were chosen to be
    // optimal for one iteration.
    A2 = 2.4045f; 
    B2 = 0.712f; 
  }

  // Standard method implies a constraint of A = B - 1
  // Kadlec's implies a constraint of A(B - 1) = 1
  // The blended constraint:
  // (1-α)(A = B + 1) + α(A(B - 1) = 1)
  float A = (1 - alpha) * A1 + alpha * A2;
  float B = (1 - alpha) * B1 + alpha * B2;

  for (int i = 1; i <= NR; i++) {
    // Two approaches to Newton's Method
    // Standard: y = y(A - Bxy^2)
    // Kadlec:  y = Ay(B - xy^2)
    // Blend the two along with params
    // Blended:  y = ((1-α)y + αAy) * (B - xy^2)
    float blinn_term = y.f * (A1 - B1 * x * y.f * y.f);
    float kadlec_term = A2 * y.f * (B2 - x * y.f * y.f);
    y.f = (1 - alpha) * blinn_term + alpha * kadlec_term;
    // 2 from our max is where we need a diff from
    if (i == NR - 2)
      prev_output = y.f;
  }

  result->input = x;
  result->output = y.f;
  result->error = fabsf(y.f - reference) / reference;
  result->diff = (NR == 1) ? y.f - initial : y.f - prev_output;
  result->iters = NR;
}

int main() {
  // Values chosen which are highest in initial error for 
  // the unmodified Blinn and Kadlec magic constants
  // This allows us to see error behavior easier
  float inputs[] = {0.0410f, 0.0313f, 0.0527f, 0.0324f, 0.05533797f};

  printf("input,output,error,diff,iters,alpha\n");

  for (int i = 0; i < 5; i++) {
    float x = inputs[i];
    for (int a = 0; a <= 10; a++) {
      float alpha = (float)a / 10.0f;
      // 8 iterations shows Kadlec stalling
      for (int NR = 1; NR <= 8; NR++) {
        ISRResult result = {0};
        interpolatedISR(x, NR, alpha, 1, &result);

        printf("%f,%f,%f,%f,%d,%d\n", result.input, result.output, result.error,
               result.diff, result.iters, a);
      }
    }
  }

  return 0;
}

kadlec-blend.R

### Compare original and adjusted values 
library(dplyr)
library(ggplot2)
library(latex2exp)


## blends from standard form to original Kadlec constants
## A = 0.70395 & B = 2.38924
original_kadlec_blend <- read.csv(text = "
input,output,error,diff,iters,alpha
0.041000,4.908667,0.006071,-0.467333,1,0
0.041000,4.936593,0.000416,0.027926,2,0
0.041000,4.938523,0.000025,0.029856,3,0
0.041000,4.938641,0.000002,0.002048,4,0
0.041000,4.938648,0.000000,0.000124,5,0
0.041000,4.938648,0.000000,0.000007,6,0
0.041000,4.938648,0.000000,0.000000,7,0
0.041000,4.938648,0.000000,0.000000,8,0
0.041000,4.896584,0.008517,-0.379440,1,1
0.041000,4.926771,0.002405,0.030187,2,1
0.041000,4.927589,0.002239,0.031005,3,1
0.041000,4.927604,0.002236,0.000834,4,1
0.041000,4.927605,0.002236,0.000016,5,1
0.041000,4.927604,0.002236,0.000000,6,1
0.041000,4.927605,0.002236,0.000000,7,1
0.041000,4.927604,0.002236,0.000000,8,1
0.041000,4.889894,0.009872,-0.286094,1,2
0.041000,4.917985,0.004184,0.028092,2,2
0.041000,4.917528,0.004277,0.027634,3,2
0.041000,4.917540,0.004274,-0.000446,4,2
0.041000,4.917539,0.004274,0.000011,5,2
0.041000,4.917540,0.004274,0.000000,6,2
0.041000,4.917539,0.004274,0.000000,7,2
0.041000,4.917540,0.004274,0.000000,8,2
0.041000,4.887794,0.010297,-0.188219,1,3
0.041000,4.909569,0.005888,0.021775,2,3
0.041000,4.908243,0.006157,0.020450,3,3
0.041000,4.908333,0.006138,-0.001236,4,3
0.041000,4.908327,0.006140,0.000084,5,3
0.041000,4.908328,0.006139,-0.000005,6,3
0.041000,4.908328,0.006139,0.000001,7,3
0.041000,4.908327,0.006140,-0.000000,8,3
0.041000,4.889552,0.009941,-0.086424,1,4
0.041000,4.900959,0.007631,0.011407,2,4
0.041000,4.899745,0.007877,0.010193,3,4
0.041000,4.899879,0.007850,-0.001080,4,4
0.041000,4.899864,0.007853,0.000119,5,4
0.041000,4.899866,0.007853,-0.000013,6,4
0.041000,4.899866,0.007853,0.000002,7,4
0.041000,4.899866,0.007853,0.000000,8,4
0.041000,4.894414,0.008957,0.018414,1,5
0.041000,4.891706,0.009505,-0.002708,2,5
0.041000,4.892119,0.009421,-0.002295,3,5
0.041000,4.892057,0.009434,0.000350,4,5
0.041000,4.892066,0.009432,-0.000053,5,5
0.041000,4.892065,0.009432,0.000008,6,5
0.041000,4.892065,0.009432,-0.000001,7,5
0.041000,4.892065,0.009432,0.000000,8,5
0.041000,4.901685,0.007485,0.125660,1,6
0.041000,4.881477,0.011576,-0.020208,2,6
0.041000,4.885504,0.010761,-0.016181,3,6
0.041000,4.884725,0.010919,0.003248,4,6
0.041000,4.884876,0.010888,-0.000628,5,6
0.041000,4.884847,0.010894,0.000122,6,6
0.041000,4.884852,0.010893,-0.000024,7,6
0.041000,4.884851,0.010893,0.000005,8,6
0.041000,4.910682,0.005663,0.234694,1,7
0.041000,4.870061,0.013888,-0.040621,2,7
0.041000,4.880054,0.011865,-0.030629,3,7
0.041000,4.877712,0.012339,0.007651,4,7
0.041000,4.878268,0.012226,-0.001786,5,7
0.041000,4.878136,0.012253,0.000424,6,7
0.041000,4.878167,0.012246,-0.000101,7,7
0.041000,4.878160,0.012248,0.000023,8,7
0.041000,4.920732,0.003628,0.344719,1,8
0.041000,4.857383,0.016455,-0.063349,2,8
0.041000,4.875918,0.012702,-0.044814,3,8
0.041000,4.870822,0.013734,0.013440,4,8
0.041000,4.872250,0.013445,-0.003668,5,8
0.041000,4.871852,0.013525,0.001029,6,8
0.041000,4.871964,0.013503,-0.000287,7,8
0.041000,4.871933,0.013509,0.000081,8,8
0.041000,4.931200,0.001508,0.455225,1,9
0.041000,4.843492,0.019268,-0.087708,2,9
0.041000,4.873209,0.013251,-0.057992,3,9
0.041000,4.863844,0.015147,0.020353,4,9
0.041000,4.866872,0.014534,-0.006336,5,9
0.041000,4.865901,0.014730,0.002056,6,9
0.041000,4.866213,0.014667,-0.000659,7,9
0.041000,4.866113,0.014687,0.000213,8,9
0.041000,4.941468,0.000571,0.565468,1,10
0.041000,4.828576,0.022288,-0.112892,2,10
0.041000,4.871974,0.013501,-0.069494,3,10
0.041000,4.856565,0.016620,0.027989,4,10
0.041000,4.862218,0.015476,-0.009756,5,10
0.041000,4.860168,0.015891,0.003603,6,10
0.041000,4.860914,0.015740,-0.001304,7,10
0.041000,4.860643,0.015795,0.000475,8,10
0.031300,5.642801,0.001687,-0.353999,1,0
0.031300,5.651740,0.000105,0.008938,2,0
0.031300,5.652299,0.000006,0.009498,3,0
0.031300,5.652333,0.000000,0.000593,4,0
0.031300,5.652334,0.000000,0.000035,5,0
0.031300,5.652334,0.000000,0.000001,6,0
0.031300,5.652334,0.000000,0.000000,7,0
0.031300,5.652334,0.000000,0.000000,8,0
0.031300,5.626687,0.004537,-0.270137,1,1
0.031300,5.639421,0.002285,0.012734,2,1
0.031300,5.639690,0.002237,0.013003,3,1
0.031300,5.639695,0.002236,0.000274,4,1
0.031300,5.639695,0.002236,0.000005,5,1
0.031300,5.639695,0.002236,0.000000,6,1
0.031300,5.639695,0.002236,0.000000,7,1
0.031300,5.639695,0.002236,0.000000,8,1
0.031300,5.616157,0.006400,-0.180631,1,2
0.031300,5.628433,0.004229,0.012276,2,2
0.031300,5.628169,0.004275,0.012012,3,2
0.031300,5.628175,0.004274,-0.000258,4,2
0.031300,5.628175,0.004274,0.000006,5,2
0.031300,5.628175,0.004274,0.000000,6,2
0.031300,5.628175,0.004274,0.000000,7,2
0.031300,5.628175,0.004274,0.000000,8,2
0.031300,5.610520,0.007398,-0.086292,1,3
0.031300,5.618095,0.006057,0.007575,2,3
0.031300,5.617600,0.006145,0.007080,3,3
0.031300,5.617634,0.006139,-0.000461,4,3
0.031300,5.617632,0.006139,0.000032,5,3
0.031300,5.617632,0.006139,-0.000002,6,3
0.031300,5.617632,0.006139,-0.000000,7,3
0.031300,5.617632,0.006139,0.000000,8,3
0.031300,5.609134,0.007643,0.012358,1,4
0.031300,5.607817,0.007876,-0.001317,2,4
0.031300,5.607962,0.007850,-0.001172,3,4
0.031300,5.607945,0.007853,0.000129,4,4
0.031300,5.607948,0.007853,-0.000014,5,4
0.031300,5.607947,0.007853,0.000002,6,4
0.031300,5.607947,0.007853,-0.000000,7,4
0.031300,5.607947,0.007853,0.000000,8,4
0.031300,5.611343,0.007252,0.114543,1,5
0.031300,5.597100,0.009772,-0.014243,2,5
0.031300,5.599310,0.009381,-0.012033,3,5
0.031300,5.598975,0.009440,0.001875,4,5
0.031300,5.599027,0.009431,-0.000284,5,5
0.031300,5.599018,0.009433,0.000043,6,5
0.031300,5.599020,0.009432,-0.000007,7,5
0.031300,5.599020,0.009432,0.000002,8,5
0.031300,5.616532,0.006334,0.219707,1,6
0.031300,5.585545,0.011816,-0.030987,2,6
0.031300,5.591769,0.010715,-0.024763,3,6
0.031300,5.590569,0.010927,0.005024,4,6
0.031300,5.590802,0.010886,-0.000968,5,6
0.031300,5.590756,0.010894,0.000188,6,6
0.031300,5.590765,0.010893,-0.000037,7,6
0.031300,5.590763,0.010893,0.000007,8,6
0.031300,5.624099,0.004995,0.327312,1,7
0.031300,5.572845,0.014063,-0.051254,2,7
0.031300,5.585500,0.011824,-0.038599,3,7
0.031300,5.582538,0.012348,0.009693,4,7
0.031300,5.583241,0.012224,-0.002259,5,7
0.031300,5.583075,0.012253,0.000536,6,7
0.031300,5.583114,0.012246,-0.000127,7,7
0.031300,5.583105,0.012248,0.000030,8,7
0.031300,5.633448,0.003341,0.436636,1,8
0.031300,5.558810,0.016546,-0.074638,2,8
0.031300,5.580677,0.012677,-0.052771,3,8
0.031300,5.574668,0.013741,0.015858,4,8
0.031300,5.576352,0.013443,-0.004325,5,8
0.031300,5.575883,0.013526,0.001215,6,8
0.031300,5.576014,0.013503,-0.000339,7,8
0.031300,5.575977,0.013509,0.000094,8,8
0.031300,5.644014,0.001472,0.547238,1,9
0.031300,5.543350,0.019281,-0.100664,2,9
0.031300,5.577460,0.013247,-0.066554,3,9
0.031300,5.566713,0.015148,0.023363,4,9
0.031300,5.570189,0.014533,-0.007271,5,9
0.031300,5.569073,0.014730,0.002360,6,9
0.031300,5.569432,0.014667,-0.000757,7,9
0.031300,5.569317,0.014687,0.000244,8,9
0.031300,5.655242,0.000514,0.658442,1,10
0.031300,5.526494,0.022263,-0.128748,2,10
0.031300,5.575979,0.013509,-0.079263,3,10
0.031300,5.558407,0.016617,0.031913,4,10
0.031300,5.564852,0.015477,-0.011127,5,10
0.031300,5.562515,0.015891,0.004108,6,10
0.031300,5.563366,0.015740,-0.001486,7,10
0.031300,5.563056,0.015795,0.000541,8,10
0.052700,4.348048,0.001841,-0.279152,1,0
0.052700,4.355567,0.000115,0.007519,2,0
0.052700,4.356039,0.000007,0.007991,3,0
0.052700,4.356067,0.000000,0.000500,4,0
0.052700,4.356068,0.000000,0.000030,5,0
0.052700,4.356068,0.000000,0.000001,6,0
0.052700,4.356068,0.000000,0.000000,7,0
0.052700,4.356068,0.000000,0.000000,8,0
0.052700,4.338266,0.004087,-0.188959,1,1
0.052700,4.346163,0.002274,0.007897,2,1
0.052700,4.346325,0.002237,0.008059,3,1
0.052700,4.346327,0.002236,0.000164,4,1
0.052700,4.346328,0.002236,0.000003,5,1
0.052700,4.346328,0.002236,0.000000,6,1
0.052700,4.346328,0.002236,0.000000,7,1
0.052700,4.346328,0.002236,0.000000,8,1
0.052700,4.332354,0.005444,-0.094834,1,2
0.052700,4.337566,0.004247,0.005212,2,2
0.052700,4.337446,0.004275,0.005092,3,2
0.052700,4.337450,0.004274,-0.000116,4,2
0.052700,4.337450,0.004274,0.000003,5,2
0.052700,4.337450,0.004274,-0.000000,6,2
0.052700,4.337450,0.004274,0.000000,7,2
0.052700,4.337450,0.004274,0.000000,8,2
0.052700,4.329465,0.006107,0.002253,1,3
0.052700,4.329315,0.006142,-0.000150,2,3
0.052700,4.329326,0.006139,-0.000139,3,3
0.052700,4.329325,0.006139,0.000010,4,3
0.052700,4.329324,0.006139,-0.000001,5,3
0.052700,4.329325,0.006139,0.000000,6,3
0.052700,4.329324,0.006139,0.000000,7,3
0.052700,4.329325,0.006139,0.000000,8,3
0.052700,4.328804,0.006259,0.101629,1,4
0.052700,4.321082,0.008032,-0.007723,2,4
0.052700,4.321946,0.007833,-0.006858,3,4
0.052700,4.321852,0.007855,0.000770,4,4
0.052700,4.321862,0.007852,-0.000084,5,4
0.052700,4.321861,0.007853,0.000010,6,4
0.052700,4.321861,0.007853,-0.000001,7,4
0.052700,4.321861,0.007853,0.000000,8,4
0.052700,4.329592,0.006078,0.202392,1,5
0.052700,4.312675,0.009961,-0.016916,2,5
0.052700,4.315329,0.009352,-0.014263,3,5
0.052700,4.314928,0.009444,0.002253,4,5
0.052700,4.314989,0.009430,-0.000340,5,5
0.052700,4.314980,0.009432,0.000051,6,5
0.052700,4.314981,0.009432,-0.000008,7,5
0.052700,4.314981,0.009432,0.000001,8,5
0.052700,4.331085,0.005735,0.303861,1,6
0.052700,4.304042,0.011943,-0.027043,2,6
0.052700,4.309499,0.010691,-0.021586,3,6
0.052700,4.308447,0.010932,0.004405,4,6
0.052700,4.308652,0.010885,-0.000847,5,6
0.052700,4.308611,0.010894,0.000165,6,6
0.052700,4.308619,0.010893,-0.000032,7,6
0.052700,4.308618,0.010893,0.000007,8,6
0.052700,4.334700,0.004905,0.371106,1,7
0.052700,4.294705,0.014087,-0.039994,2,7
0.052700,4.304585,0.011819,-0.030114,3,7
0.052700,4.302273,0.012349,0.007568,4,7
0.052700,4.302822,0.012224,-0.001764,5,7
0.052700,4.302692,0.012253,0.000419,6,7
0.052700,4.302723,0.012246,-0.000099,7,7
0.052700,4.302715,0.012248,0.000023,8,7
0.052700,4.340531,0.003567,0.426925,1,8
0.052700,4.284305,0.016474,-0.056226,2,8
0.052700,4.300761,0.012697,-0.039771,3,8
0.052700,4.296237,0.013735,0.011932,4,8
0.052700,4.297505,0.013444,-0.003255,5,8
0.052700,4.297151,0.013525,0.000914,6,8
0.052700,4.297250,0.013503,-0.000255,7,8
0.052700,4.297223,0.013509,0.000072,8,8
0.052700,4.347874,0.001881,0.484286,1,9
0.052700,4.272745,0.019128,-0.075129,2,9
0.052700,4.298163,0.013293,-0.049710,3,9
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0.055338,4.169494,0.019167,-0.073930,2,9
0.055338,4.194516,0.013281,-0.048908,3,9
0.055338,4.186628,0.015137,0.017134,4,9
0.055338,4.189178,0.014537,-0.005337,5,9
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0.055338,4.156880,0.022135,-0.095011,2,10
0.055338,4.193364,0.013552,-0.058527,3,10
0.055338,4.180403,0.016601,0.023523,4,10
0.055338,4.185156,0.015483,-0.008208,5,10
0.055338,4.183432,0.015888,0.003029,6,10
0.055338,4.184060,0.015741,-0.001096,7,10
0.055338,4.183831,0.015795,0.000399,8,10
")
original_kadlec_blend$source <- "Unmodified"

## blends from standard form to adjusted Kadlec constants
## that meet the constraint A(B - 1) = 1
## A = 0.712 & B = 2.4045
adjusted_kadlec_blend <- read.csv(text = "
input,output,error,diff,iters,alpha
0.041000,4.908667,0.006071,-0.467333,1,0
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0.041000,4.920674,0.003640,-0.155338,1,3
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0.041000,4.939375,0.000147,0.006186,2,4
0.041000,4.938559,0.000018,0.005370,3,4
0.041000,4.938668,0.000004,-0.000707,4,4
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0.031300,5.642801,0.001687,-0.353999,1,0
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0.031300,5.652714,0.000067,0.011481,2,2
0.031300,5.652325,0.000002,0.011092,3,2
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0.031300,5.647980,0.000770,-0.048832,1,3
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0.031300,5.652310,0.000004,0.004330,3,3
0.031300,5.652343,0.000002,-0.000364,4,3
0.031300,5.652340,0.000001,0.000031,5,3
0.031300,5.652341,0.000001,-0.000002,6,3
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0.055338,4.341019,0.021182,0.611834,1,10
0.055338,4.208709,0.009942,-0.132310,2,10
0.055338,4.268016,0.004009,-0.073003,3,10
0.055338,4.243618,0.001730,0.034910,4,10
0.055338,4.254082,0.000731,-0.013934,5,10
0.055338,4.249668,0.000307,0.006050,6,10
0.055338,4.251543,0.000134,-0.002539,7,10
0.055338,4.250749,0.000053,0.001081,8,10
")
adjusted_kadlec_blend$source <- "Modified to meet A(B - 1) =1"

kadlec <- rbind(adjusted_kadlec_blend, original_kadlec_blend)
rm(adjusted_kadlec_blend, original_kadlec_blend)

subt <- TeX(r"(Blending factor $0 \Rightarrow \frac{1}{\sqrt{x}} (A - B x (\frac{1}{\sqrt{x}})^2)$, $10 \Rightarrow \frac{A}{\sqrt{x}} (B - x (\frac{1}{\sqrt{x}})^2)$)")

kadlec %>%
  ggplot(aes(x = iters, y = error, color = factor(alpha), group = alpha)) +
  geom_line() + facet_wrap(~source) +
  labs(title = "Approximation stalls without modified values",
       subtitle = subt,
       color = "Blending\nfactor",
       y = "Relative error")