_divideComplex, real part

Average Error: 25.6 → 12.9

Time: 18.2s

Precision: 64

Math

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.086126503497258 \cdot 10^{+188}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 6.204871618360054 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2461077 = x_re;
        double r2461078 = y_re;
        double r2461079 = r2461077 * r2461078;
        double r2461080 = x_im;
        double r2461081 = y_im;
        double r2461082 = r2461080 * r2461081;
        double r2461083 = r2461079 + r2461082;
        double r2461084 = r2461078 * r2461078;
        double r2461085 = r2461081 * r2461081;
        double r2461086 = r2461084 + r2461085;
        double r2461087 = r2461083 / r2461086;
        return r2461087;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r2461088 = y_re;
        double r2461089 = -5.086126503497258e+188;
        bool r2461090 = r2461088 <= r2461089;
        double r2461091 = x_re;
        double r2461092 = -r2461091;
        double r2461093 = y_im;
        double r2461094 = hypot(r2461093, r2461088);
        double r2461095 = r2461092 / r2461094;
        double r2461096 = 6.204871618360054e+176;
        bool r2461097 = r2461088 <= r2461096;
        double r2461098 = x_im;
        double r2461099 = r2461093 * r2461098;
        double r2461100 = fma(r2461091, r2461088, r2461099);
        double r2461101 = r2461100 / r2461094;
        double r2461102 = r2461101 / r2461094;
        double r2461103 = r2461091 / r2461094;
        double r2461104 = r2461097 ? r2461102 : r2461103;
        double r2461105 = r2461090 ? r2461095 : r2461104;
        return r2461105;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 3 regimes

2. if y.re < -5.086126503497258e+188

   1. Initial program 41.2

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

   2. Simplified 41.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 41.2

      \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]

   5. Applied associate-/r* 41.2

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
   6. Using strategy rm

   7. Applied clear-num 41.2

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}}}\]

   8. Simplified 29.6

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)} \cdot \mathsf{hypot}\left(y.im, y.re\right)}}\]
   9. Using strategy rm

   10. Applied associate-/r* 29.0

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   11. Simplified 29.0

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   12. Taylor expanded around -inf 12.0

       \[\leadsto \frac{\color{blue}{-1 \cdot x.re}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   13. Simplified 12.0

       \[\leadsto \frac{\color{blue}{-x.re}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   if -5.086126503497258e+188 < y.re < 6.204871618360054e+176

   1. Initial program 20.9

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

   2. Simplified 20.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 20.9

      \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]

   5. Applied associate-/r* 20.8

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
   6. Using strategy rm

   7. Applied clear-num 21.1

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}}}\]

   8. Simplified 13.6

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)} \cdot \mathsf{hypot}\left(y.im, y.re\right)}}\]
   9. Using strategy rm

   10. Applied associate-/r* 13.1

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   11. Simplified 13.0

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   if 6.204871618360054e+176 < y.re

   1. Initial program 44.6

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

   2. Simplified 44.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 44.6

      \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} \cdot \sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]

   5. Applied associate-/r* 44.6

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}\]
   6. Using strategy rm

   7. Applied clear-num 44.6

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\sqrt{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}}}}}\]

   8. Simplified 31.9

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)} \cdot \mathsf{hypot}\left(y.im, y.re\right)}}\]
   9. Using strategy rm

   10. Applied associate-/r* 31.4

       \[\leadsto \color{blue}{\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{\mathsf{fma}\left(x.im, y.im, y.re \cdot x.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}}\]

   11. Simplified 31.4

       \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}}{\mathsf{hypot}\left(y.im, y.re\right)}\]

   12. Taylor expanded around inf 12.5

       \[\leadsto \frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.im, y.re\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.086126503497258 \cdot 10^{+188}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 6.204871618360054 \cdot 10^{+176}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))