_divideComplex, real part

Average Error: 25.9 → 14.1

Time: 14.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.2360569206326496 \cdot 10^{252}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le -4.92872222777350137 \cdot 10^{189}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le -9.9314823771901832 \cdot 10^{103}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 7.287774565713375 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r47107 = x_re;
        double r47108 = y_re;
        double r47109 = r47107 * r47108;
        double r47110 = x_im;
        double r47111 = y_im;
        double r47112 = r47110 * r47111;
        double r47113 = r47109 + r47112;
        double r47114 = r47108 * r47108;
        double r47115 = r47111 * r47111;
        double r47116 = r47114 + r47115;
        double r47117 = r47113 / r47116;
        return r47117;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r47118 = y_re;
        double r47119 = -1.2360569206326496e+252;
        bool r47120 = r47118 <= r47119;
        double r47121 = x_re;
        double r47122 = -r47121;
        double r47123 = y_im;
        double r47124 = hypot(r47118, r47123);
        double r47125 = r47122 / r47124;
        double r47126 = -4.928722227773501e+189;
        bool r47127 = r47118 <= r47126;
        double r47128 = x_im;
        double r47129 = r47128 * r47123;
        double r47130 = fma(r47121, r47118, r47129);
        double r47131 = r47130 / r47124;
        double r47132 = 3.0;
        double r47133 = pow(r47131, r47132);
        double r47134 = cbrt(r47133);
        double r47135 = r47134 / r47124;
        double r47136 = -9.931482377190183e+103;
        bool r47137 = r47118 <= r47136;
        double r47138 = 7.287774565713375e+101;
        bool r47139 = r47118 <= r47138;
        double r47140 = r47131 / r47124;
        double r47141 = r47121 / r47124;
        double r47142 = r47139 ? r47140 : r47141;
        double r47143 = r47137 ? r47125 : r47142;
        double r47144 = r47127 ? r47135 : r47143;
        double r47145 = r47120 ? r47125 : r47144;
        return r47145;
}

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 4 regimes

2. if y.re < -1.2360569206326496e+252 or -4.928722227773501e+189 < y.re < -9.931482377190183e+103

   1. Initial program 36.1

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 36.1

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 36.1

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

   5. Applied times-frac 36.1

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 36.1

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

   7. Simplified 26.0

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

   9. Applied *-un-lft-identity 26.0

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

   10. Applied associate-*l* 26.0

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

   11. Simplified 26.0

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

   12. Taylor expanded around -inf 16.2

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

   13. Simplified 16.2

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

   if -1.2360569206326496e+252 < y.re < -4.928722227773501e+189

   1. Initial program 45.4

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 45.4

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 45.4

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

   5. Applied times-frac 45.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 45.4

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

   7. Simplified 28.3

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

   9. Applied *-un-lft-identity 28.3

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

   10. Applied associate-*l* 28.3

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

   11. Simplified 28.3

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

   13. Applied add-cbrt-cube 28.6

       \[\leadsto 1 \cdot \frac{\color{blue}{\sqrt[3]{\left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}}{\mathsf{hypot}\left(y.re, y.im\right)}\]

   14. Simplified 28.6

       \[\leadsto 1 \cdot \frac{\sqrt[3]{\color{blue}{{\left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}}}{\mathsf{hypot}\left(y.re, y.im\right)}\]

   if -9.931482377190183e+103 < y.re < 7.287774565713375e+101

   1. Initial program 18.5

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 18.5

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 18.5

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

   5. Applied times-frac 18.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 18.5

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

   7. Simplified 11.6

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

   9. Applied *-un-lft-identity 11.6

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

   10. Applied associate-*l* 11.6

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

   11. Simplified 11.4

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

   if 7.287774565713375e+101 < y.re

   1. Initial program 40.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 40.0

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity 40.0

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

   5. Applied times-frac 40.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified 40.0

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

   7. Simplified 26.2

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

   9. Applied *-un-lft-identity 26.2

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

   10. Applied associate-*l* 26.2

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

   11. Simplified 26.1

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

   12. Taylor expanded around inf 17.8

       \[\leadsto 1 \cdot \frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.2360569206326496 \cdot 10^{252}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le -4.92872222777350137 \cdot 10^{189}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le -9.9314823771901832 \cdot 10^{103}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 7.287774565713375 \cdot 10^{101}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +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))))