powComplex, real part

Average Error: 33.8 → 7.0

Time: 20.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.6507641711952303 \cdot 10^{52}:\\ \;\;\;\;e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\sqrt[3]{\log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}^{3}\right)\right)}\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r26903 = x_re;
        double r26904 = r26903 * r26903;
        double r26905 = x_im;
        double r26906 = r26905 * r26905;
        double r26907 = r26904 + r26906;
        double r26908 = sqrt(r26907);
        double r26909 = log(r26908);
        double r26910 = y_re;
        double r26911 = r26909 * r26910;
        double r26912 = atan2(r26905, r26903);
        double r26913 = y_im;
        double r26914 = r26912 * r26913;
        double r26915 = r26911 - r26914;
        double r26916 = exp(r26915);
        double r26917 = r26909 * r26913;
        double r26918 = r26912 * r26910;
        double r26919 = r26917 + r26918;
        double r26920 = cos(r26919);
        double r26921 = r26916 * r26920;
        return r26921;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r26922 = y_re;
        double r26923 = -1.6507641711952303e+52;
        bool r26924 = r26922 <= r26923;
        double r26925 = x_re;
        double r26926 = x_im;
        double r26927 = hypot(r26925, r26926);
        double r26928 = pow(r26927, r26922);
        double r26929 = log(r26927);
        double r26930 = y_im;
        double r26931 = atan2(r26926, r26925);
        double r26932 = r26931 * r26922;
        double r26933 = fma(r26929, r26930, r26932);
        double r26934 = cos(r26933);
        double r26935 = r26928 * r26934;
        double r26936 = log(r26935);
        double r26937 = r26931 * r26930;
        double r26938 = r26936 - r26937;
        double r26939 = exp(r26938);
        double r26940 = expm1(r26934);
        double r26941 = 3.0;
        double r26942 = pow(r26940, r26941);
        double r26943 = expm1(r26942);
        double r26944 = log1p(r26943);
        double r26945 = exp(r26944);
        double r26946 = log(r26945);
        double r26947 = cbrt(r26946);
        double r26948 = log1p(r26947);
        double r26949 = r26928 * r26948;
        double r26950 = exp(r26937);
        double r26951 = r26949 / r26950;
        double r26952 = r26924 ? r26939 : r26951;
        return r26952;
}

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 2 regimes

2. if y.re < -1.6507641711952303e+52

   1. Initial program 37.8

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Simplified 10.9

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\]
   3. Using strategy rm

   4. Applied add-exp-log 35.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \color{blue}{e^{\log \left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   5. Applied add-exp-log 35.4

      \[\leadsto \frac{{\color{blue}{\left(e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}}^{y.re} \cdot e^{\log \left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   6. Applied pow-exp 35.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}} \cdot e^{\log \left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   7. Applied prod-exp 35.4

      \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re + \log \left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   8. Applied div-exp 30.5

      \[\leadsto \color{blue}{e^{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re + \log \left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   9. Simplified 2.2

      \[\leadsto e^{\color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   if -1.6507641711952303e+52 < y.re

   1. Initial program 32.6

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   2. Simplified 8.4

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}}\]
   3. Using strategy rm

   4. Applied log1p-expm1-u 8.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
   5. Using strategy rm

   6. Applied add-cbrt-cube 8.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\color{blue}{\sqrt[3]{\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) \cdot \mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right) \cdot \mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]

   7. Simplified 8.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{{\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}^{3}}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
   8. Using strategy rm

   9. Applied add-log-exp 8.4

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\sqrt[3]{\color{blue}{\log \left(e^{{\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}^{3}}\right)}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
   10. Using strategy rm

   11. Applied log1p-expm1-u 8.4

       \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\sqrt[3]{\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}^{3}\right)\right)}}\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.6507641711952303 \cdot 10^{52}:\\ \;\;\;\;e^{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \mathsf{log1p}\left(\sqrt[3]{\log \left(e^{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{expm1}\left(\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right)\right)}^{3}\right)\right)}\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))