powComplex, imaginary part

Average Error: 31.1 → 11.3

Time: 4.0m

Precision: 64

• could not determine a ground truth for program body

  1. x.re = 1.9275448083637878e-225
  2. x.im = -4.632749300641585e+255
  3. y.re = -2.260830684783954e-154
  4. y.im = -2.947752396708778e+142

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -6700130532588.583:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log \left(-x.re\right)} \cdot y.im\right) \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\sqrt[3]{\log \left(-x.re\right)} \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right)}\right)\right)\\ \mathbf{elif}\;y.re \le -5.469308494101399 \cdot 10^{-269}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|\sqrt[3]{x.im \cdot x.im + x.re \cdot x.re}\right| \cdot \sqrt{\sqrt[3]{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(-x.re\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r552936 = x_re;
        double r552937 = r552936 * r552936;
        double r552938 = x_im;
        double r552939 = r552938 * r552938;
        double r552940 = r552937 + r552939;
        double r552941 = sqrt(r552940);
        double r552942 = log(r552941);
        double r552943 = y_re;
        double r552944 = r552942 * r552943;
        double r552945 = atan2(r552938, r552936);
        double r552946 = y_im;
        double r552947 = r552945 * r552946;
        double r552948 = r552944 - r552947;
        double r552949 = exp(r552948);
        double r552950 = r552942 * r552946;
        double r552951 = r552945 * r552943;
        double r552952 = r552950 + r552951;
        double r552953 = sin(r552952);
        double r552954 = r552949 * r552953;
        return r552954;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r552955 = y_re;
        double r552956 = -6700130532588.583;
        bool r552957 = r552955 <= r552956;
        double r552958 = x_im;
        double r552959 = r552958 * r552958;
        double r552960 = x_re;
        double r552961 = r552960 * r552960;
        double r552962 = r552959 + r552961;
        double r552963 = sqrt(r552962);
        double r552964 = log(r552963);
        double r552965 = r552964 * r552955;
        double r552966 = atan2(r552958, r552960);
        double r552967 = y_im;
        double r552968 = r552966 * r552967;
        double r552969 = r552965 - r552968;
        double r552970 = exp(r552969);
        double r552971 = r552966 * r552955;
        double r552972 = -r552960;
        double r552973 = log(r552972);
        double r552974 = cbrt(r552973);
        double r552975 = r552974 * r552967;
        double r552976 = r552974 * r552974;
        double r552977 = r552974 * r552976;
        double r552978 = cbrt(r552977);
        double r552979 = r552974 * r552978;
        double r552980 = r552975 * r552979;
        double r552981 = r552971 + r552980;
        double r552982 = sin(r552981);
        double r552983 = r552970 * r552982;
        double r552984 = -5.469308494101399e-269;
        bool r552985 = r552955 <= r552984;
        double r552986 = cbrt(r552962);
        double r552987 = fabs(r552986);
        double r552988 = sqrt(r552986);
        double r552989 = r552987 * r552988;
        double r552990 = log(r552989);
        double r552991 = r552990 * r552967;
        double r552992 = r552991 + r552971;
        double r552993 = sin(r552992);
        double r552994 = r552970 * r552993;
        double r552995 = r552955 * r552973;
        double r552996 = r552995 - r552968;
        double r552997 = exp(r552996);
        double r552998 = r552967 * r552973;
        double r552999 = r552998 + r552971;
        double r553000 = sin(r552999);
        double r553001 = r552997 * r553000;
        double r553002 = r552985 ? r552994 : r553001;
        double r553003 = r552957 ? r552983 : r553002;
        return r553003;
}

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 < -6700130532588.583

   1. Initial program 36.2

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

   2. Taylor expanded around -inf 0.1

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

   3. Simplified 0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right) \cdot \sqrt[3]{\log \left(-x.re\right)}\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   6. Applied associate-*l* 0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right) \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot y.im\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 0.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\left(\sqrt[3]{\log \left(-x.re\right)} \cdot \color{blue}{\sqrt[3]{\left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right) \cdot \sqrt[3]{\log \left(-x.re\right)}}}\right) \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot y.im\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   if -6700130532588.583 < y.re < -5.469308494101399e-269

   1. Initial program 32.5

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

   3. Applied add-cube-cbrt 32.5

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{\color{blue}{\left(\sqrt[3]{x.re \cdot x.re + x.im \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot \sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   4. Applied sqrt-prod 32.5

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(\sqrt{\sqrt[3]{x.re \cdot x.re + x.im \cdot x.im} \cdot \sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}} \cdot \sqrt{\sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}}\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   5. Simplified 32.5

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\color{blue}{\left|\sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}\right|} \cdot \sqrt{\sqrt[3]{x.re \cdot x.re + x.im \cdot x.im}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

   if -5.469308494101399e-269 < y.re

   1. Initial program 24.2

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

   2. Taylor expanded around -inf 26.8

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

   3. Simplified 26.8

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

   4. Taylor expanded around -inf 17.2

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

   5. Simplified 17.2

      \[\leadsto e^{\log \color{blue}{\left(-x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6700130532588.583:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log \left(-x.re\right)} \cdot y.im\right) \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\sqrt[3]{\log \left(-x.re\right)} \cdot \left(\sqrt[3]{\log \left(-x.re\right)} \cdot \sqrt[3]{\log \left(-x.re\right)}\right)}\right)\right)\\ \mathbf{elif}\;y.re \le -5.469308494101399 \cdot 10^{-269}:\\ \;\;\;\;e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\left|\sqrt[3]{x.im \cdot x.im + x.re \cdot x.re}\right| \cdot \sqrt{\sqrt[3]{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(-x.re\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))