powComplex, real part

Average Error: 33.3 → 5.9

Time: 31.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le 0.999999999861857947:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r25005 = x_re;
        double r25006 = r25005 * r25005;
        double r25007 = x_im;
        double r25008 = r25007 * r25007;
        double r25009 = r25006 + r25008;
        double r25010 = sqrt(r25009);
        double r25011 = log(r25010);
        double r25012 = y_re;
        double r25013 = r25011 * r25012;
        double r25014 = atan2(r25007, r25005);
        double r25015 = y_im;
        double r25016 = r25014 * r25015;
        double r25017 = r25013 - r25016;
        double r25018 = exp(r25017);
        double r25019 = r25011 * r25015;
        double r25020 = r25014 * r25012;
        double r25021 = r25019 + r25020;
        double r25022 = cos(r25021);
        double r25023 = r25018 * r25022;
        return r25023;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r25024 = x_re;
        double r25025 = r25024 * r25024;
        double r25026 = x_im;
        double r25027 = r25026 * r25026;
        double r25028 = r25025 + r25027;
        double r25029 = sqrt(r25028);
        double r25030 = log(r25029);
        double r25031 = y_re;
        double r25032 = r25030 * r25031;
        double r25033 = atan2(r25026, r25024);
        double r25034 = y_im;
        double r25035 = r25033 * r25034;
        double r25036 = r25032 - r25035;
        double r25037 = exp(r25036);
        double r25038 = r25030 * r25034;
        double r25039 = r25033 * r25031;
        double r25040 = r25038 + r25039;
        double r25041 = cos(r25040);
        double r25042 = r25037 * r25041;
        double r25043 = 0.999999999861858;
        bool r25044 = r25042 <= r25043;
        double r25045 = hypot(r25024, r25026);
        double r25046 = log(r25045);
        double r25047 = fma(r25046, r25034, r25039);
        double r25048 = cos(r25047);
        double r25049 = cbrt(r25048);
        double r25050 = r25049 * r25049;
        double r25051 = r25050 * r25049;
        double r25052 = pow(r25045, r25031);
        double r25053 = cbrt(r25034);
        double r25054 = r25053 * r25053;
        double r25055 = r25054 * r25033;
        double r25056 = r25055 * r25053;
        double r25057 = exp(r25056);
        double r25058 = r25052 / r25057;
        double r25059 = r25051 * r25058;
        double r25060 = r25044 ? r25042 : r25059;
        return r25060;
}

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 (* (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)))) < 0.999999999861858

   1. Initial program 4.5

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

   if 0.999999999861858 < (* (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))))

   1. Initial program 46.6

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

   2. Simplified 6.6

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

   4. Applied add-cube-cbrt 6.6

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

   5. Applied associate-*r* 6.6

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

   6. Simplified 6.6

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

   8. Applied add-cube-cbrt 6.6

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \le 0.999999999861857947:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right) \cdot \sqrt[3]{\cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\left(\left(\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sqrt[3]{y.im}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  (* (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)))))