Average Error: 32.4 → 3.5
Time: 34.3s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\]
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)
↓
e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r514082 = x_re;
        double r514083 = r514082 * r514082;
        double r514084 = x_im;
        double r514085 = r514084 * r514084;
        double r514086 = r514083 + r514085;
        double r514087 = sqrt(r514086);
        double r514088 = log(r514087);
        double r514089 = y_re;
        double r514090 = r514088 * r514089;
        double r514091 = atan2(r514084, r514082);
        double r514092 = y_im;
        double r514093 = r514091 * r514092;
        double r514094 = r514090 - r514093;
        double r514095 = exp(r514094);
        double r514096 = r514088 * r514092;
        double r514097 = r514091 * r514089;
        double r514098 = r514096 + r514097;
        double r514099 = sin(r514098);
        double r514100 = r514095 * r514099;
        return r514100;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r514101 = x_re;
        double r514102 = x_im;
        double r514103 = hypot(r514101, r514102);
        double r514104 = log(r514103);
        double r514105 = y_re;
        double r514106 = r514104 * r514105;
        double r514107 = y_im;
        double r514108 = atan2(r514102, r514101);
        double r514109 = r514107 * r514108;
        double r514110 = r514106 - r514109;
        double r514111 = exp(r514110);
        double r514112 = r514107 * r514104;
        double r514113 = sin(r514112);
        double r514114 = expm1(r514113);
        double r514115 = log1p(r514114);
        double r514116 = r514105 * r514108;
        double r514117 = cos(r514116);
        double r514118 = r514115 * r514117;
        double r514119 = cos(r514112);
        double r514120 = sin(r514116);
        double r514121 = r514119 * r514120;
        double r514122 = r514118 + r514121;
        double r514123 = r514111 * r514122;
        return r514123;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  2. Simplified3.5

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

  4. Applied fma-udef3.5

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

  5. Applied sin-sum3.5

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

  7. Applied log1p-expm1-u3.5

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

  8. Final simplification3.5

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

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(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)))))