Average Error: 33.2 → 4.3
Time: 8.4s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\]
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)
↓
e^{\log \left(1 \cdot \mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15039 = x_re;
        double r15040 = r15039 * r15039;
        double r15041 = x_im;
        double r15042 = r15041 * r15041;
        double r15043 = r15040 + r15042;
        double r15044 = sqrt(r15043);
        double r15045 = log(r15044);
        double r15046 = y_re;
        double r15047 = r15045 * r15046;
        double r15048 = atan2(r15041, r15039);
        double r15049 = y_im;
        double r15050 = r15048 * r15049;
        double r15051 = r15047 - r15050;
        double r15052 = exp(r15051);
        double r15053 = r15045 * r15049;
        double r15054 = r15048 * r15046;
        double r15055 = r15053 + r15054;
        double r15056 = cos(r15055);
        double r15057 = r15052 * r15056;
        return r15057;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r15058 = 1.0;
        double r15059 = x_re;
        double r15060 = x_im;
        double r15061 = hypot(r15059, r15060);
        double r15062 = r15058 * r15061;
        double r15063 = log(r15062);
        double r15064 = y_re;
        double r15065 = r15063 * r15064;
        double r15066 = atan2(r15060, r15059);
        double r15067 = y_im;
        double r15068 = r15066 * r15067;
        double r15069 = r15065 - r15068;
        double r15070 = exp(r15069);
        return r15070;
}

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 33.2

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

  2. Taylor expanded around 0 20.0

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

  4. Applied *-un-lft-identity20.0

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

  5. Applied sqrt-prod20.0

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

  6. Simplified20.0

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

  7. Simplified4.3

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

  8. Final simplification4.3

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

Reproduce

herbie shell --seed 2020057 +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)))))