Average Error: 32.0 → 0.4
Time: 7.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
double f(double re, double im, double base) {
        double r54641 = im;
        double r54642 = re;
        double r54643 = atan2(r54641, r54642);
        double r54644 = base;
        double r54645 = log(r54644);
        double r54646 = r54643 * r54645;
        double r54647 = r54642 * r54642;
        double r54648 = r54641 * r54641;
        double r54649 = r54647 + r54648;
        double r54650 = sqrt(r54649);
        double r54651 = log(r54650);
        double r54652 = 0.0;
        double r54653 = r54651 * r54652;
        double r54654 = r54646 - r54653;
        double r54655 = r54645 * r54645;
        double r54656 = r54652 * r54652;
        double r54657 = r54655 + r54656;
        double r54658 = r54654 / r54657;
        return r54658;
}
↓
double f(double re, double im, double base) {
        double r54659 = im;
        double r54660 = re;
        double r54661 = atan2(r54659, r54660);
        double r54662 = base;
        double r54663 = log(r54662);
        double r54664 = r54661 * r54663;
        double r54665 = 1.0;
        double r54666 = hypot(r54660, r54659);
        double r54667 = r54665 * r54666;
        double r54668 = log(r54667);
        double r54669 = 0.0;
        double r54670 = r54668 * r54669;
        double r54671 = r54664 - r54670;
        double r54672 = r54663 * r54663;
        double r54673 = r54669 * r54669;
        double r54674 = r54672 + r54673;
        double r54675 = r54671 / r54674;
        return r54675;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

    \[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity32.0

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{\color{blue}{1 \cdot \left(re \cdot re + im \cdot im\right)}}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  4. Applied sqrt-prod32.0

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \color{blue}{\left(\sqrt{1} \cdot \sqrt{re \cdot re + im \cdot im}\right)} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  5. Simplified32.0

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\color{blue}{1} \cdot \sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  6. Simplified0.4

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(1 \cdot \color{blue}{\mathsf{hypot}\left(re, im\right)}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

  7. Final simplification0.4

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(1 \cdot \mathsf{hypot}\left(re, im\right)\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  :precision binary64
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))