Average Error: 31.4 → 0.3
Time: 17.1s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re} \cdot \log base - \log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]
\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}}{\log base}
double f(double re, double im, double base) {
        double r100310 = im;
        double r100311 = re;
        double r100312 = atan2(r100310, r100311);
        double r100313 = base;
        double r100314 = log(r100313);
        double r100315 = r100312 * r100314;
        double r100316 = r100311 * r100311;
        double r100317 = r100310 * r100310;
        double r100318 = r100316 + r100317;
        double r100319 = sqrt(r100318);
        double r100320 = log(r100319);
        double r100321 = 0.0;
        double r100322 = r100320 * r100321;
        double r100323 = r100315 - r100322;
        double r100324 = r100314 * r100314;
        double r100325 = r100321 * r100321;
        double r100326 = r100324 + r100325;
        double r100327 = r100323 / r100326;
        return r100327;
}
↓
double f(double re, double im, double base) {
        double r100328 = im;
        double r100329 = re;
        double r100330 = atan2(r100328, r100329);
        double r100331 = base;
        double r100332 = log(r100331);
        double r100333 = r100330 / r100332;
        return r100333;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  2. Simplified0.4

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

  3. Taylor expanded around -inf 64.0

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

  4. Simplified0.3

    \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{im}{re}}{0 + \log base}}\]

  5. Final simplification0.3

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\log base}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, imaginary part"
  (/ (- (* (atan2 im re) (log base)) (* (log (sqrt (+ (* re re) (* im im)))) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))