Average Error: 32.1 → 0.4
Time: 21.7s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), 0.0 \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}\]
\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}
↓
\frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), 0.0 \cdot \tan^{-1}_* \frac{im}{re}\right)}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}}{\sqrt{\mathsf{fma}\left(0.0, 0.0, \log base \cdot \log base\right)}}
double f(double re, double im, double base) {
        double r2275552 = re;
        double r2275553 = r2275552 * r2275552;
        double r2275554 = im;
        double r2275555 = r2275554 * r2275554;
        double r2275556 = r2275553 + r2275555;
        double r2275557 = sqrt(r2275556);
        double r2275558 = log(r2275557);
        double r2275559 = base;
        double r2275560 = log(r2275559);
        double r2275561 = r2275558 * r2275560;
        double r2275562 = atan2(r2275554, r2275552);
        double r2275563 = 0.0;
        double r2275564 = r2275562 * r2275563;
        double r2275565 = r2275561 + r2275564;
        double r2275566 = r2275560 * r2275560;
        double r2275567 = r2275563 * r2275563;
        double r2275568 = r2275566 + r2275567;
        double r2275569 = r2275565 / r2275568;
        return r2275569;
}
↓
double f(double re, double im, double base) {
        double r2275570 = base;
        double r2275571 = log(r2275570);
        double r2275572 = re;
        double r2275573 = im;
        double r2275574 = hypot(r2275572, r2275573);
        double r2275575 = log(r2275574);
        double r2275576 = 0.0;
        double r2275577 = atan2(r2275573, r2275572);
        double r2275578 = r2275576 * r2275577;
        double r2275579 = fma(r2275571, r2275575, r2275578);
        double r2275580 = r2275571 * r2275571;
        double r2275581 = fma(r2275576, r2275576, r2275580);
        double r2275582 = sqrt(r2275581);
        double r2275583 = r2275579 / r2275582;
        double r2275584 = r2275583 / r2275582;
        return r2275584;
}

Error

Bits error versus re

Bits error versus im

Bits error versus base

Derivation

  1. Initial program 32.1

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

  2. Simplified0.5

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

  4. Applied add-sqr-sqrt0.5

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

  5. Applied associate-/r*0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))