Average Error: 32.1 → 0.4
Time: 8.5s
Precision: 64
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0.0}{\log base \cdot \log base + 0.0 \cdot 0.0}\]
↓
\[\frac{\frac{\mathsf{fma}\left(\log base, \log \left(\mathsf{hypot}\left(re, im\right)\right), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}\]
\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), \tan^{-1}_* \frac{im}{re} \cdot 0.0\right)}{\mathsf{hypot}\left(\log base, 0.0\right) \cdot 1}}{\sqrt{\log base \cdot \log base + 0.0 \cdot 0.0}}
double f(double re, double im, double base) {
        double r50250 = re;
        double r50251 = r50250 * r50250;
        double r50252 = im;
        double r50253 = r50252 * r50252;
        double r50254 = r50251 + r50253;
        double r50255 = sqrt(r50254);
        double r50256 = log(r50255);
        double r50257 = base;
        double r50258 = log(r50257);
        double r50259 = r50256 * r50258;
        double r50260 = atan2(r50252, r50250);
        double r50261 = 0.0;
        double r50262 = r50260 * r50261;
        double r50263 = r50259 + r50262;
        double r50264 = r50258 * r50258;
        double r50265 = r50261 * r50261;
        double r50266 = r50264 + r50265;
        double r50267 = r50263 / r50266;
        return r50267;
}
↓
double f(double re, double im, double base) {
        double r50268 = base;
        double r50269 = log(r50268);
        double r50270 = re;
        double r50271 = im;
        double r50272 = hypot(r50270, r50271);
        double r50273 = log(r50272);
        double r50274 = atan2(r50271, r50270);
        double r50275 = 0.0;
        double r50276 = r50274 * r50275;
        double r50277 = fma(r50269, r50273, r50276);
        double r50278 = hypot(r50269, r50275);
        double r50279 = 1.0;
        double r50280 = r50278 * r50279;
        double r50281 = r50277 / r50280;
        double r50282 = r50269 * r50269;
        double r50283 = r50275 * r50275;
        double r50284 = r50282 + r50283;
        double r50285 = sqrt(r50284);
        double r50286 = r50281 / r50285;
        return r50286;
}

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. Using strategy rm

  3. Applied *-un-lft-identity32.1

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

  4. Applied sqrt-prod32.1

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

  5. Simplified32.1

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

  6. Simplified0.5

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

  8. Applied add-sqr-sqrt0.5

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

  9. Applied associate-/r*0.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (re im base)
  :name "math.log/2 on complex, real part"
  :precision binary64
  (/ (+ (* (log (sqrt (+ (* re re) (* im im)))) (log base)) (* (atan2 im re) 0.0)) (+ (* (log base) (log base)) (* 0.0 0.0))))