Average Error: 31.1 → 0.4

Time: 1.4m

Precision: 64

Internal Precision: 128

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]

↓

\[\frac{\log \left((e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^* \cdot (e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^*\right)}{\log base}\]

Error

The X axis uses an exponential scale — Bits error versus `re`

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Derivation

Initial program 31.1
\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right) \cdot \log base + \tan^{-1}_* \frac{im}{re} \cdot 0}{\log base \cdot \log base + 0 \cdot 0}\]
Initial simplification0.4
\[\leadsto \frac{\log \left(\sqrt{re^2 + im^2}^*\right)}{\log base}\]
Using strategy rm
Applied add-cube-cbrt0.4
\[\leadsto \frac{\log \color{blue}{\left(\left(\sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right) \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}}{\log base}\]
Using strategy rm
Applied expm1-log1p-u0.4
\[\leadsto \frac{\log \left(\color{blue}{(e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*}\right)}{\log base}\]
Using strategy rm
Applied expm1-log1p-u0.4
\[\leadsto \frac{\log \left((e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^* \cdot \color{blue}{(e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^*}\right)}{\log base}\]
Final simplification0.4
\[\leadsto \frac{\log \left((e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*} \cdot \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^* \cdot (e^{\log_* (1 + \sqrt[3]{\sqrt{re^2 + im^2}^*})} - 1)^*\right)}{\log base}\]

Runtime

herbie shell --seed 2018349 +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)) (+ (* (log base) (log base)) (* 0 0))))