Average Error: 26.3 → 18.1
Time: 44.2s
Precision: 64
Internal Precision: 384
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -9.476119299999374 \cdot 10^{+106}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \le 1.8299841684748482 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if y.im < -9.476119299999374e+106 or 1.8299841684748482e+136 < y.im

    1. Initial program 40.8

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-cbrt-cube46.0

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    4. Applied add-cbrt-cube53.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)\right) \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}\]

    5. Applied cbrt-undiv53.3

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x.im \cdot y.re - x.re \cdot y.im\right) \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)\right) \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    6. Applied simplify43.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)}^{3}}}\]

    7. Taylor expanded around 0 62.9

      \[\leadsto \color{blue}{e^{\left(\log -1 + \log x.re\right) - \log y.im}}\]

    8. Applied simplify15.8

      \[\leadsto \color{blue}{\frac{-x.re}{y.im}}\]

    if -9.476119299999374e+106 < y.im < 1.8299841684748482e+136

    1. Initial program 19.2

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt19.2

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

    4. Applied associate-/r*19.1

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 44.2s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))