Average Error: 25.7 → 11.0
Time: 1.3m
Precision: 64
Internal Precision: 384
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -2.591897792400522 \cdot 10^{+175}:\\ \;\;\;\;\frac{(\left(\frac{x.re}{y.re}\right) \cdot y.im + \left(-x.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{if}\;y.re \le 2.356201000260532 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{(x.im \cdot y.re + \left(-x.re \cdot y.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{x.re}{y.re}\right) \cdot \left(-y.im\right) + x.im)_*}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

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 3 regimes

  2. if y.re < -2.591897792400522e+175

    1. Initial program 44.0

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify44.0

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt44.0

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

    5. Applied *-un-lft-identity44.0

      \[\leadsto \frac{\color{blue}{1 \cdot \left(y.re \cdot x.im - x.re \cdot y.im\right)}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    6. Applied times-frac44.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

    7. Applied simplify44.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    8. Applied simplify30.0

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt30.2

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\left(\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}}\]

    11. Applied associate-/r*30.2

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}}\]

    12. Taylor expanded around -inf 11.4

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\left(\frac{x.re \cdot y.im}{y.re} - x.im\right)}\]

    13. Applied simplify5.6

      \[\leadsto \color{blue}{\frac{(\left(\frac{x.re}{y.re}\right) \cdot y.im + \left(-x.im\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]

    if -2.591897792400522e+175 < y.re < 2.356201000260532e+102

    1. Initial program 19.8

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify19.8

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt19.8

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

    5. Applied *-un-lft-identity19.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(y.re \cdot x.im - x.re \cdot y.im\right)}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    6. Applied times-frac19.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

    7. Applied simplify19.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    8. Applied simplify12.2

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}\]
    9. Using strategy rm

    10. Applied fma-neg12.2

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{\color{blue}{(x.im \cdot y.re + \left(-x.re \cdot y.im\right))_*}}{\sqrt{y.im^2 + y.re^2}^*}\]

    if 2.356201000260532e+102 < y.re

    1. Initial program 37.7

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify37.7

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt37.7

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

    5. Applied *-un-lft-identity37.7

      \[\leadsto \frac{\color{blue}{1 \cdot \left(y.re \cdot x.im - x.re \cdot y.im\right)}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    6. Applied times-frac37.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

    7. Applied simplify37.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

    8. Applied simplify25.1

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}\]

    9. Taylor expanded around 0 29.1

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{y.re}}\]

    10. Applied simplify9.9

      \[\leadsto \color{blue}{\frac{(\left(\frac{x.re}{y.re}\right) \cdot \left(-y.im\right) + x.im)_*}{\sqrt{y.im^2 + y.re^2}^*}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(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))))