Average Error: 26.5 → 13.6
Time: 37.8s
Precision: 64
Internal Precision: 384
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -1.1311343419872614 \cdot 10^{+58}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{if}\;y.re \le 4.330565057877467 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

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 < -1.1311343419872614e+58

    1. Initial program 36.2

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

    2. Applied simplify36.2

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

    4. Applied add-sqr-sqrt36.2

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

    5. Applied associate-/r*36.2

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

    7. Applied fma-udef36.2

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

    8. Applied hypot-def36.2

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

    9. Taylor expanded around -inf 18.3

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

    10. Applied simplify18.3

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

    if -1.1311343419872614e+58 < y.re < 4.330565057877467e+130

    1. Initial program 19.1

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

    2. Applied simplify19.1

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

    4. Applied add-sqr-sqrt19.1

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

    5. Applied associate-/r*19.0

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

    7. Applied fma-udef19.0

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

    8. Applied hypot-def19.0

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

    10. Applied fma-udef19.0

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

    11. Applied hypot-def11.6

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

    if 4.330565057877467e+130 < y.re

    1. Initial program 42.3

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

    2. Applied simplify42.3

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

    4. Applied add-sqr-sqrt42.3

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

    5. Applied associate-/r*42.3

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

    7. Applied fma-udef42.3

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

    8. Applied hypot-def42.3

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

    9. Taylor expanded around inf 15.1

      \[\leadsto \frac{\color{blue}{x.im}}{\sqrt{y.im^2 + y.re^2}^*}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 37.8s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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))))