Average Error: 25.8 → 25.2
Time: 21.8s
Precision: 64
Internal Precision: 384
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -5.712982657475991 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{(\left(\frac{x.im}{y.im}\right) \cdot y.re + \left(-x.re\right))_*}{-1}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\\ \end{array}\]

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 < -5.712982657475991e+84

    1. Initial program 38.1

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

    2. Applied simplify38.1

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

    4. Applied add-sqr-sqrt38.1

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

    5. Applied associate-/r*38.0

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

    6. Taylor expanded around -inf 38.9

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

    7. Applied simplify35.1

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

    if -5.712982657475991e+84 < y.im

    1. Initial program 23.0

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

    2. Applied simplify23.0

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

    4. Applied add-sqr-sqrt23.0

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

    5. Applied associate-/r*22.9

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

Runtime

Time bar (total: 21.8s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' +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))))