Average Error: 26.1 → 13.4
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im} \cdot \frac{\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im}}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \frac{\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im}}{\sqrt{y.re^2 + y.im^2}^*} \le -1.7761824129243311 \cdot 10^{+308}:\\ \;\;\;\;\frac{(\left(\frac{y.re}{y.im}\right) \cdot x.im + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{if}\;\left(\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im} \cdot \frac{\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im}}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \frac{\sqrt[3]{y.re \cdot x.im - x.re \cdot y.im}}{\sqrt{y.re^2 + y.im^2}^*} \le 1.7795226370784402 \cdot 10^{+308}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{y.re}{y.im}\right) \cdot x.im + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

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 2 regimes

  2. if (* (* (cbrt (- (* y.re x.im) (* x.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im))) < -1.7761824129243311e+308 or 1.7795226370784402e+308 < (* (* (cbrt (- (* y.re x.im) (* x.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im)))

    1. Initial program 62.5

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

    3. Applied add-sqr-sqrt62.5

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

    4. Applied *-un-lft-identity62.5

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

    5. Applied times-frac62.5

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

    6. Applied simplify62.5

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

    7. Applied simplify62.0

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

    9. Applied associate-*r/62.0

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

    10. Applied simplify62.0

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

    11. Taylor expanded around 0 62.8

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

    12. Applied simplify47.7

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

    if -1.7761824129243311e+308 < (* (* (cbrt (- (* y.re x.im) (* x.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im))) (/ (cbrt (- (* y.re x.im) (* x.re y.im))) (hypot y.re y.im))) < 1.7795226370784402e+308

    1. Initial program 13.5

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

    3. Applied add-sqr-sqrt13.5

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

    4. Applied *-un-lft-identity13.5

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

    5. Applied times-frac13.5

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

    6. Applied simplify13.5

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

    7. Applied simplify1.6

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

    9. Applied associate-*r/1.6

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

    10. Applied simplify1.5

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018178 +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))))