Average Error: 25.5 → 12.9
Time: 1.1m
Precision: 64
Internal Precision: 320
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \le -1.7770805725111145 \cdot 10^{+308}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\sqrt[3]{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \le 1.7752851147652889 \cdot 10^{+308}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (* (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (hypot y.re y.im)))) < -1.7770805725111145e+308 or 1.7752851147652889e+308 < (* (/ (* (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (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. Taylor expanded around 0 48.8

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

    9. Applied simplify48.8

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

    if -1.7770805725111145e+308 < (* (/ (* (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))) (cbrt (hypot y.re y.im)))) < 1.7752851147652889e+308

    1. Initial program 13.6

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

    3. Applied add-sqr-sqrt13.6

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

    4. Applied *-un-lft-identity13.6

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

    5. Applied times-frac13.6

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

    6. Applied simplify13.6

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

    7. Applied simplify1.5

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

    9. Applied associate-*r/1.4

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

    10. Applied simplify1.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018167 +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))))