Average Error: 25.8 → 7.6
Time: 42.1s
Precision: 64
Internal Precision: 576
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}} \le -1.9988990777228786 \cdot 10^{+296}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}} \le 7.145262622020625 \cdot 10^{+256}:\\ \;\;\;\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (- (/ (/ x.im (hypot y.im y.re)) (/ (hypot y.im y.re) y.re)) (/ (/ (* x.re y.im) (hypot y.im y.re)) (* (sqrt (hypot y.im y.re)) (sqrt (hypot y.im y.re))))) < -1.9988990777228786e+296 or 7.145262622020625e+256 < (- (/ (/ x.im (hypot y.im y.re)) (/ (hypot y.im y.re) y.re)) (/ (/ (* x.re y.im) (hypot y.im y.re)) (* (sqrt (hypot y.im y.re)) (sqrt (hypot y.im y.re)))))

    1. Initial program 54.3

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

    2. Initial simplification54.3

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

    4. Applied add-sqr-sqrt54.3

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

    5. Applied *-un-lft-identity54.3

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

    6. Applied times-frac54.3

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

    7. Simplified54.3

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

    8. Simplified52.1

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

    10. Applied associate-*l/52.1

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

    11. Simplified52.1

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

    12. Taylor expanded around 0 40.4

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

    13. Simplified40.4

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

    if -1.9988990777228786e+296 < (- (/ (/ x.im (hypot y.im y.re)) (/ (hypot y.im y.re) y.re)) (/ (/ (* x.re y.im) (hypot y.im y.re)) (* (sqrt (hypot y.im y.re)) (sqrt (hypot y.im y.re))))) < 7.145262622020625e+256

    1. Initial program 20.4

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

    2. Initial simplification20.4

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

    4. Applied add-sqr-sqrt20.4

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

    5. Applied *-un-lft-identity20.4

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

    6. Applied times-frac20.4

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

    7. Simplified20.4

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

    8. Simplified9.8

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

    10. Applied associate-*l/9.7

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

    11. Simplified9.7

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

    13. Applied add-sqr-sqrt9.9

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

    15. Applied div-sub9.9

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

    16. Applied div-sub9.9

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

    17. Simplified1.3

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

  4. Final simplification7.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}} \le -1.9988990777228786 \cdot 10^{+296}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}} \le 7.145262622020625 \cdot 10^{+256}:\\ \;\;\;\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 42.1s)Debug logProfile

herbie shell --seed 2018225 +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))))