Average Error: 25.5 → 1.8
Time: 45.9s
Precision: 64
Internal Precision: 128
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \cdot y.re \le -2.8997370585315772 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{\sqrt{y.im^2 + y.re^2}^*}{x.im}}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;x.im \cdot y.re \le 5.856336305743566 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

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 3 regimes

  2. if (* x.im y.re) < -2.8997370585315772e+301

    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. Initial simplification62.5

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

    4. Applied add-sqr-sqrt62.5

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

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

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

    6. Applied times-frac62.5

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

    7. Simplified62.5

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

    8. Simplified60.1

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

    10. Applied associate-*l/60.1

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

    11. Simplified60.1

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

    13. Applied div-sub60.1

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

    14. Applied div-sub60.1

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

    16. Applied associate-/l*6.1

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

    if -2.8997370585315772e+301 < (* x.im y.re) < 5.856336305743566e+295

    1. Initial program 19.7

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

    2. Initial simplification19.7

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

    4. Applied add-sqr-sqrt19.7

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

    5. Applied *-un-lft-identity19.7

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

    6. Applied times-frac19.7

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

    7. Simplified19.7

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

    8. Simplified9.4

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

    10. Applied associate-*l/9.3

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

    11. Simplified9.3

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

    13. Applied div-sub9.3

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

    14. Applied div-sub9.3

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

    16. Applied *-un-lft-identity9.3

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

    17. Applied times-frac1.0

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

    18. Simplified1.0

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

    if 5.856336305743566e+295 < (* x.im y.re)

    1. Initial program 60.7

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

    2. Initial simplification60.7

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

    4. Applied add-sqr-sqrt60.7

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

    5. Applied *-un-lft-identity60.7

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

    6. Applied times-frac60.7

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

    7. Simplified60.7

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

    8. Simplified58.7

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

    10. Applied associate-*l/58.7

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

    11. Simplified58.7

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

    13. Applied div-sub58.7

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

    14. Applied div-sub58.7

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

    16. Applied add-sqr-sqrt58.7

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

    17. Applied times-frac8.3

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \cdot y.re \le -2.8997370585315772 \cdot 10^{+301}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{\sqrt{y.im^2 + y.re^2}^*}{x.im}}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;x.im \cdot y.re \le 5.856336305743566 \cdot 10^{+295}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{y.im \cdot x.re}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 45.9s)Debug logProfile

herbie shell --seed 2018274 +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))))