Average Error: 25.7 → 17.4
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-1}{y.re} \le -3.331236333471601 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{if}\;\frac{-1}{y.re} \le -1921186529010587.2:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{if}\;\frac{-1}{y.re} \le -2.0882375893266236 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{if}\;\frac{-1}{y.re} \le 1.4713495521679513 \cdot 10^{-122}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{if}\;\frac{-1}{y.re} \le 4.304808412481382 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

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 (/ -1 y.re) < -3.331236333471601e+125 or -1921186529010587.2 < (/ -1 y.re) < -2.0882375893266236e-38 or 4.304808412481382e+139 < (/ -1 y.re)

    1. Initial program 21.5

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

    3. Applied add-cbrt-cube39.1

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    4. Applied add-cbrt-cube50.0

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}\]

    5. Applied cbrt-undiv51.7

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    6. Applied simplify40.9

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{y.re \cdot x.re + y.im \cdot x.im}{y.im \cdot y.im + y.re \cdot y.re}\right)}^{3}}}\]

    7. Taylor expanded around 0 52.8

      \[\leadsto \color{blue}{e^{\log x.im - \log y.im}}\]

    8. Applied simplify19.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im}}\]

    if -3.331236333471601e+125 < (/ -1 y.re) < -1921186529010587.2 or 1.4713495521679513e-122 < (/ -1 y.re) < 4.304808412481382e+139

    1. Initial program 14.6

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

    3. Applied add-sqr-sqrt14.6

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

    4. Applied associate-/r*14.5

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

    if -2.0882375893266236e-38 < (/ -1 y.re) < 1.4713495521679513e-122

    1. Initial program 37.8

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

    3. Applied add-cbrt-cube47.2

      \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    4. Applied add-cbrt-cube53.9

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}}{\sqrt[3]{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}\]

    5. Applied cbrt-undiv54.1

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(x.re \cdot y.re + x.im \cdot y.im\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\right) \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\left(\left(y.re \cdot y.re + y.im \cdot y.im\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)\right) \cdot \left(y.re \cdot y.re + y.im \cdot y.im\right)}}}\]

    6. Applied simplify43.4

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{y.re \cdot x.re + y.im \cdot x.im}{y.im \cdot y.im + y.re \cdot y.re}\right)}^{3}}}\]

    7. Taylor expanded around inf 51.3

      \[\leadsto \color{blue}{e^{\log \left(\frac{1}{y.re}\right) - \log \left(\frac{1}{x.re}\right)}}\]

    8. Applied simplify18.2

      \[\leadsto \color{blue}{\frac{x.re}{y.re}}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2020178 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))