Average Error: 26.0 → 13.6
Time: 53.8s
Precision: 64
Internal Precision: 576
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le +\infty:\\ \;\;\;\;\frac{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 2 regimes

  2. if (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))) < +inf.0

    1. Initial program 16.1

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

    2. Applied simplify16.1

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

    4. Applied add-sqr-sqrt16.1

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

    5. Applied *-un-lft-identity16.1

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

    6. Applied times-frac16.1

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

    7. Applied simplify16.1

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

    8. Applied simplify4.5

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

    10. Applied associate-*r/4.5

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

    11. Applied simplify4.4

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

    if +inf.0 < (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))

    1. Initial program 62.9

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

    2. Applied simplify62.9

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

    4. Applied add-sqr-sqrt62.9

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

    5. Applied *-un-lft-identity62.9

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

    6. Applied times-frac62.9

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

    7. Applied simplify62.9

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

    8. Applied simplify62.2

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

    9. Taylor expanded around 0 47.9

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

    10. Applied simplify47.8

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

Runtime

Time bar (total: 53.8s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +o rules:numerics
(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))))