Average Error: 25.9 → 13.1
Time: 1.1m
Precision: 64
Internal Precision: 576
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(\frac{\sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*} \cdot \sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right) \le -1.7776580676770771 \cdot 10^{+308}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{if}\;\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(\frac{\sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*} \cdot \sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\sqrt[3]{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right) \le 1.7822238799756147 \cdot 10^{+308}:\\ \;\;\;\;\frac{\frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 3 regimes

  2. if (* (/ 1 (hypot y.re y.im)) (* (/ (* (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (fma x.im y.im (* y.re x.re)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (hypot y.re y.im))))) < -1.7776580676770771e+308

    1. Initial program 63.0

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

    3. Applied add-sqr-sqrt63.0

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

    4. Applied *-un-lft-identity63.0

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

    5. Applied times-frac63.0

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

    6. Applied simplify63.0

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

    7. Applied simplify62.0

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

    8. Taylor expanded around -inf 48.0

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

    9. Applied simplify47.9

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

    if -1.7776580676770771e+308 < (* (/ 1 (hypot y.re y.im)) (* (/ (* (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (fma x.im y.im (* y.re x.re)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (hypot y.re y.im))))) < 1.7822238799756147e+308

    1. Initial program 13.7

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

    3. Applied add-sqr-sqrt13.7

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

    4. Applied *-un-lft-identity13.7

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

    5. Applied times-frac13.7

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

    6. Applied simplify13.7

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

    7. Applied simplify1.5

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

    9. Applied associate-*r/1.4

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

    10. Applied simplify1.3

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

    if 1.7822238799756147e+308 < (* (/ 1 (hypot y.re y.im)) (* (/ (* (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (fma x.im y.im (* y.re x.re)))) (* (cbrt (hypot y.re y.im)) (cbrt (hypot y.re y.im)))) (/ (cbrt (fma x.im y.im (* y.re x.re))) (cbrt (hypot y.re y.im)))))

    1. Initial program 62.0

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

    3. Applied add-sqr-sqrt62.0

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

    4. Applied *-un-lft-identity62.0

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

    5. Applied times-frac62.0

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

    6. Applied simplify62.0

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

    7. Applied simplify62.0

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

    8. Taylor expanded around 0 49.0

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

    9. Applied simplify48.9

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018199 +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))))