Average Error: 26.2 → 13.2
Time: 35.2s
Precision: 64
Internal Precision: 576
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \le -1.7753108952350828 \cdot 10^{+308}:\\ \;\;\;\;x.im \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;\left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \le 1.7777210658548273 \cdot 10^{+308}:\\ \;\;\;\;\frac{(x.im \cdot y.re + \left(-y.im \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{y.re^2 + y.im^2}^*} \cdot x.re\\ \end{array}\]

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 (* (sqrt (/ 1 (hypot y.re y.im))) (* (sqrt (/ 1 (hypot y.re y.im))) (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im)))) < -1.7753108952350828e+308

    1. Initial program 63.0

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

    3. Applied add-sqr-sqrt63.0

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

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

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

    5. Applied times-frac63.0

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

    6. Simplified63.0

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

    7. Simplified61.9

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

    8. Taylor expanded around inf 48.2

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

    if -1.7753108952350828e+308 < (* (sqrt (/ 1 (hypot y.re y.im))) (* (sqrt (/ 1 (hypot y.re y.im))) (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im)))) < 1.7777210658548273e+308

    1. Initial program 13.9

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

    3. Applied add-sqr-sqrt13.9

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

    4. Applied *-un-lft-identity13.9

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

    5. Applied times-frac13.9

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

    6. Simplified13.9

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

    7. Simplified1.5

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

    9. Applied fma-neg1.5

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

    if 1.7777210658548273e+308 < (* (sqrt (/ 1 (hypot y.re y.im))) (* (sqrt (/ 1 (hypot y.re y.im))) (/ (- (* x.im y.re) (* x.re y.im)) (hypot y.re y.im))))

    1. Initial program 62.0

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

    3. Applied add-sqr-sqrt62.0

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

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

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

    5. Applied times-frac62.0

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

    6. Simplified62.0

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

    7. Simplified62.0

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

    8. Taylor expanded around 0 47.6

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

    9. Simplified47.6

      \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(-x.re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \le -1.7753108952350828 \cdot 10^{+308}:\\ \;\;\;\;x.im \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;\left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \le 1.7777210658548273 \cdot 10^{+308}:\\ \;\;\;\;\frac{(x.im \cdot y.re + \left(-y.im \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sqrt{y.re^2 + y.im^2}^*} \cdot x.re\\ \end{array}\]

Runtime

Time bar (total: 35.2s)Debug logProfile

herbie shell --seed 2018217 +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))))