Average Error: 25.8 → 13.3
Time: 1.4m
Precision: 64
Internal Precision: 576
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le -9.54297242402449 \cdot 10^{+218}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{if}\;y.re \le 2.2312818577631614 \cdot 10^{+139}:\\ \;\;\;\;\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{1}{\frac{\sqrt{y.re^2 + y.im^2}^*}{y.re \cdot x.im - x.re \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

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 y.re < -9.54297242402449e+218

    1. Initial program 43.2

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

    3. Applied add-sqr-sqrt43.2

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

    4. Applied *-un-lft-identity43.2

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

    5. Applied times-frac43.2

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

    6. Applied simplify43.2

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

    7. Applied simplify35.3

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

    8. Taylor expanded around -inf 10.0

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

    9. Applied simplify9.8

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

    if -9.54297242402449e+218 < y.re < 2.2312818577631614e+139

    1. Initial program 21.0

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

    3. Applied add-sqr-sqrt21.0

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

    4. Applied *-un-lft-identity21.0

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

    5. Applied times-frac21.1

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

    6. Applied simplify21.1

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

    7. Applied simplify13.3

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

    9. Applied clear-num13.4

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

    if 2.2312818577631614e+139 < y.re

    1. Initial program 42.9

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

    3. Applied add-sqr-sqrt42.9

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

    4. Applied *-un-lft-identity42.9

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

    5. Applied times-frac42.9

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

    6. Applied simplify42.9

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

    7. Applied simplify26.7

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

    8. Taylor expanded around inf 14.8

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

    9. Applied simplify14.6

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed '#(1071821486 549052472 3784827256 1559736200 3548510075 881134285)' +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))))