Average Error: 25.8 → 13.6
Time: 33.2s
Precision: 64
Internal Precision: 384
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le 9.190386957669971 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(y.re \cdot \left(\frac{x.im}{y.im}\right) + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

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 2 regimes

  2. if y.im < 9.190386957669971e+115

    1. Initial program 23.2

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

    3. Applied add-sqr-sqrt23.2

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

    4. Applied *-un-lft-identity23.2

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

    5. Applied times-frac23.2

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

    6. Applied simplify23.2

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

    7. Applied simplify14.5

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

    9. Applied pow114.5

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

    10. Applied pow114.5

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

    11. Applied pow-prod-down14.5

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

    12. Applied simplify14.4

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

    if 9.190386957669971e+115 < y.im

    1. Initial program 38.7

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

    3. Applied add-sqr-sqrt38.7

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

    4. Applied *-un-lft-identity38.7

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

    5. Applied times-frac38.7

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

    6. Applied simplify38.7

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

    7. Applied simplify26.4

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

    9. Applied pow126.4

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

    10. Applied pow126.4

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

    11. Applied pow-prod-down26.4

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

    12. Applied simplify26.3

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

    13. Taylor expanded around 0 29.4

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

    14. Applied simplify9.6

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

  4. Applied simplify13.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;y.im \le 9.190386957669971 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{(y.re \cdot \left(\frac{x.im}{y.im}\right) + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}}\]

Runtime

Time bar (total: 33.2s)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +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))))