Average Error: 25.9 → 14.4
Time: 20.5s
Precision: 64
Internal Precision: 384
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \le 9.263907382191759 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{(x.im \cdot y.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

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 2 regimes

  2. if y.re < 9.263907382191759e+167

    1. Initial program 23.4

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

    3. Applied add-sqr-sqrt23.4

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

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

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

    5. Applied times-frac23.4

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

    6. Applied simplify23.4

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

    7. Applied simplify14.9

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

    9. Applied pow114.9

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

    10. Applied pow114.9

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

    11. Applied pow-prod-down14.9

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

    12. Applied simplify14.8

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

    if 9.263907382191759e+167 < y.re

    1. Initial program 43.3

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

    3. Applied add-sqr-sqrt43.3

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

    4. Applied *-un-lft-identity43.3

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

    5. Applied times-frac43.3

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

    6. Applied simplify43.3

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

    7. Applied simplify27.2

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

    8. Taylor expanded around 0 11.8

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

    9. Applied simplify11.6

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

  4. Applied simplify14.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;y.re \le 9.263907382191759 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{(x.im \cdot y.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}}\]

Runtime

Time bar (total: 20.5s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))))