Average Error: 25.7 → 13.2
Time: 42.2s
Precision: 64
Internal Precision: 576
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -2.102080422642774 \cdot 10^{+82}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{if}\;y.im \le 7.460001049181088 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

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.im < -2.102080422642774e+82

    1. Initial program 38.0

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify38.0

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

    4. Applied add-sqr-sqrt38.0

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

    5. Applied *-un-lft-identity38.0

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

    6. Applied times-frac38.0

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

    7. Applied simplify38.0

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

    8. Applied simplify24.5

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

    9. Taylor expanded around -inf 18.8

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

    10. Applied simplify18.7

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

    if -2.102080422642774e+82 < y.im < 7.460001049181088e+157

    1. Initial program 18.7

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify18.7

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

    4. Applied add-sqr-sqrt18.7

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

    5. Applied *-un-lft-identity18.7

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

    6. Applied times-frac18.7

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

    7. Applied simplify18.7

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

    8. Applied simplify11.8

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

    10. Applied associate-*r/11.8

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

    11. Applied simplify11.7

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

    if 7.460001049181088e+157 < y.im

    1. Initial program 44.3

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

    2. Applied simplify44.3

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

    4. Applied add-sqr-sqrt44.3

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

    5. Applied *-un-lft-identity44.3

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

    6. Applied times-frac44.3

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

    7. Applied simplify44.3

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

    8. Applied simplify28.2

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

    9. Taylor expanded around inf 12.6

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

    10. Applied simplify12.4

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

Runtime

Time bar (total: 42.2s)Debug logProfile

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +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))))