Average Error: 25.7 → 13.2
Time: 18.5s
Precision: 64
Internal Precision: 384
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \le -1.343292130239259 \cdot 10^{+105}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{if}\;y.im \le 3.10201271226388 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \left(\frac{1}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

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 < -1.343292130239259e+105

    1. Initial program 39.3

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

    2. Applied simplify39.3

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

    4. Applied add-sqr-sqrt39.3

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

    5. Applied *-un-lft-identity39.3

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

    6. Applied times-frac39.3

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

    8. Applied fma-udef39.3

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

    9. Applied hypot-def39.3

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

    11. Applied fma-udef39.3

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

    12. Applied hypot-def26.2

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

    13. Taylor expanded around -inf 17.2

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

    14. Applied simplify17.0

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

    if -1.343292130239259e+105 < y.im < 3.10201271226388e+144

    1. Initial program 18.5

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

    2. Applied simplify18.5

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

    4. Applied add-sqr-sqrt18.5

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

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

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

    6. Applied times-frac18.5

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

    8. Applied fma-udef18.5

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

    9. Applied hypot-def18.5

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

    11. Applied fma-udef18.5

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

    12. Applied hypot-def11.3

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

    14. Applied add-cube-cbrt12.0

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}}\]

    15. Applied *-un-lft-identity12.0

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{\color{blue}{1 \cdot (x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\left(\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}\]

    16. Applied times-frac12.0

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.im^2 + y.re^2}^*}}\right)}\]

    if 3.10201271226388e+144 < y.im

    1. Initial program 43.8

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

    2. Applied simplify43.8

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

    4. Applied add-sqr-sqrt43.8

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

    5. Applied *-un-lft-identity43.8

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

    6. Applied times-frac43.8

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

    8. Applied fma-udef43.8

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

    9. Applied hypot-def43.8

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

    11. Applied fma-udef43.8

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

    12. Applied hypot-def26.2

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

    13. Taylor expanded around inf 14.6

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

    14. Applied simplify14.5

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

Runtime

Time bar (total: 18.5s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))