Average Error: 37.5 → 11.0
Time: 12.5s
Precision: 64
Internal Precision: 3392
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -16281288.137872282:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(2.0 \cdot re\right))_*}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Target

Original37.5
Target32.6
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -16281288.137872282

    1. Initial program 55.7

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Initial simplification38.7

      \[\leadsto 0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\]
    3. Using strategy rm

    4. Applied add-exp-log39.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)}}}\]
    5. Using strategy rm

    6. Applied pow139.9

      \[\leadsto 0.5 \cdot \sqrt{e^{\log \color{blue}{\left({\left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)}^{1}\right)}}}\]

    7. Applied log-pow39.9

      \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{1 \cdot \log \left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)}}}\]

    8. Applied exp-prod40.0

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left(e^{1}\right)}^{\left(\log \left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)\right)}}}\]

    9. Simplified40.0

      \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{e}}^{\left(\log \left((\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*\right)\right)}}\]

    10. Taylor expanded around -inf 46.6

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(\log \left(\frac{-1}{re}\right) + \log 1.0\right) - 2 \cdot \log \left(\frac{-1}{im}\right)}}}\]

    11. Simplified29.1

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}}}\]

    if -16281288.137872282 < re

    1. Initial program 31.9

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    2. Initial simplification5.4

      \[\leadsto 0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -16281288.137872282:\\ \;\;\;\;\sqrt{\frac{\frac{-1}{re}}{\frac{-1}{im}} \cdot \frac{1.0}{\frac{-1}{im}}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(2.0 \cdot re\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 12.5s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))