Error

Target

Derivation

1. Split input into 2 regimes

2. if re < -4.834562283134786e+206

   1. Initial program 62.3

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

   2. Simplified43.7

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

   4. Applied add-sqr-sqrt43.7

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

   6. Applied add-sqr-sqrt43.7

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

   7. Applied sqrt-prod43.7

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

   8. Applied associate-*r*43.7

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

   10. Applied pow143.7

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

   11. Applied add-sqr-sqrt43.7

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

   12. Applied pow343.7

       \[\leadsto 0.5 \cdot \left(\color{blue}{{\left(\sqrt{\sqrt{\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}}}\right)}^{3}} \cdot {\left(\sqrt{\sqrt{\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}}}\right)}^{1}\right)\]

   13. Applied pow-prod-up43.7

       \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{\sqrt{\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}}}\right)}^{\left(3 + 1\right)}}\]

   14. Simplified43.7

       \[\leadsto 0.5 \cdot {\left(\sqrt{\sqrt{\sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}}}\right)}^{\color{blue}{4}}\]

   15. Taylor expanded around -inf 37.0

       \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{\frac{1}{8} \cdot \left(\left(\log \left(\frac{-1}{re}\right) + \log 1.0\right) - 2 \cdot \log \left(\frac{-1}{im}\right)\right)}\right)}^{4}}\]

   16. Simplified37.0

       \[\leadsto 0.5 \cdot \color{blue}{{\left(e^{(\left(\log 1.0 + \log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{8} + \left(\frac{-1}{4} \cdot \log \left(\frac{-1}{im}\right)\right))_*}\right)}^{4}}\]

   if -4.834562283134786e+206 < re

   1. Initial program 35.3

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

   2. Simplified10.8

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

4. Final simplification13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -4.834562283134786 \cdot 10^{+206}:\\ \;\;\;\;0.5 \cdot {\left(e^{(\left(\log 1.0 + \log \left(\frac{-1}{re}\right)\right) \cdot \frac{1}{8} + \left(\log \left(\frac{-1}{im}\right) \cdot \frac{-1}{4}\right))_*}\right)}^{4}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{(\left(\sqrt{re^2 + im^2}^*\right) \cdot 2.0 + \left(re \cdot 2.0\right))_*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 +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)))))

Details

Time bar (total: 13.4s)Debug log