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Target

Derivation

1. Split input into 4 regimes

2. if re < -1.6958975798724648e+132

   1. Initial program 61.2

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

   3. Applied sqrt-prod61.2

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2.0} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]
   4. Using strategy rm

   5. Applied flip-+61.2

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \sqrt{\color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\]

   6. Applied sqrt-div61.2

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\]

   7. Simplified46.1

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   8. Taylor expanded around -inf 9.5

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{\color{blue}{-2 \cdot re}}}\right)\]

   if -1.6958975798724648e+132 < re < -1.0124855282781271e-294

   1. Initial program 38.8

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

   3. Applied sqrt-prod39.0

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2.0} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} + re}\right)}\]
   4. Using strategy rm

   5. Applied flip-+38.9

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \sqrt{\color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\]

   6. Applied sqrt-div38.9

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\]

   7. Simplified20.6

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]
   8. Using strategy rm

   9. Applied add-sqr-sqrt20.7

      \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \frac{\color{blue}{\sqrt{\left|im\right|} \cdot \sqrt{\left|im\right|}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\right)\]

   10. Applied associate-/l*20.6

       \[\leadsto 0.5 \cdot \left(\sqrt{2.0} \cdot \color{blue}{\frac{\sqrt{\left|im\right|}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}}\right)\]
   11. Using strategy rm

   12. Applied add-cube-cbrt20.6

       \[\leadsto 0.5 \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \sqrt[3]{\sqrt{2.0}}\right)} \cdot \frac{\sqrt{\left|im\right|}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}\right)\]

   13. Applied associate-*l*20.6

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \left(\sqrt[3]{\sqrt{2.0}} \cdot \frac{\sqrt{\left|im\right|}}{\frac{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}{\sqrt{\left|im\right|}}}\right)\right)}\]

   if -1.0124855282781271e-294 < re < 8.234804025380906e+145

   1. Initial program 20.4

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

   3. Applied sqrt-prod20.7

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

   if 8.234804025380906e+145 < re

   1. Initial program 59.4

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

   2. Taylor expanded around inf 8.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{re} + re\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.6958975798724648 \cdot 10^{+132}:\\ \;\;\;\;\left(\sqrt{2.0} \cdot \frac{\left|im\right|}{\sqrt{re \cdot -2}}\right) \cdot 0.5\\ \mathbf{elif}\;re \le -1.0124855282781271 \cdot 10^{-294}:\\ \;\;\;\;\left(\left(\frac{\sqrt{\left|im\right|}}{\frac{\sqrt{\sqrt{im \cdot im + re \cdot re} - re}}{\sqrt{\left|im\right|}}} \cdot \sqrt[3]{\sqrt{2.0}}\right) \cdot \left(\sqrt[3]{\sqrt{2.0}} \cdot \sqrt[3]{\sqrt{2.0}}\right)\right) \cdot 0.5\\ \mathbf{elif}\;re \le 8.234804025380906 \cdot 10^{+145}:\\ \;\;\;\;\left(\sqrt{\sqrt{im \cdot im + re \cdot re} + re} \cdot \sqrt{2.0}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + re\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 
(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: 26.3s)Debug log