math.sqrt on complex, imaginary part, im greater than 0 branch

Average Error: 38.9 → 21.3

Time: 3.8s

Precision: binary64

Math

\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -1969972166812.6543:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -7.46472792011176953 \cdot 10^{-134}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.29424588234639691 \cdot 10^{91}:\\ \;\;\;\;0.5 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{\left|im\right|}\right) \cdot \sqrt{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{re + re}}\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 4 regimes

2. if re < -1969972166812.6543

   1. Initial program 41.5

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

   2. Taylor expanded around -inf 14.6

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

   if -1969972166812.6543 < re < -7.46472792011176953e-134

   1. Initial program 16.7

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

   3. Applied flip-- 40.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 40.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]

   5. Taylor expanded around -inf 40.0

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

   if -7.46472792011176953e-134 < re < 1.29424588234639691e91

   1. Initial program 34.5

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

   3. Applied flip-- 35.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 30.9

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied associate-*r/ 30.9

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(0 + {im}^{2}\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   7. Applied sqrt-div 30.9

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]
   8. Using strategy rm

   9. Applied sqrt-prod 31.0

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{0 + {im}^{2}}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

   10. Simplified 24.2

       \[\leadsto 0.5 \cdot \frac{\sqrt{2} \cdot \color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
   11. Using strategy rm

   12. Applied add-sqr-sqrt 24.2

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

   13. Applied associate-*r* 24.2

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

   if 1.29424588234639691e91 < re

   1. Initial program 61.3

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

   3. Applied flip-- 61.4

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   4. Simplified 45.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{0 + {im}^{2}}}{\sqrt{re \cdot re + im \cdot im} + re}}\]
   5. Using strategy rm

   6. Applied associate-*r/ 45.0

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2 \cdot \left(0 + {im}^{2}\right)}{\sqrt{re \cdot re + im \cdot im} + re}}}\]

   7. Applied sqrt-div 43.5

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\]
   8. Using strategy rm

   9. Applied sqrt-prod 43.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2} \cdot \sqrt{0 + {im}^{2}}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

   10. Simplified 41.0

       \[\leadsto 0.5 \cdot \frac{\sqrt{2} \cdot \color{blue}{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

   11. Taylor expanded around inf 10.9

       \[\leadsto 0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{\color{blue}{re} + re}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 21.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1969972166812.6543:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -7.46472792011176953 \cdot 10^{-134}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-\left(re + im\right)\right)}\\ \mathbf{elif}\;re \le 1.29424588234639691 \cdot 10^{91}:\\ \;\;\;\;0.5 \cdot \frac{\left(\sqrt{2} \cdot \sqrt{\left|im\right|}\right) \cdot \sqrt{\left|im\right|}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2} \cdot \left|im\right|}{\sqrt{re + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020150 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))