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Target

Derivation

1. Split input into 2 regimes

2. if c < 2.2233373112832877e+107

   1. Initial program 22.7

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

   2. Initial simplification22.7

      \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt22.7

      \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   5. Applied associate-/r*22.6

      \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]

   if 2.2233373112832877e+107 < c

   1. Initial program 40.4

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\]

   2. Initial simplification40.4

      \[\leadsto \frac{b \cdot d + a \cdot c}{c \cdot c + d \cdot d}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt40.4

      \[\leadsto \frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]

   5. Applied associate-/r*40.4

      \[\leadsto \color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}}\]
   6. Using strategy rm

   7. Applied add-sqr-sqrt40.5

      \[\leadsto \frac{\frac{b \cdot d + a \cdot c}{\color{blue}{\sqrt{\sqrt{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{c \cdot c + d \cdot d}}\]

   8. Applied associate-/r*40.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{b \cdot d + a \cdot c}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}{\sqrt{\sqrt{c \cdot c + d \cdot d}}}}}{\sqrt{c \cdot c + d \cdot d}}\]

   9. Taylor expanded around 0 40.7

      \[\leadsto \frac{\color{blue}{a}}{\sqrt{c \cdot c + d \cdot d}}\]
3. Recombined 2 regimes into one program.

4. Final simplification25.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le 2.2233373112832877 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{b \cdot d + c \cdot a}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Runtime

Time bar (total: 45.6s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (a b c d)
  :name "Complex division, real part"

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))