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Target

Derivation

1. Split input into 2 regimes

2. if d < 5.1781749674137205e+95

   1. Initial program 22.6

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

   2. Initial simplification22.6

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

   4. Applied add-sqr-sqrt22.6

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

   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 5.1781749674137205e+95 < d

   1. Initial program 39.0

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

   2. Initial simplification39.0

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

   4. Applied add-sqr-sqrt39.0

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

   5. Applied associate-/r*39.0

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

   6. Taylor expanded around inf 38.5

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

4. Final simplification25.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 5.1781749674137205 \cdot 10^{+95}:\\ \;\;\;\;\frac{\frac{a \cdot c + b \cdot d}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Runtime

Time bar (total: 19.5s)Debug logProfile

herbie shell --seed 2018235 
(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))))