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Target

Derivation

1. Split input into 2 regimes

2. if (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) < 2.993017549186881e+304

   1. Initial program 14.2

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

   2. Initial simplification14.2

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

   if 2.993017549186881e+304 < (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))

   1. Initial program 62.3

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

   2. Initial simplification62.3

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

   4. Applied add-sqr-sqrt62.3

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

   5. Applied *-un-lft-identity62.3

      \[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   6. Applied times-frac62.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]

   7. Taylor expanded around inf 60.5

      \[\leadsto \frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \color{blue}{b}\]
3. Recombined 2 regimes into one program.

4. Final simplification25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \le 2.993017549186881 \cdot 10^{+304}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot b\\ \end{array}\]

Runtime

Time bar (total: 21.5s)Debug logProfile

herbie shell --seed 2018257 
(FPCore (a b c d)
  :name "Complex division, imag part"

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

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