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Target

Derivation

1. Split input into 2 regimes

2. if d < 3.052453129606842e+67

   1. Initial program 23.0

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

   2. Initial simplification23.0

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

   4. Applied add-sqr-sqrt23.0

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

   5. Applied associate-/r*22.9

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

   7. Applied div-inv23.0

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

   if 3.052453129606842e+67 < d

   1. Initial program 37.9

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

   2. Initial simplification37.9

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

   4. Applied add-sqr-sqrt37.9

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

   5. Applied associate-/r*37.9

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

   7. Applied div-inv37.9

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

   8. Taylor expanded around inf 37.8

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

4. Final simplification26.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 3.052453129606842 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{d \cdot d + c \cdot c}} \cdot \left(a \cdot c + b \cdot d\right)}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d \cdot d + c \cdot c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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))))

Details

Time bar (total: 12.1s)Debug log