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Target

Derivation

1. Split input into 3 regimes

2. if c < -3.499229178127668e+134

   1. Initial program 42.8

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

   2. Initial simplification42.8

      \[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt42.8

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

   5. Applied *-un-lft-identity42.8

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

   6. Applied times-frac42.8

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

   7. Simplified42.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

   8. Simplified28.0

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/27.9

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

   11. Simplified27.9

       \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]

   12. Taylor expanded around -inf 13.9

       \[\leadsto \frac{\color{blue}{-1 \cdot b}}{\sqrt{d^2 + c^2}^*}\]

   13. Simplified13.9

       \[\leadsto \frac{\color{blue}{-b}}{\sqrt{d^2 + c^2}^*}\]

   if -3.499229178127668e+134 < c < 9.744182275133171e+98

   1. Initial program 17.6

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

   2. Initial simplification17.6

      \[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt17.6

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

   5. Applied *-un-lft-identity17.6

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

   6. Applied times-frac17.6

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

   7. Simplified17.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

   8. Simplified11.0

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/10.9

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

   11. Simplified10.9

       \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]

   if 9.744182275133171e+98 < c

   1. Initial program 38.5

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

   2. Initial simplification38.5

      \[\leadsto \frac{b \cdot c - a \cdot d}{(d \cdot d + \left(c \cdot c\right))_*}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt38.5

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

   5. Applied *-un-lft-identity38.5

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

   6. Applied times-frac38.5

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

   7. Simplified38.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{d^2 + c^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{(d \cdot d + \left(c \cdot c\right))_*}}\]

   8. Simplified24.8

      \[\leadsto \frac{1}{\sqrt{d^2 + c^2}^*} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/24.7

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{b \cdot c - a \cdot d}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}}\]

   11. Simplified24.7

       \[\leadsto \frac{\color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{d^2 + c^2}^*}}}{\sqrt{d^2 + c^2}^*}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity24.7

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b - a \cdot d\right)}}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\]

   14. Applied associate-/l*24.7

       \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{d^2 + c^2}^*}{c \cdot b - a \cdot d}}}}{\sqrt{d^2 + c^2}^*}\]

   15. Taylor expanded around 0 15.1

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

4. Final simplification12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -3.499229178127668 \cdot 10^{+134}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 9.744182275133171 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 42.9s)Debug logProfile

herbie shell --seed 2018258 +o rules:numerics
(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))))