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Target

Derivation

1. Split input into 3 regimes

2. if c < -1.2124496835135948e+160

   1. Initial program 44.8

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

   2. Initial simplification44.8

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

   4. Applied add-sqr-sqrt44.8

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

   5. Applied *-un-lft-identity44.8

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

   6. Applied times-frac44.8

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

   7. Simplified44.8

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

   8. Simplified28.9

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

   10. Applied associate-*l/28.9

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

   11. Simplified28.9

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

   12. Taylor expanded around -inf 12.8

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

   13. Simplified12.8

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

   if -1.2124496835135948e+160 < c < 1.3202977940269371e+175

   1. Initial program 19.6

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

   2. Initial simplification19.6

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

   4. Applied add-sqr-sqrt19.6

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

   5. Applied *-un-lft-identity19.6

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

   6. Applied times-frac19.6

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

   7. Simplified19.6

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

   8. Simplified12.4

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

   10. Applied associate-*l/12.2

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

   11. Simplified12.2

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

   13. Applied div-inv12.3

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

   if 1.3202977940269371e+175 < c

   1. Initial program 45.3

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

   2. Initial simplification45.3

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

   4. Applied add-sqr-sqrt45.3

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

   5. Applied *-un-lft-identity45.3

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

   6. Applied times-frac45.3

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

   7. Simplified45.3

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

   8. Simplified30.8

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

   10. Applied associate-*l/30.8

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

   11. Simplified30.8

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

   13. Applied div-inv30.8

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

   14. Taylor expanded around inf 12.1

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

4. Final simplification12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -1.2124496835135948 \cdot 10^{+160}:\\ \;\;\;\;\frac{-b}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;c \le 1.3202977940269371 \cdot 10^{+175}:\\ \;\;\;\;\frac{\left(b \cdot c - d \cdot a\right) \cdot \frac{1}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 35.7s)Debug logProfile

herbie shell --seed 2018234 +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))))