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Target

Derivation

1. Split input into 2 regimes

2. if d < 4.4205857620523035e+121

   1. Initial program 23.1

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

   3. Applied add-sqr-sqrt23.1

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

   4. Applied *-un-lft-identity23.1

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

   5. Applied times-frac23.1

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

   6. Simplified23.1

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

   7. Simplified14.5

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

   9. Applied associate-*l/14.4

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

   10. Simplified14.4

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

   if 4.4205857620523035e+121 < d

   1. Initial program 41.1

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

   3. Applied add-sqr-sqrt41.1

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

   4. Applied *-un-lft-identity41.1

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

   5. Applied times-frac41.1

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

   6. Simplified41.1

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

   7. Simplified26.3

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

   9. Applied associate-*l/26.2

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

   10. Simplified26.2

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

   12. Applied *-un-lft-identity26.2

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

   13. Applied associate-/l*26.3

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

   14. Taylor expanded around 0 15.1

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

   15. Simplified15.1

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

4. Final simplification14.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 4.4205857620523035 \cdot 10^{+121}:\\ \;\;\;\;\frac{\frac{c \cdot b - d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019008 +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))))

Details

Time bar (total: 24.2s)Debug log