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Target

Derivation

1. Initial program 25.5

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

3. Applied add-sqr-sqrt25.5

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

4. Applied associate-/r*25.4

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

6. Applied div-sub25.5

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

8. Applied *-un-lft-identity25.5

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

9. Applied times-frac23.6

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

10. Simplified23.6

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

12. Applied *-un-lft-identity23.6

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

13. Applied sqrt-prod23.6

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

14. Applied times-frac21.9

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

15. Simplified21.9

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

16. Final simplification21.9

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

Runtime

Time bar (total: 24.9s)Debug logProfile

herbie shell --seed 2018274 
(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))))