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Target

Derivation

1. Split input into 2 regimes

2. if b < -3.7549061336177695e+50 or 3.160825419620763e+142 < b

   1. Initial program 33.6

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

   3. Applied add-sqr-sqrt33.6

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

   4. Applied *-un-lft-identity33.6

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

   5. Applied times-frac33.6

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

   6. Simplified33.6

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

   7. Simplified28.1

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

   9. Applied associate-*l/28.0

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

   10. Simplified28.0

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

   12. Applied div-sub28.0

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

   14. Applied *-un-lft-identity28.0

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

   15. Applied times-frac10.2

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

   16. Simplified10.2

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

   if -3.7549061336177695e+50 < b < 3.160825419620763e+142

   1. Initial program 21.1

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

   3. Applied add-sqr-sqrt21.1

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

   4. Applied *-un-lft-identity21.1

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

   5. Applied times-frac21.1

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

   6. Simplified21.1

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

   7. Simplified10.6

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

   9. Applied associate-*l/10.5

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

   10. Simplified10.5

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

   12. Applied div-sub10.5

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

   14. Applied *-un-lft-identity10.5

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

   15. Applied times-frac2.3

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

   16. Simplified2.3

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

4. Final simplification4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.7549061336177695 \cdot 10^{+50} \lor \neg \left(b \le 3.160825419620763 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{\sqrt{c^2 + d^2}^*} - \frac{d \cdot a}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - a \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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

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