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Target

Derivation

1. Split input into 4 regimes

2. if b < -2.146582560147369e+98

   1. Initial program 34.3

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

   2. Initial simplification34.3

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

   4. Applied add-sqr-sqrt34.3

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

   5. Applied *-un-lft-identity34.3

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

   6. Applied times-frac34.3

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

   7. Simplified34.3

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

   8. Simplified29.0

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

   10. Applied associate-*l/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. Using strategy rm

   13. Applied div-sub28.9

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

   14. Applied div-sub28.9

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

   16. Applied add-sqr-sqrt29.1

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

   17. Applied add-sqr-sqrt29.2

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

   18. Applied times-frac8.9

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

   19. Applied times-frac10.1

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

   20. Simplified9.8

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

   21. Simplified9.5

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

   if -2.146582560147369e+98 < b < 5.022503988430891e+208

   1. Initial program 22.7

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

   2. Initial simplification22.7

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

   4. Applied add-sqr-sqrt22.7

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

   5. Applied *-un-lft-identity22.7

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

   6. Applied times-frac22.7

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

   7. Simplified22.7

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

   8. Simplified12.5

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

   10. Applied associate-*l/12.4

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

   11. Simplified12.4

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

   13. Applied div-sub12.4

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

   14. Applied div-sub12.4

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

   16. Applied associate-/l*4.1

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

   if 5.022503988430891e+208 < b < 2.6543417870726475e+277

   1. Initial program 38.1

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

   2. Initial simplification38.1

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

   4. Applied add-sqr-sqrt38.1

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

   5. Applied *-un-lft-identity38.1

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

   6. Applied times-frac38.1

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

   7. Simplified38.1

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

   8. Simplified33.4

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

   10. Applied associate-*l/33.3

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

   11. Simplified33.3

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

   13. Applied div-sub33.3

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

   14. Applied div-sub33.3

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

   16. Applied *-un-lft-identity33.3

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

   17. Applied times-frac12.1

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

   18. Simplified12.1

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

   if 2.6543417870726475e+277 < b

   1. Initial program 46.9

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

   2. Initial simplification46.9

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

   4. Applied add-sqr-sqrt46.9

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

   5. Applied *-un-lft-identity46.9

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

   6. Applied times-frac46.9

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

   7. Simplified46.9

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

   8. Simplified43.5

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

   10. Applied associate-*l/43.5

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

   11. Simplified43.5

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

   13. Applied div-sub43.5

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

   14. Applied div-sub43.5

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

   16. Applied *-un-lft-identity43.5

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

   17. Applied times-frac40.5

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

   18. Applied associate-/l*40.5

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

   19. Simplified40.5

       \[\leadsto \frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\color{blue}{a}}{\frac{\sqrt{d^2 + c^2}^*}{\frac{d}{\sqrt{d^2 + c^2}^*}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.146582560147369 \cdot 10^{+98}:\\ \;\;\;\;\frac{b}{\sqrt{d^2 + c^2}^*} \cdot \frac{c}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;b \le 5.022503988430891 \cdot 10^{+208}:\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{a}{\frac{\sqrt{d^2 + c^2}^*}{d}}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{elif}\;b \le 2.6543417870726475 \cdot 10^{+277}:\\ \;\;\;\;\frac{c \cdot \frac{b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{\frac{d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*} - \frac{a}{\frac{\sqrt{d^2 + c^2}^*}{\frac{d}{\sqrt{d^2 + c^2}^*}}}\\ \end{array}\]

Runtime

Time bar (total: 37.1s)Debug logProfile

herbie shell --seed 2018251 +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))))