Error

Target

Derivation

1. Split input into 2 regimes

2. if d < 1.6480088990828942e+161

   1. Initial program 23.2

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

   2. Simplified23.2

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

   4. Applied add-sqr-sqrt23.2

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

   5. Applied associate-/r*23.1

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

   7. Applied fma-udef23.1

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

   8. Applied hypot-def23.1

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

   10. Applied fma-udef23.1

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

   11. Applied hypot-def15.1

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

   12. Taylor expanded around inf 15.1

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

   if 1.6480088990828942e+161 < d

   1. Initial program 44.0

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

   2. Simplified44.0

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

   4. Applied add-sqr-sqrt44.0

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

   5. Applied associate-/r*44.0

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

   7. Applied fma-udef44.0

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

   8. Applied hypot-def44.0

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

   10. Applied fma-udef44.0

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

   11. Applied hypot-def27.1

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

   12. Taylor expanded around 0 13.8

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

   13. Simplified13.8

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

4. Final simplification15.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 1.6480088990828942 \cdot 10^{+161}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\sqrt{d^2 + c^2}^*}}{\sqrt{d^2 + c^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\sqrt{d^2 + c^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019091 +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))))