Average Error: 25.7 → 22.8
Time: 47.0s
Precision: 64
Internal Precision: 576
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.7
Target0.5
Herbie22.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

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

  2. Initial simplification25.7

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

  4. Applied add-sqr-sqrt25.7

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

  5. Applied associate-/r*25.6

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

  7. Applied div-sub25.6

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

  8. Applied div-sub25.6

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

  9. Simplified25.7

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

  11. Applied *-un-lft-identity25.7

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

  12. Applied times-frac24.0

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

  13. Simplified24.0

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

  15. Applied *-un-lft-identity24.0

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

  16. Applied times-frac22.8

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

  17. Simplified22.8

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

  18. Final simplification22.8

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

Runtime

Time bar (total: 47.0s)Debug logProfile

herbie shell --seed 2018250 
(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))))