Average Error: 25.3 → 25.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Results

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Target

Original25.3
Target0.4
Herbie25.3
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.3

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

  3. Applied add-sqr-sqrt25.3

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

  4. Applied associate-/r*25.2

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

  5. Taylor expanded around -inf 25.2

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

  7. Applied *-un-lft-identity25.2

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

  8. Applied *-un-lft-identity25.2

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

  9. Applied distribute-lft-out25.2

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

  10. Applied sqrt-prod25.2

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

  11. Applied *-un-lft-identity25.2

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

  12. Applied *-un-lft-identity25.2

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

  13. Applied times-frac25.2

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

  14. Applied times-frac25.2

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

  15. Simplified25.2

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

  16. Simplified25.3

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

  17. Final simplification25.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018252 
(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))))