Average Error: 25.7 → 0.9
Time: 4.0s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right)}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right)}
double code(double a, double b, double c, double d) {
	return (((b * c) - (a * d)) / ((c * c) + (d * d)));
}
↓
double code(double a, double b, double c, double d) {
	return ((fma((b / sqrt(hypot(c, d))), (c / sqrt(hypot(c, d))), -((d / sqrt(hypot(c, d))) * (a / sqrt(hypot(c, d))))) + ((d / sqrt(hypot(c, d))) * (-(a / sqrt(hypot(c, d))) + (a / sqrt(hypot(c, d)))))) / hypot(c, d));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original25.7
Target0.4
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.7

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

  3. Applied add-sqr-sqrt25.7

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

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

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

  5. Applied times-frac25.7

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

  6. Simplified25.7

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

  7. Simplified16.4

    \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}\]
  8. Using strategy rm

  9. Applied pow116.4

    \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \color{blue}{{\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

  10. Applied pow116.4

    \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]

  11. Applied pow-prod-down16.4

    \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(c, d\right) \cdot 1} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}}\]

  12. Simplified16.3

    \[\leadsto {\color{blue}{\left(\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\right)}}^{1}\]
  13. Using strategy rm

  14. Applied div-sub16.3

    \[\leadsto {\left(\frac{\color{blue}{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}}{\mathsf{hypot}\left(c, d\right)}\right)}^{1}\]
  15. Using strategy rm

  16. Applied add-sqr-sqrt16.4

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

  17. Applied times-frac9.2

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

  18. Applied add-sqr-sqrt9.3

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

  19. Applied times-frac0.9

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

  20. Applied prod-diff0.9

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

  21. Simplified0.9

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

  22. Final simplification0.9

    \[\leadsto \frac{\mathsf{fma}\left(\frac{b}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, \frac{c}{\sqrt{\mathsf{hypot}\left(c, d\right)}}, -\frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{d}{\sqrt{\mathsf{hypot}\left(c, d\right)}} \cdot \left(\left(-\frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right) + \frac{a}{\sqrt{\mathsf{hypot}\left(c, d\right)}}\right)}{\mathsf{hypot}\left(c, d\right)}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :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))))