Average Error: 26.2 → 0.9
Time: 12.9s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot b - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot b - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}
double f(double a, double b, double c, double d) {
        double r120631 = b;
        double r120632 = c;
        double r120633 = r120631 * r120632;
        double r120634 = a;
        double r120635 = d;
        double r120636 = r120634 * r120635;
        double r120637 = r120633 - r120636;
        double r120638 = r120632 * r120632;
        double r120639 = r120635 * r120635;
        double r120640 = r120638 + r120639;
        double r120641 = r120637 / r120640;
        return r120641;
}
↓
double f(double a, double b, double c, double d) {
        double r120642 = c;
        double r120643 = d;
        double r120644 = hypot(r120642, r120643);
        double r120645 = r120642 / r120644;
        double r120646 = b;
        double r120647 = r120645 * r120646;
        double r120648 = a;
        double r120649 = r120648 / r120644;
        double r120650 = r120643 * r120649;
        double r120651 = r120647 - r120650;
        double r120652 = r120651 / r120644;
        return r120652;
}

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

Original26.2
Target0.5
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 26.2

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

  2. Simplified26.2

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

  4. Applied add-sqr-sqrt26.2

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

  5. Applied *-un-lft-identity26.2

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

  6. Applied times-frac26.2

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

  7. Simplified26.2

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

  8. Simplified17.0

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

  10. Applied *-un-lft-identity17.0

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

  11. Applied associate-*l*17.0

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

  12. Simplified16.9

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

  14. Applied div-sub16.9

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

  15. Simplified9.9

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

  16. Simplified1.3

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

  18. Applied associate-/r/0.9

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

  19. Final simplification0.9

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

Reproduce

herbie shell --seed 2019325 +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))))