Average Error: 26.2 → 0.9
Time: 12.8s
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 r80910 = b;
        double r80911 = c;
        double r80912 = r80910 * r80911;
        double r80913 = a;
        double r80914 = d;
        double r80915 = r80913 * r80914;
        double r80916 = r80912 - r80915;
        double r80917 = r80911 * r80911;
        double r80918 = r80914 * r80914;
        double r80919 = r80917 + r80918;
        double r80920 = r80916 / r80919;
        return r80920;
}
↓
double f(double a, double b, double c, double d) {
        double r80921 = c;
        double r80922 = d;
        double r80923 = hypot(r80921, r80922);
        double r80924 = r80921 / r80923;
        double r80925 = b;
        double r80926 = r80924 * r80925;
        double r80927 = a;
        double r80928 = r80927 / r80923;
        double r80929 = r80922 * r80928;
        double r80930 = r80926 - r80929;
        double r80931 = r80930 / r80923;
        return r80931;
}

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 pow117.0

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

  11. Applied pow117.0

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

  12. Applied pow-prod-down17.0

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

  13. Simplified16.9

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

  15. Applied div-sub16.9

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

  16. Simplified9.9

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

  17. Simplified1.3

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

  19. Applied associate-/r/0.9

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

  20. 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))))