Average Error: 25.9 → 13.8
Time: 15.0s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right| \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a}{\frac{c}{\frac{d}{c}} + d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right| \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a}{\frac{c}{\frac{d}{c}} + d}
double f(double a, double b, double c, double d) {
        double r189522 = b;
        double r189523 = c;
        double r189524 = r189522 * r189523;
        double r189525 = a;
        double r189526 = d;
        double r189527 = r189525 * r189526;
        double r189528 = r189524 - r189527;
        double r189529 = r189523 * r189523;
        double r189530 = r189526 * r189526;
        double r189531 = r189529 + r189530;
        double r189532 = r189528 / r189531;
        return r189532;
}
↓
double f(double a, double b, double c, double d) {
        double r189533 = b;
        double r189534 = c;
        double r189535 = r189534 * r189534;
        double r189536 = d;
        double r189537 = r189536 * r189536;
        double r189538 = r189535 + r189537;
        double r189539 = sqrt(r189538);
        double r189540 = r189539 / r189534;
        double r189541 = r189533 / r189540;
        double r189542 = cbrt(r189538);
        double r189543 = fabs(r189542);
        double r189544 = sqrt(r189542);
        double r189545 = r189543 * r189544;
        double r189546 = r189541 / r189545;
        double r189547 = a;
        double r189548 = r189536 / r189534;
        double r189549 = r189534 / r189548;
        double r189550 = r189549 + r189536;
        double r189551 = r189547 / r189550;
        double r189552 = r189546 - r189551;
        return r189552;
}

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

Original25.9
Target0.4
Herbie13.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.9

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

  3. Applied div-sub25.9

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

  4. Simplified24.3

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

  5. Taylor expanded around 0 17.5

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

  6. Simplified16.0

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

  8. Applied add-sqr-sqrt16.0

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

  9. Applied associate-/r*15.9

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

  10. Simplified13.7

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

  12. Applied add-cube-cbrt13.8

    \[\leadsto \frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\sqrt{\color{blue}{\left(\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}\right) \cdot \sqrt[3]{c \cdot c + d \cdot d}}}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\]

  13. Applied sqrt-prod13.8

    \[\leadsto \frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\color{blue}{\sqrt{\sqrt[3]{c \cdot c + d \cdot d} \cdot \sqrt[3]{c \cdot c + d \cdot d}} \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\]

  14. Simplified13.8

    \[\leadsto \frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\color{blue}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right|} \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a}{d + \frac{c}{\frac{d}{c}}}\]

  15. Final simplification13.8

    \[\leadsto \frac{\frac{b}{\frac{\sqrt{c \cdot c + d \cdot d}}{c}}}{\left|\sqrt[3]{c \cdot c + d \cdot d}\right| \cdot \sqrt{\sqrt[3]{c \cdot c + d \cdot d}}} - \frac{a}{\frac{c}{\frac{d}{c}} + d}\]

Reproduce

herbie shell --seed 2019195 
(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))))