Average Error: 25.8 → 25.7
Time: 25.9s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{c \cdot b - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{c \cdot b - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r4753620 = b;
        double r4753621 = c;
        double r4753622 = r4753620 * r4753621;
        double r4753623 = a;
        double r4753624 = d;
        double r4753625 = r4753623 * r4753624;
        double r4753626 = r4753622 - r4753625;
        double r4753627 = r4753621 * r4753621;
        double r4753628 = r4753624 * r4753624;
        double r4753629 = r4753627 + r4753628;
        double r4753630 = r4753626 / r4753629;
        return r4753630;
}
↓
double f(double a, double b, double c, double d) {
        double r4753631 = 1.0;
        double r4753632 = c;
        double r4753633 = r4753632 * r4753632;
        double r4753634 = d;
        double r4753635 = r4753634 * r4753634;
        double r4753636 = r4753633 + r4753635;
        double r4753637 = sqrt(r4753636);
        double r4753638 = b;
        double r4753639 = r4753632 * r4753638;
        double r4753640 = a;
        double r4753641 = r4753634 * r4753640;
        double r4753642 = r4753639 - r4753641;
        double r4753643 = r4753637 / r4753642;
        double r4753644 = r4753631 / r4753643;
        double r4753645 = r4753644 / r4753637;
        return r4753645;
}

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

Original25.8
Target0.4
Herbie25.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.8

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

  3. Applied add-sqr-sqrt25.8

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

  4. Applied associate-/r*25.7

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

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

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

  7. Applied associate-/l*25.7

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

  8. Final simplification25.7

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

Reproduce

herbie shell --seed 2019151 
(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))))