Average Error: 25.4 → 25.4
Time: 12.9s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{\frac{1}{\frac{\sqrt{c \cdot c + d \cdot d}}{b \cdot c - d \cdot a}}}{\sqrt{c \cdot c + d \cdot d}}
double f(double a, double b, double c, double d) {
        double r113617 = b;
        double r113618 = c;
        double r113619 = r113617 * r113618;
        double r113620 = a;
        double r113621 = d;
        double r113622 = r113620 * r113621;
        double r113623 = r113619 - r113622;
        double r113624 = r113618 * r113618;
        double r113625 = r113621 * r113621;
        double r113626 = r113624 + r113625;
        double r113627 = r113623 / r113626;
        return r113627;
}
↓
double f(double a, double b, double c, double d) {
        double r113628 = 1.0;
        double r113629 = c;
        double r113630 = r113629 * r113629;
        double r113631 = d;
        double r113632 = r113631 * r113631;
        double r113633 = r113630 + r113632;
        double r113634 = sqrt(r113633);
        double r113635 = b;
        double r113636 = r113635 * r113629;
        double r113637 = a;
        double r113638 = r113631 * r113637;
        double r113639 = r113636 - r113638;
        double r113640 = r113634 / r113639;
        double r113641 = r113628 / r113640;
        double r113642 = r113641 / r113634;
        return r113642;
}

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

Original25.4
Target0.5
Herbie25.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 25.4

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

  3. Applied add-sqr-sqrt25.4

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

  4. Applied associate-/r*25.3

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

  6. Applied clear-num25.4

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

  7. Simplified25.4

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

  8. Final simplification25.4

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

Reproduce

herbie shell --seed 2019212 
(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))))