Average Error: 26.5 → 26.5
Time: 9.3s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
double f(double a, double b, double c, double d) {
        double r62590 = b;
        double r62591 = c;
        double r62592 = r62590 * r62591;
        double r62593 = a;
        double r62594 = d;
        double r62595 = r62593 * r62594;
        double r62596 = r62592 - r62595;
        double r62597 = r62591 * r62591;
        double r62598 = r62594 * r62594;
        double r62599 = r62597 + r62598;
        double r62600 = r62596 / r62599;
        return r62600;
}
↓
double f(double a, double b, double c, double d) {
        double r62601 = b;
        double r62602 = c;
        double r62603 = r62601 * r62602;
        double r62604 = a;
        double r62605 = d;
        double r62606 = r62604 * r62605;
        double r62607 = r62603 - r62606;
        double r62608 = r62602 * r62602;
        double r62609 = r62605 * r62605;
        double r62610 = r62608 + r62609;
        double r62611 = r62607 / r62610;
        return r62611;
}

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

Original26.5
Target0.4
Herbie26.5
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.5

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

  3. Applied add-sqr-sqrt26.5

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

  4. Applied associate-/r*26.4

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

  5. Final simplification26.5

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

Reproduce

herbie shell --seed 2019297 
(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))))