Average Error: 26.4 → 26.6
Time: 2.8s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}
double code(double a, double b, double c, double d) {
	return (((b * c) - (a * d)) / ((c * c) + (d * d)));
}
↓
double code(double a, double b, double c, double d) {
	return (((b * c) - (a * d)) * (1.0 / ((c * c) + (d * d))));
}

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

Original26.4
Target0.5
Herbie26.6
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.4

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

  3. Applied div-inv26.6

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

  4. Final simplification26.6

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

Reproduce

herbie shell --seed 2020078 
(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))))