Average Error: 27.0 → 23.7
Time: 3.6s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
b \cdot \frac{c}{c \cdot c + d \cdot d} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}
double code(double a, double b, double c, double d) {
	return ((double) (((double) (((double) (b * c)) - ((double) (a * d)))) / ((double) (((double) (c * c)) + ((double) (d * d))))));
}
↓
double code(double a, double b, double c, double d) {
	return ((double) (((double) (b * ((double) (c / ((double) (((double) (c * c)) + ((double) (d * d)))))))) - ((double) (((double) (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d)))))))) * ((double) (d / ((double) sqrt(((double) (((double) (c * c)) + ((double) (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

Original27.0
Target0.5
Herbie23.7
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 27.0

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

  3. Applied div-sub27.0

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

  4. Simplified25.5

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

  5. Simplified23.6

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

  7. Applied add-sqr-sqrt23.6

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

  8. Applied *-un-lft-identity23.6

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

  9. Applied times-frac23.6

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

  10. Applied associate-*r*23.8

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

  11. Simplified23.7

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

  12. Final simplification23.7

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

Reproduce

herbie shell --seed 2020181 
(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)))) (/ (+ (neg a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))