Average Error: 26.2 → 22.9
Time: 3.4s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{c \cdot c + d \cdot d}} - \frac{a}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{d}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{c}{\sqrt{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 (((b * c) - (a * d)) / ((c * c) + (d * d)));
}
↓
double code(double a, double b, double c, double d) {
	return (((b / sqrt(((c * c) + (d * d)))) * (c / sqrt(((c * c) + (d * d))))) - ((a / sqrt(((c * c) + (d * d)))) * (d / sqrt(((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

Original26.2
Target0.4
Herbie22.9
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.2

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

  3. Applied div-sub26.2

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

  5. Applied add-sqr-sqrt26.2

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

  6. Applied times-frac24.7

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

  8. Applied add-sqr-sqrt24.7

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

  9. Applied times-frac22.9

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

  10. Final simplification22.9

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

Reproduce

herbie shell --seed 2020049 
(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))))