Average Error: 26.2 → 14.2
Time: 5.1s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{b}{c + d \cdot \frac{d}{c}} - a \cdot \frac{d}{c \cdot c + d \cdot d}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{b}{c + d \cdot \frac{d}{c}} - a \cdot \frac{d}{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) (d * ((double) (d / c)))))))) - ((double) (a * ((double) (d / ((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

Original26.2
Target0.5
Herbie14.2
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

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. Simplified24.9

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

  5. Simplified23.3

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

  7. Applied clear-num23.3

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

  8. Taylor expanded around 0 15.9

    \[\leadsto b \cdot \frac{1}{\color{blue}{\frac{{d}^{2}}{c} + c}} - a \cdot \frac{d}{c \cdot c + d \cdot d}\]

  9. Simplified14.3

    \[\leadsto b \cdot \frac{1}{\color{blue}{c + d \cdot \frac{d}{c}}} - a \cdot \frac{d}{c \cdot c + d \cdot d}\]
  10. Using strategy rm

  11. Applied un-div-inv14.2

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

  12. Final simplification14.2

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

Reproduce

herbie shell --seed 2020184 
(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))))