Average Error: 26.0 → 3.4
Time: 2.2s
Precision: 64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[b \cdot \frac{1}{\mathsf{fma}\left(\frac{d}{c}, d, c\right)} - \frac{a}{\mathsf{fma}\left(\frac{c}{d}, c, d\right)}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
b \cdot \frac{1}{\mathsf{fma}\left(\frac{d}{c}, d, c\right)} - \frac{a}{\mathsf{fma}\left(\frac{c}{d}, c, d\right)}
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 * (1.0 / fma((d / c), d, c))) - (a / fma((c / d), c, 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

Original26.0
Target0.5
Herbie3.4
\[\begin{array}{l} \mathbf{if}\;\left|d\right| \lt \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program 26.0

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

  3. Applied div-sub26.0

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

  4. Simplified24.8

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

  5. Simplified23.2

    \[\leadsto \frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \color{blue}{\frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}}\]

  6. Taylor expanded around 0 15.7

    \[\leadsto \frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\color{blue}{d + \frac{{c}^{2}}{d}}}\]

  7. Simplified14.0

    \[\leadsto \frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, c, d\right)}}\]

  8. Taylor expanded around 0 5.7

    \[\leadsto \frac{b}{\color{blue}{\frac{{d}^{2}}{c} + c}} - \frac{a}{\mathsf{fma}\left(\frac{c}{d}, c, d\right)}\]

  9. Simplified3.3

    \[\leadsto \frac{b}{\color{blue}{\mathsf{fma}\left(\frac{d}{c}, d, c\right)}} - \frac{a}{\mathsf{fma}\left(\frac{c}{d}, c, d\right)}\]
  10. Using strategy rm

  11. Applied div-inv3.4

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

  12. Final simplification3.4

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

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))))