Average Error: 26.0 → 22.8
Time: 3.7s
Precision: binary64
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
↓
\[\frac{c \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\]
\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}
↓
\frac{c \cdot \frac{b}{\sqrt{c \cdot c + d \cdot d}} - d \cdot \frac{a}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}
(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
↓
(FPCore (a b c d)
 :precision binary64
 (/
  (-
   (* c (/ b (sqrt (+ (* c c) (* d d)))))
   (* d (/ a (sqrt (+ (* c c) (* d d))))))
  (sqrt (+ (* c c) (* d d)))))
double code(double a, double b, double c, double d) {
	return (((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) (c * (b / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * d))))))))) - ((double) (d * (a / ((double) sqrt(((double) (((double) (c * c)) + ((double) (d * 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

Original26.0
Target0.4
Herbie22.8
\[\begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array}\]

Derivation

  1. Initial program Error: 26.0 bits

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

  3. Applied add-sqr-sqrtError: 26.0 bits

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

  4. Applied associate-/r*Error: 25.9 bits

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

  6. Applied div-subError: 25.9 bits

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

  7. SimplifiedError: 24.2 bits

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

  8. SimplifiedError: 22.8 bits

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

  9. Final simplificationError: 22.8 bits

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

Reproduce

herbie shell --seed 2020203 
(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))))