Complex division, imag part

Average Error: 27.0 → 8.0

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -6.5063855279161376 \cdot 10^{152} \lor \neg \left(d \le 2.9116788974845384 \cdot 10^{85}\right):\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \end{array}\]

C

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) {
	double VAR;
	if (((d <= -6.506385527916138e+152) || !(d <= 2.9116788974845384e+85))) {
		VAR = ((double) (((double) (b / ((double) (((double) fma(c, c, ((double) (d * d)))) / c)))) - ((double) (a / ((double) (((double) hypot(c, d)) * ((double) (((double) hypot(c, d)) / d))))))));
	} else {
		VAR = ((double) (((double) (b / ((double) (((double) hypot(c, d)) * ((double) (((double) hypot(c, d)) / c)))))) - ((double) (a / ((double) (((double) fma(c, c, ((double) (d * d)))) / d))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	27.0
Target	0.5
Herbie	8.0

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

Derivation

1. Split input into 2 regimes

2. if d < -6.506385527916138e+152 or 2.9116788974845384e+85 < d

   1. Initial program 42.3

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

   3. Applied div-sub 42.3

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

   4. Simplified 42.2

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

   5. Simplified 39.5

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

   7. Applied *-un-lft-identity 39.5

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

   8. Applied add-sqr-sqrt 39.5

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

   9. Applied times-frac 39.5

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

   10. Simplified 39.5

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

   11. Simplified 11.9

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

   if -6.506385527916138e+152 < d < 2.9116788974845384e+85

   1. Initial program 19.7

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

   3. Applied div-sub 19.7

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

   4. Simplified 17.5

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

   5. Simplified 16.4

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

   7. Applied *-un-lft-identity 16.4

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

   8. Applied add-sqr-sqrt 16.4

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

   9. Applied times-frac 16.4

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

   10. Simplified 16.4

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

   11. Simplified 6.0

       \[\leadsto \frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \color{blue}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -6.5063855279161376 \cdot 10^{152} \lor \neg \left(d \le 2.9116788974845384 \cdot 10^{85}\right):\\ \;\;\;\;\frac{b}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{c}} - \frac{a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}} - \frac{a}{\frac{\mathsf{fma}\left(c, c, d \cdot d\right)}{d}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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))))