Complex division, imag part

Average Error: 26.3 → 12.5

Time: 9.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -7.275365746467164 \cdot 10^{+183}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(c, b, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{if}\;d \leq -5.175827438536771 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.532329616296603 \cdot 10^{-115}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 3.4182045595690496 \cdot 10^{+70}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))

↓

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7.275365746467164e+183)
   (- (/ a d))
   (let* ((t_0 (/ (/ (fma c b (- (* d a))) (hypot d c)) (hypot d c))))
     (if (<= d -5.175827438536771e-73)
       t_0
       (if (<= d 2.532329616296603e-115)
         (- (/ b c) (/ (* d a) (* c c)))
         (if (<= d 3.4182045595690496e+70)
           t_0
           (/ (- (/ (* c b) d) a) (hypot d c))))))))

C

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) {
	double tmp;
	if (d <= -7.275365746467164e+183) {
		tmp = -(a / d);
	} else {
		double t_0 = (fma(c, b, -(d * a)) / hypot(d, c)) / hypot(d, c);
		double tmp_1;
		if (d <= -5.175827438536771e-73) {
			tmp_1 = t_0;
		} else if (d <= 2.532329616296603e-115) {
			tmp_1 = (b / c) - ((d * a) / (c * c));
		} else if (d <= 3.4182045595690496e+70) {
			tmp_1 = t_0;
		} else {
			tmp_1 = (((c * b) / d) - a) / hypot(d, c);
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.3
Target	0.4
Herbie	12.5

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

Derivation

1. Split input into 4 regimes

2. if d < -7.27536574646716366e183

   1. Initial program 43.8

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

   2. Simplified 43.8

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

   3. Taylor expanded in c around 0 11.2

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

   if -7.27536574646716366e183 < d < -5.1758274385367712e-73 or 2.5323296162966029e-115 < d < 3.4182045595690496e70

   1. Initial program 18.9

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

   2. Simplified 18.9

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

   3. Applied add-sqr-sqrt_binary64 18.9

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

   4. Applied *-un-lft-identity_binary64 18.9

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

   5. Applied times-frac_binary64 18.9

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

   6. Simplified 18.9

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

   7. Simplified 12.9

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

   8. Applied associate-*l/_binary64 12.8

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

   9. Simplified 12.8

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

   10. Applied fma-neg_binary64 12.8

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

   if -5.1758274385367712e-73 < d < 2.5323296162966029e-115

   1. Initial program 21.2

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

   2. Simplified 21.2

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

   3. Taylor expanded in c around inf 11.7

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

   4. Simplified 11.7

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

   if 3.4182045595690496e70 < d

   1. Initial program 37.7

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

   2. Simplified 37.7

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

   3. Applied add-sqr-sqrt_binary64 37.7

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

   4. Applied *-un-lft-identity_binary64 37.7

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

   5. Applied times-frac_binary64 37.7

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

   6. Simplified 37.7

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

   7. Simplified 25.4

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

   8. Applied associate-*l/_binary64 25.3

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

   9. Simplified 25.3

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

   10. Taylor expanded in c around 0 13.7

       \[\leadsto \frac{\color{blue}{\frac{c \cdot b}{d} - a}}{\mathsf{hypot}\left(d, c\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.275365746467164 \cdot 10^{+183}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq -5.175827438536771 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, b, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq 2.532329616296603 \cdot 10^{-115}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 3.4182045595690496 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, b, -d \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2022024 
(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))))