Complex division, imag part

Average Error: 25.6 → 10.8

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq 1.8860901164613604 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(d, c\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= t_0 1.8860901164613604e+292)
     (/ (- (/ (* b c) (hypot d c)) (/ (* a d) (hypot d c))) (hypot d c))
     (if (<= t_0 INFINITY) (/ b c) (/ (- a) d)))))

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 t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (t_0 <= 1.8860901164613604e+292) {
		tmp = (((b * c) / hypot(d, c)) - ((a * d) / hypot(d, c))) / hypot(d, c);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.6
Target	0.5
Herbie	10.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.88609011646136039e292

   1. Initial program 13.9

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

   2. Simplified 13.9

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

   3. Applied add-sqr-sqrt_binary64 13.9

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

   4. Applied *-un-lft-identity_binary64 13.9

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

   5. Applied times-frac_binary64 13.9

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

   6. Simplified 13.9

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

   7. Simplified 2.8

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

   8. Applied associate-*r/_binary64 2.8

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

   9. Applied sub-neg_binary64 2.8

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

   10. Applied distribute-rgt-in_binary64 2.8

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

   11. Simplified 2.7

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

   12. Simplified 2.7

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

   if 1.88609011646136039e292 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 58.5

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

   2. Simplified 58.5

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

   3. Taylor expanded in c around inf 37.8

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if +inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded in c around 0 36.3

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 1.8860901164613604 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(d, c\right)} - \frac{a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Reproduce

herbie shell --seed 2021313 
(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))))