Complex division, real part

Average Error: 25.7 → 10.6

Time: 8.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \leq -1.9291899314496838 \cdot 10^{+93}:\\ \;\;\;\;\frac{c \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{if}\;c \leq -7.366462231695976 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.578520467817402 \cdot 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 204889703146200.6:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \end{array}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.9291899314496838e+93)
   (/ (* c (/ a (hypot d c))) (hypot d c))
   (let* ((t_0 (/ (/ (fma d b (* c a)) (hypot d c)) (hypot d c))))
     (if (<= c -7.366462231695976e-163)
       t_0
       (if (<= c 4.578520467817402e-188)
         (fma (/ c d) (/ a d) (/ b d))
         (if (<= c 204889703146200.6)
           t_0
           (/ (fma (/ d c) b a) (hypot d c))))))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

↓

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.9291899314496838e+93) {
		tmp = (c * (a / hypot(d, c))) / hypot(d, c);
	} else {
		double t_0 = (fma(d, b, (c * a)) / hypot(d, c)) / hypot(d, c);
		double tmp_1;
		if (c <= -7.366462231695976e-163) {
			tmp_1 = t_0;
		} else if (c <= 4.578520467817402e-188) {
			tmp_1 = fma((c / d), (a / d), (b / d));
		} else if (c <= 204889703146200.6) {
			tmp_1 = t_0;
		} else {
			tmp_1 = fma((d / c), b, 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	25.7
Target	0.5
Herbie	10.6

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

Derivation

1. Split input into 4 regimes

2. if c < -1.92918993144968375e93

   1. Initial program 38.5

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

   3. Applied add-sqr-sqrt_binary64 38.5

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

   4. Applied *-un-lft-identity_binary64 38.5

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

   5. Applied times-frac_binary64 38.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

   6. Simplified 38.5

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

   7. Simplified 25.0

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

   8. Taylor expanded in d around 0 32.5

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

   9. Simplified 13.1

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

   if -1.92918993144968375e93 < c < -7.3664622316959762e-163 or 4.578520467817402e-188 < c < 204889703146200.594

   1. Initial program 15.4

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

   3. Applied add-sqr-sqrt_binary64 15.4

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

   4. Applied *-un-lft-identity_binary64 15.4

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

   5. Applied times-frac_binary64 15.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

   6. Simplified 15.4

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

   7. Simplified 10.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \color{blue}{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}} \]
   8. Using strategy rm

   9. Applied *-un-lft-identity_binary64 10.2

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

   10. Applied associate-*l*_binary64 10.2

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

   11. Simplified 10.1

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

   if -7.3664622316959762e-163 < c < 4.578520467817402e-188

   1. Initial program 24.3

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

   2. Taylor expanded in c around 0 8.2

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

   3. Simplified 5.6

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

   if 204889703146200.594 < c

   1. Initial program 32.5

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

   3. Applied add-sqr-sqrt_binary64 32.5

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

   4. Applied *-un-lft-identity_binary64 32.5

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

   5. Applied times-frac_binary64 32.5

      \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

   6. Simplified 32.5

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

   7. Simplified 22.6

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

   8. Taylor expanded in d around 0 16.4

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

   9. Simplified 13.5

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9291899314496838 \cdot 10^{+93}:\\ \;\;\;\;\frac{c \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq -7.366462231695976 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 4.578520467817402 \cdot 10^{-188}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;c \leq 204889703146200.6:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))