Complex division, real part

Average Error: 25.9 → 10.3

Time: 7.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \leq -9.06783613733096 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq -3.44433832248687 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.7560140316565585 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}\\ \mathbf{elif}\;c \leq 3.5151647360045185 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{d \cdot b}{\mathsf{hypot}\left(d, c\right)} + \frac{c \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)\\ \end{array} \]

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 -9.06783613733096e+79)
   (/ (- (fma (/ d c) b a)) (hypot d c))
   (if (<= c -3.44433832248687e-174)
     (/ (/ (fma d b (* c a)) (hypot d c)) (hypot d c))
     (if (<= c 2.7560140316565585e-220)
       (+ (/ b d) (/ (* c a) (pow d 2.0)))
       (if (<= c 3.5151647360045185e+96)
         (/ (+ (/ (* d b) (hypot d c)) (/ (* c a) (hypot d c))) (hypot d c))
         (fma (/ b c) (/ d c) (/ a 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 <= -9.06783613733096e+79) {
		tmp = -fma((d / c), b, a) / hypot(d, c);
	} else if (c <= -3.44433832248687e-174) {
		tmp = (fma(d, b, (c * a)) / hypot(d, c)) / hypot(d, c);
	} else if (c <= 2.7560140316565585e-220) {
		tmp = (b / d) + ((c * a) / pow(d, 2.0));
	} else if (c <= 3.5151647360045185e+96) {
		tmp = (((d * b) / hypot(d, c)) + ((c * a) / hypot(d, c))) / hypot(d, c);
	} else {
		tmp = fma((b / c), (d / c), (a / c));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.9
Target	0.5
Herbie	10.3

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

Derivation

1. Split input into 5 regimes

2. if c < -9.06783613733096029e79

   1. Initial program 38.3

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

   2. Simplified 38.3

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

   3. Applied add-sqr-sqrt_binary64 38.3

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

   4. Applied *-un-lft-identity_binary64 38.3

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

   5. Applied times-frac_binary64 38.3

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

   6. Simplified 38.3

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

   7. Simplified 26.8

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

   8. Applied associate-*r/_binary64 26.8

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

   9. Taylor expanded in c around -inf 14.0

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

   10. Simplified 10.3

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

   if -9.06783613733096029e79 < c < -3.4443383224868699e-174

   1. Initial program 16.7

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

   2. Simplified 16.7

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

   3. Applied add-sqr-sqrt_binary64 16.7

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

   4. Applied *-un-lft-identity_binary64 16.7

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

   5. Applied times-frac_binary64 16.7

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

   6. Simplified 16.7

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

   7. Simplified 11.7

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

   8. Applied associate-*l/_binary64 11.6

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

   if -3.4443383224868699e-174 < c < 2.756014031656558e-220

   1. Initial program 23.0

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

   2. Simplified 23.0

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

   3. Taylor expanded in c around 0 8.1

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

   if 2.756014031656558e-220 < c < 3.5151647360045185e96

   1. Initial program 16.7

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

   2. Simplified 16.7

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

   3. Applied add-sqr-sqrt_binary64 16.7

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

   4. Applied *-un-lft-identity_binary64 16.7

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

   5. Applied times-frac_binary64 16.7

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

   6. Simplified 16.7

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

   7. Simplified 10.4

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

   8. Applied associate-*r/_binary64 10.3

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

   9. Applied fma-udef_binary64 10.3

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

   10. Applied distribute-rgt-in_binary64 10.3

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

   11. Simplified 10.3

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

   12. Simplified 10.2

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

   if 3.5151647360045185e96 < c

   1. Initial program 38.3

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

   2. Simplified 38.3

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

   3. Applied add-sqr-sqrt_binary64 38.3

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

   4. Applied *-un-lft-identity_binary64 38.3

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

   5. Applied times-frac_binary64 38.3

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

   6. Simplified 38.3

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

   7. Simplified 26.7

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

   8. Taylor expanded in d around 0 16.4

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

   9. Simplified 10.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.06783613733096 \cdot 10^{+79}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq -3.44433832248687 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.7560140316565585 \cdot 10^{-220}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot a}{{d}^{2}}\\ \mathbf{elif}\;c \leq 3.5151647360045185 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{d \cdot b}{\mathsf{hypot}\left(d, c\right)} + \frac{c \cdot a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021220 
(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))))