Complex division, real part

Average Error: 25.8 → 12.2

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -7.553879560626287 \cdot 10^{+93}:\\ \;\;\;\;\frac{b}{d} + \frac{1}{d} \cdot \frac{a}{\frac{d}{c}}\\ \mathbf{elif}\;d \leq -7.875830746956647 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 2.5365321914959593 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 7.162264487896466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ \end{array}\]

FPCore

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

↓

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7.553879560626287e+93)
   (+ (/ b d) (* (/ 1.0 d) (/ a (/ d c))))
   (if (<= d -7.875830746956647e-107)
     (/
      (/ (+ (* a c) (* d b)) (sqrt (+ (* c c) (* d d))))
      (sqrt (+ (* c c) (* d d))))
     (if (<= d 2.5365321914959593e-134)
       (+ (/ a c) (/ (* d b) (pow c 2.0)))
       (if (<= d 7.162264487896466e+81)
         (/
          (/ (+ (* a c) (* d b)) (sqrt (+ (* c c) (* d d))))
          (sqrt (+ (* c c) (* d d))))
         (+ (/ b d) (/ (/ a d) (/ 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 (d <= -7.553879560626287e+93) {
		tmp = (b / d) + ((1.0 / d) * (a / (d / c)));
	} else if (d <= -7.875830746956647e-107) {
		tmp = (((a * c) + (d * b)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else if (d <= 2.5365321914959593e-134) {
		tmp = (a / c) + ((d * b) / pow(c, 2.0));
	} else if (d <= 7.162264487896466e+81) {
		tmp = (((a * c) + (d * b)) / sqrt((c * c) + (d * d))) / sqrt((c * c) + (d * d));
	} else {
		tmp = (b / d) + ((a / d) / (d / c));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	25.8
Target	0.5
Herbie	12.2

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

Derivation

1. Split input into 4 regimes

2. if d < -7.5538795606262866e93

   1. Initial program 40.4

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

   3. Applied add-sqr-sqrt_binary64_1123 40.4

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

   4. Applied *-un-lft-identity_binary64_1101 40.4

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

   5. Applied times-frac_binary64_1107 40.4

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

   6. Simplified 40.4

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

   7. Taylor expanded around 0 16.4

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

   8. Simplified 15.7

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_1101 15.7

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

   11. Applied times-frac_binary64_1107 13.6

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

   12. Applied *-un-lft-identity_binary64_1101 13.6

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

   13. Applied times-frac_binary64_1107 10.3

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

   14. Simplified 10.3

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

   if -7.5538795606262866e93 < d < -7.87583074695664708e-107 or 2.5365321914959593e-134 < d < 7.16226448789646589e81

   1. Initial program 15.1

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

   3. Applied add-sqr-sqrt_binary64_1123 15.1

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

   4. Applied associate-/r*_binary64_1045 15.0

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

   if -7.87583074695664708e-107 < d < 2.5365321914959593e-134

   1. Initial program 21.9

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

   2. Taylor expanded around inf 10.6

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

   if 7.16226448789646589e81 < d

   1. Initial program 37.4

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

   3. Applied add-sqr-sqrt_binary64_1123 37.4

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

   4. Applied *-un-lft-identity_binary64_1101 37.4

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

   5. Applied times-frac_binary64_1107 37.4

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

   6. Simplified 37.4

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

   7. Taylor expanded around 0 16.7

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

   8. Simplified 16.1

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_1101 16.1

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

   11. Applied times-frac_binary64_1107 14.5

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

   12. Applied associate-/r*_binary64_1045 11.5

       \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a}{\frac{d}{1}}}{\frac{d}{c}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.553879560626287 \cdot 10^{+93}:\\ \;\;\;\;\frac{b}{d} + \frac{1}{d} \cdot \frac{a}{\frac{d}{c}}\\ \mathbf{elif}\;d \leq -7.875830746956647 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{elif}\;d \leq 2.5365321914959593 \cdot 10^{-134}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{{c}^{2}}\\ \mathbf{elif}\;d \leq 7.162264487896466 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{a \cdot c + d \cdot b}{\sqrt{c \cdot c + d \cdot d}}}{\sqrt{c \cdot c + d \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a}{d}}{\frac{d}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))