Complex division, real part

Average Error: 26.3 → 15.3

Time: 9.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \leq -2.5033965365909377 \cdot 10^{+145}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -1.1669216222378627 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{d \cdot b + a \cdot c}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{elif}\;d \leq 1.0481161816497777 \cdot 10^{-159}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{c \cdot c}\\ \mathbf{elif}\;d \leq 5.454863883134953 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{d \cdot b + a \cdot c}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{elif}\;d \leq 4.762697302129139 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c \cdot c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \end{array}\]

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 -2.5033965365909377e+145)
   (+ (/ b d) (/ a (/ (* d d) c)))
   (if (<= d -1.1669216222378627e-135)
     (/
      (/ (+ (* d b) (* a c)) (sqrt (+ (* d d) (* c c))))
      (sqrt (+ (* d d) (* c c))))
     (if (<= d 1.0481161816497777e-159)
       (+ (/ a c) (/ (* d b) (* c c)))
       (if (<= d 5.454863883134953e+45)
         (/
          (/ (+ (* d b) (* a c)) (sqrt (+ (* d d) (* c c))))
          (sqrt (+ (* d d) (* c c))))
         (if (<= d 4.762697302129139e+91)
           (+ (/ a c) (/ b (/ (* c c) 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 <= -2.5033965365909377e+145) {
		tmp = (b / d) + (a / ((d * d) / c));
	} else if (d <= -1.1669216222378627e-135) {
		tmp = (((d * b) + (a * c)) / sqrt((d * d) + (c * c))) / sqrt((d * d) + (c * c));
	} else if (d <= 1.0481161816497777e-159) {
		tmp = (a / c) + ((d * b) / (c * c));
	} else if (d <= 5.454863883134953e+45) {
		tmp = (((d * b) + (a * c)) / sqrt((d * d) + (c * c))) / sqrt((d * d) + (c * c));
	} else if (d <= 4.762697302129139e+91) {
		tmp = (a / c) + (b / ((c * c) / 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	26.3
Target	0.5
Herbie	15.3

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

Derivation

1. Split input into 4 regimes

2. if d < -2.5033965365909377e145 or 4.76269730212913932e91 < d

   1. Initial program 40.9

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

   3. Applied div-inv_binary64_2121 40.9

      \[\leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt_binary64_2146 40.9

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

   6. Applied add-cube-cbrt_binary64_2159 40.9

      \[\leadsto \left(a \cdot c + b \cdot d\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   7. Applied times-frac_binary64_2130 41.0

      \[\leadsto \left(a \cdot c + b \cdot d\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right)}\]

   8. Applied associate-*r*_binary64_2064 40.9

      \[\leadsto \color{blue}{\left(\left(a \cdot c + b \cdot d\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}}\]

   9. Simplified 40.9

      \[\leadsto \color{blue}{\left(\left(d \cdot b + c \cdot a\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right)} \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\]

   10. Taylor expanded around inf 16.4

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

   11. Simplified 15.2

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

   if -2.5033965365909377e145 < d < -1.16692162223786273e-135 or 1.0481161816497777e-159 < d < 5.4548638831349529e45

   1. Initial program 16.7

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

   3. Applied add-sqr-sqrt_binary64_2146 16.7

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

   4. Applied associate-/r*_binary64_2068 16.6

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

   5. Simplified 16.6

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

   if -1.16692162223786273e-135 < d < 1.0481161816497777e-159

   1. Initial program 23.2

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

   2. Taylor expanded around inf 9.4

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

   3. Simplified 9.4

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

   if 5.4548638831349529e45 < d < 4.76269730212913932e91

   1. Initial program 19.7

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

   3. Applied div-inv_binary64_2121 19.8

      \[\leadsto \color{blue}{\left(a \cdot c + b \cdot d\right) \cdot \frac{1}{c \cdot c + d \cdot d}}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt_binary64_2146 19.8

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

   6. Applied add-cube-cbrt_binary64_2159 19.8

      \[\leadsto \left(a \cdot c + b \cdot d\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]

   7. Applied times-frac_binary64_2130 19.9

      \[\leadsto \left(a \cdot c + b \cdot d\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right)}\]

   8. Applied associate-*r*_binary64_2064 19.8

      \[\leadsto \color{blue}{\left(\left(a \cdot c + b \cdot d\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}}\]

   9. Simplified 19.8

      \[\leadsto \color{blue}{\left(\left(d \cdot b + c \cdot a\right) \cdot \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\right)} \cdot \frac{\sqrt[3]{1}}{\sqrt{c \cdot c + d \cdot d}}\]

   10. Taylor expanded around 0 38.8

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

   11. Simplified 37.2

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

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5033965365909377 \cdot 10^{+145}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;d \leq -1.1669216222378627 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{d \cdot b + a \cdot c}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{elif}\;d \leq 1.0481161816497777 \cdot 10^{-159}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot b}{c \cdot c}\\ \mathbf{elif}\;d \leq 5.454863883134953 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{d \cdot b + a \cdot c}{\sqrt{d \cdot d + c \cdot c}}}{\sqrt{d \cdot d + c \cdot c}}\\ \mathbf{elif}\;d \leq 4.762697302129139 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{c \cdot c}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{d \cdot d}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021025 
(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))))