Complex division, imag part

Average Error: 26.5 → 4.8

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;c \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(c \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}\\ \end{array}\]

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 temp;
	if (((c <= -2.2616143508082277e+79) || !(c <= 1.0382708421545201e-35))) {
		temp = (((b / (hypot(c, d) / c)) / hypot(c, d)) - (((a * d) / hypot(c, d)) / hypot(c, d)));
	} else {
		temp = ((((b * c) / hypot(c, d)) / hypot(c, d)) - ((a / 1.0) / (hypot(c, d) / (d / hypot(c, d)))));
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Target

Original	26.5
Target	0.4
Herbie	4.8

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

Derivation

1. Split input into 2 regimes

2. if c < -2.2616143508082277e+79 or 1.0382708421545201e-35 < c

   1. Initial program 33.6

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

   3. Applied add-sqr-sqrt 33.6

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

   4. Applied *-un-lft-identity 33.6

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

   5. Applied times-frac 33.6

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

   6. Simplified 33.6

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

   7. Simplified 23.3

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

   9. Applied associate-*r/ 23.2

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

   10. Simplified 23.2

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

   12. Applied div-sub 23.2

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

   13. Applied div-sub 23.2

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

   15. Applied associate-/l* 7.8

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

   if -2.2616143508082277e+79 < c < 1.0382708421545201e-35

   1. Initial program 19.6

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

   3. Applied add-sqr-sqrt 19.6

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

   4. Applied *-un-lft-identity 19.6

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

   5. Applied times-frac 19.6

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

   6. Simplified 19.6

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

   7. Simplified 11.8

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

   9. Applied associate-*r/ 11.7

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

   10. Simplified 11.7

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

   12. Applied div-sub 11.7

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

   13. Applied div-sub 11.7

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

   15. Applied *-un-lft-identity 11.7

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

   16. Applied times-frac 1.7

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

   17. Applied associate-/l* 1.9

       \[\leadsto \frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \color{blue}{\frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(c \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a \cdot d}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} - \frac{\frac{a}{1}}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{d}{\mathsf{hypot}\left(c, d\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(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))))