Kahan p9 Example

Average Error: 20.5 → 5.4

Time: 1.9s

Precision: binary64

\[0 < x \land x < 1 \land y < 1\]

Math

\[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3482160583740896 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.601462366399571 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \leq 3.499968286662803 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y + x}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.3482160583740896e+154)
   -1.0
   (if (<= y -1.601462366399571e-162)
     (cbrt (pow (/ (- (* x x) (* y y)) (+ (* x x) (* y y))) 3.0))
     (if (<= y 3.499968286662803e-175)
       1.0
       (*
        (/ (- x y) (sqrt (+ (* x x) (* y y))))
        (/ (+ y x) (sqrt (+ (* x x) (* y y)))))))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

↓

double code(double x, double y) {
	double tmp;
	if (y <= -1.3482160583740896e+154) {
		tmp = -1.0;
	} else if (y <= -1.601462366399571e-162) {
		tmp = cbrt(pow((((x * x) - (y * y)) / ((x * x) + (y * y))), 3.0));
	} else if (y <= 3.499968286662803e-175) {
		tmp = 1.0;
	} else {
		tmp = ((x - y) / sqrt((x * x) + (y * y))) * ((y + x) / sqrt((x * x) + (y * y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	20.5
Target	0.1
Herbie	5.4

\[\begin{array}{l} \mathbf{if}\;0.5 < \left|\frac{x}{y}\right| \land \left|\frac{x}{y}\right| < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if y < -1.3482160583740896e154

   1. Initial program 64.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

   2. Taylor expanded around 0 0

      \[\leadsto \color{blue}{-1}\]

   if -1.3482160583740896e154 < y < -1.6014623663995711e-162

   1. Initial program 0.0

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-cbrt-cube_binary64 39.2

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   4. Applied add-cbrt-cube_binary64 39.6

      \[\leadsto \frac{\left(x - y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   5. Applied add-cbrt-cube_binary64 39.7

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)}} \cdot \sqrt[3]{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   6. Applied cbrt-unprod_binary64 39.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}}}{\sqrt[3]{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}\]

   7. Applied cbrt-undiv_binary64 39.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\left(x - y\right) \cdot \left(x - y\right)\right) \cdot \left(x - y\right)\right) \cdot \left(\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + y\right)\right)}{\left(\left(x \cdot x + y \cdot y\right) \cdot \left(x \cdot x + y \cdot y\right)\right) \cdot \left(x \cdot x + y \cdot y\right)}}}\]

   8. Simplified 0.1

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}}\]

   if -1.6014623663995711e-162 < y < 3.49996828666280287e-175

   1. Initial program 31.2

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]

   2. Taylor expanded around inf 15.7

      \[\leadsto \color{blue}{1}\]

   if 3.49996828666280287e-175 < y

   1. Initial program 2.6

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt_binary64 2.6

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}}}\]

   4. Applied times-frac_binary64 3.1

      \[\leadsto \color{blue}{\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{x + y}{\sqrt{x \cdot x + y \cdot y}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 5.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3482160583740896 \cdot 10^{+154}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq -1.601462366399571 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{x \cdot x - y \cdot y}{x \cdot x + y \cdot y}\right)}^{3}}\\ \mathbf{elif}\;y \leq 3.499968286662803 \cdot 10^{-175}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\sqrt{x \cdot x + y \cdot y}} \cdot \frac{y + x}{\sqrt{x \cdot x + y \cdot y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (< 0.0 x 1.0) (< y 1.0))

  :herbie-target
  (if (< 0.5 (fabs (/ x y)) 2.0) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))