Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, D

Average Error: 22.4 → 0.3

Time: 5.9s

Precision: binary64

Math

\[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.999425018399054:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000201864133105:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \end{array}\]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 0.999425018399054)
   (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
   (if (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 1.0000201864133105)
     (-
      (+ (/ x (* y y)) (+ x (+ (/ 1.0 y) (/ 1.0 (pow y 3.0)))))
      (+ (/ x (pow y 3.0)) (+ (/ x y) (/ 1.0 (* y y)))))
     (* x (/ y (+ 1.0 y))))))

C

double code(double x, double y) {
	return 1.0 - (((1.0 - x) * y) / (y + 1.0));
}

↓

double code(double x, double y) {
	double tmp;
	if ((((1.0 - x) * y) / (1.0 + y)) <= 0.999425018399054) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else if ((((1.0 - x) * y) / (1.0 + y)) <= 1.0000201864133105) {
		tmp = ((x / (y * y)) + (x + ((1.0 / y) + (1.0 / pow(y, 3.0))))) - ((x / pow(y, 3.0)) + ((x / y) + (1.0 / (y * y))));
	} else {
		tmp = x * (y / (1.0 + y));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.4
Target	0.2
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.999425018399053955

   1. Initial program 7.6

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around 0 7.6

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{y + 1} - \frac{x \cdot y}{y + 1}\right)}\]

   3. Simplified 0.0

      \[\leadsto 1 - \color{blue}{\frac{y}{y + 1} \cdot \left(1 - x\right)}\]

   if 0.999425018399053955 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.00002018641331047

   1. Initial program 58.6

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\left(\frac{x}{{y}^{2}} + \left(\frac{1}{{y}^{3}} + \left(\frac{1}{y} + x\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{{y}^{2}} + \frac{x}{y}\right)\right)}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)}\]

   if 1.00002018641331047 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 21.5

      \[1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\]

   2. Taylor expanded around 0 21.5

      \[\leadsto 1 - \color{blue}{\left(\frac{y}{y + 1} - \frac{x \cdot y}{y + 1}\right)}\]

   3. Simplified 0.2

      \[\leadsto 1 - \color{blue}{\frac{y}{y + 1} \cdot \left(1 - x\right)}\]

   4. Taylor expanded around inf 22.8

      \[\leadsto \color{blue}{\frac{x \cdot y}{y + 1}}\]

   5. Simplified 1.4

      \[\leadsto \color{blue}{x \cdot \frac{y}{1 + y}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.999425018399054:\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000201864133105:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{1 + y}\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1.0 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))) (- (/ 1.0 y) (- (/ x y) x))))

  (- 1.0 (/ (* (- 1.0 x) y) (+ y 1.0))))