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

Average Error: 21.9 → 0.1

Time: 4.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.9195687097649881 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.006962404202967\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 0.9195687097649881)
         (not (<= (/ (* (- 1.0 x) y) (+ 1.0 y)) 1.006962404202967)))
   (- 1.0 (* (- 1.0 x) (/ y (+ 1.0 y))))
   (-
    (+ (/ x (* y y)) (+ x (+ (/ 1.0 y) (/ 1.0 (pow y 3.0)))))
    (+ (/ x (pow y 3.0)) (+ (/ x y) (/ 1.0 (* y 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)) <= 0.9195687097649881) || !((((1.0 - x) * y) / (1.0 + y)) <= 1.006962404202967)) {
		tmp = 1.0 - ((1.0 - x) * (y / (1.0 + y)));
	} else {
		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))));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	21.9
Target	0.2
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.9195687097649881 or 1.00696240420296701 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 10.5

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

   2. Taylor expanded around 0 10.5

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

   3. Simplified 0.0

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

   if 0.9195687097649881 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.00696240420296701

   1. Initial program 57.8

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.9195687097649881 \lor \neg \left(\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.006962404202967\right):\\ \;\;\;\;1 - \left(1 - x\right) \cdot \frac{y}{1 + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y \cdot y} + \left(x + \left(\frac{1}{y} + \frac{1}{{y}^{3}}\right)\right)\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{x}{y} + \frac{1}{y \cdot y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))