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

Average Error: 22.6 → 0.1

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\\ \mathbf{if}\;y \leq -151487198.04354337:\\ \;\;\;\;t_0 + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 41952.18459881466:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\frac{x}{y} + \left(\frac{1}{y \cdot y} - \frac{1}{y}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x (/ x (* y y))) (/ 1.0 (pow y 3.0)))))
   (if (<= y -151487198.04354337)
     (+ t_0 (/ (- 1.0 x) y))
     (if (<= y 41952.18459881466)
       (- 1.0 (/ (* y (- 1.0 x)) (+ y 1.0)))
       (- t_0 (+ (/ x y) (- (/ 1.0 (* y 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 t_0 = (x + (x / (y * y))) + (1.0 / pow(y, 3.0));
	double tmp;
	if (y <= -151487198.04354337) {
		tmp = t_0 + ((1.0 - x) / y);
	} else if (y <= 41952.18459881466) {
		tmp = 1.0 - ((y * (1.0 - x)) / (y + 1.0));
	} else {
		tmp = t_0 - ((x / y) + ((1.0 / (y * y)) - (1.0 / y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.6
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 3 regimes

2. if y < -151487198.043543369

   1. Initial program 45.5

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

   2. Simplified 28.8

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Taylor expanded in y around -inf 0.1

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

   6. Simplified 0.1

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

   if -151487198.043543369 < y < 41952.1845988146597

   1. Initial program 0.1

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

   if 41952.1845988146597 < y

   1. Initial program 45.6

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

   2. Simplified 29.6

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Taylor expanded in y around inf 0.0

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

   6. Simplified 0.0

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -151487198.04354337:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) + \frac{1 - x}{y}\\ \mathbf{elif}\;y \leq 41952.18459881466:\\ \;\;\;\;1 - \frac{y \cdot \left(1 - x\right)}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{y} + \left(\frac{1}{y \cdot y} - \frac{1}{y}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021329 
(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))))