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

Average Error: 22.5 → 0.0

Time: 17.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{y \cdot y} + \frac{x + -1}{y}\\ t_1 := x + \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -115615.37911247845:\\ \;\;\;\;\left(t_1 + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + t_0\right)\\ \mathbf{elif}\;y \leq 11126.107645742042:\\ \;\;\;\;1 + \frac{y - y \cdot x}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(\frac{1 - x}{{y}^{3}} - t_0\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 (* y y)) (/ (+ x -1.0) y))) (t_1 (+ x (/ x (* y y)))))
   (if (<= y -115615.37911247845)
     (- (+ t_1 (/ 1.0 (pow y 3.0))) (+ (/ x (pow y 3.0)) t_0))
     (if (<= y 11126.107645742042)
       (+ 1.0 (* (/ (- y (* y x)) (fma y y -1.0)) (- 1.0 y)))
       (+ t_1 (- (/ (- 1.0 x) (pow y 3.0)) t_0))))))

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 = (1.0 / (y * y)) + ((x + -1.0) / y);
	double t_1 = x + (x / (y * y));
	double tmp;
	if (y <= -115615.37911247845) {
		tmp = (t_1 + (1.0 / pow(y, 3.0))) - ((x / pow(y, 3.0)) + t_0);
	} else if (y <= 11126.107645742042) {
		tmp = 1.0 + (((y - (y * x)) / fma(y, y, -1.0)) * (1.0 - y));
	} else {
		tmp = t_1 + (((1.0 - x) / pow(y, 3.0)) - t_0);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.5
Target	0.2
Herbie	0.0

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

Derivation

1. Split input into 3 regimes

2. if y < -115615.379112478447

   1. Initial program 45.6

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

   2. Simplified 28.0

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

   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

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

   if -115615.379112478447 < y < 11126.107645742042

   1. Initial program 0.1

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

   2. Applied flip-+_binary64 0.1

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

   3. Applied associate-/r/_binary64 0.1

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

   4. Applied cancel-sign-sub-inv_binary64 0.1

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

   5. Simplified 0.1

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

   if 11126.107645742042 < y

   1. Initial program 45.4

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

   2. Simplified 29.7

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

   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. Applied associate--l+_binary64 0.0

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

   6. Simplified 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -115615.37911247845:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 11126.107645742042:\\ \;\;\;\;1 + \frac{y - y \cdot x}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(1 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) + \left(\frac{1 - x}{{y}^{3}} - \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))