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

Average Error: 22.8 → 7.4

Time: 3.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -6097841703286955 \lor \neg \left(y \le 4.7030060381657199 \cdot 10^{33}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y - 1\right) \cdot \frac{y}{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)\right), x - 1, 1\right)\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if (((y <= -6097841703286955.0) || !(y <= 4.70300603816572e+33))) {
		VAR = fma(1.0, ((x / pow(y, 2.0)) - (x / y)), x);
	} else {
		VAR = fma((((y - 1.0) * (y / (pow((y * y), 3.0) - pow((1.0 * 1.0), 3.0)))) * (((y * y) * (y * y)) + (((1.0 * 1.0) * (1.0 * 1.0)) + ((y * y) * (1.0 * 1.0))))), (x - 1.0), 1.0);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.8
Target	0.2
Herbie	7.4

\[\begin{array}{l} \mathbf{if}\;y \lt -3693.84827882972468:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y \lt 6799310503.41891003:\\ \;\;\;\;1 - \frac{\left(1 - x\right) \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} - \left(\frac{x}{y} - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -6097841703286955.0 or 4.70300603816572e+33 < y

   1. Initial program 47.1

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

   2. Simplified 30.0

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

   4. Applied flip-+ 47.6

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

   5. Applied associate-/r/ 47.6

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

   6. Taylor expanded around inf 14.1

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

   7. Simplified 14.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)}\]

   if -6097841703286955.0 < y < 4.70300603816572e+33

   1. Initial program 1.6

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

   2. Simplified 1.5

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

   4. Applied flip-+ 1.5

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

   5. Applied associate-/r/ 1.5

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{y \cdot y - 1 \cdot 1} \cdot \left(y - 1\right)}, x - 1, 1\right)\]
   6. Using strategy rm

   7. Applied clear-num 1.5

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

   8. Applied associate-*l/ 1.5

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

   9. Simplified 1.5

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y - 1}}{\frac{y \cdot y - 1 \cdot 1}{y}}, x - 1, 1\right)\]
   10. Using strategy rm

   11. Applied clear-num 1.5

       \[\leadsto \mathsf{fma}\left(\frac{y - 1}{\color{blue}{\frac{1}{\frac{y}{y \cdot y - 1 \cdot 1}}}}, x - 1, 1\right)\]
   12. Using strategy rm

   13. Applied flip3-- 1.5

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

   14. Applied associate-/r/ 1.6

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

   15. Applied associate-/r* 1.6

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

   16. Applied associate-/r/ 1.6

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

   17. Simplified 1.5

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

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -6097841703286955 \lor \neg \left(y \le 4.7030060381657199 \cdot 10^{33}\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(y - 1\right) \cdot \frac{y}{{\left(y \cdot y\right)}^{3} - {\left(1 \cdot 1\right)}^{3}}\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(y \cdot y\right) \cdot \left(1 \cdot 1\right)\right)\right), x - 1, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ 1 y) (- (/ x y) x)) (if (< y 6799310503.41891) (- 1 (/ (* (- 1 x) y) (+ y 1))) (- (/ 1 y) (- (/ x y) x))))

  (- 1 (/ (* (- 1 x) y) (+ y 1))))