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

Average Error: 22.0 → 0.1

Time: 4.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + -1}{y}\\ \mathbf{if}\;y \leq -387431.34117193345:\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \left({y}^{-2} + t_0\right)\\ \mathbf{elif}\;y \leq 338143.07044951775:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + t_0 \cdot \left(-1 + \frac{1}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ x -1.0) y)))
   (if (<= y -387431.34117193345)
     (- (+ x (/ x (* y y))) (+ (pow y -2.0) t_0))
     (if (<= y 338143.07044951775)
       (fma y (/ (+ x -1.0) (+ y 1.0)) 1.0)
       (+ x (* t_0 (+ -1.0 (/ 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 + -1.0) / y;
	double tmp;
	if (y <= -387431.34117193345) {
		tmp = (x + (x / (y * y))) - (pow(y, -2.0) + t_0);
	} else if (y <= 338143.07044951775) {
		tmp = fma(y, ((x + -1.0) / (y + 1.0)), 1.0);
	} else {
		tmp = x + (t_0 * (-1.0 + (1.0 / y)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.0
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 < -387431.34117193345

   1. Initial program 45.3

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

   2. Simplified 28.7

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Applied pow2_binary64 0.0

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

   6. Applied pow-flip_binary64 0.0

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

   7. Simplified 0.0

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

   if -387431.34117193345 < y < 338143.070449517749

   1. Initial program 0.1

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

   2. Simplified 0.1

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

   if 338143.070449517749 < y

   1. Initial program 45.6

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

   2. Simplified 28.7

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

   3. Taylor expanded in y around inf 0.0

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

   4. Simplified 0.0

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

   5. Applied add-cube-cbrt_binary64 0.3

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

   6. Applied add-cube-cbrt_binary64 1.4

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

   7. Applied prod-diff_binary64 1.4

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

   8. Simplified 0.0

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

   9. Simplified 0.0

      \[\leadsto \left(x + \frac{x + -1}{y} \cdot \left(\frac{1}{y} + -1\right)\right) + \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -387431.34117193345:\\ \;\;\;\;\left(x + \frac{x}{y \cdot y}\right) - \left({y}^{-2} + \frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 338143.07044951775:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x + -1}{y + 1}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x + -1}{y} \cdot \left(-1 + \frac{1}{y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022068 
(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))))