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

Average Error: 22.1 → 0.2

Time: 7.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x, \frac{y}{1 + y}, 1\right)\\ t_1 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_2 := \frac{y}{1 - y \cdot y}\\ t_3 := \mathsf{fma}\left(y + -1, t_2, \left(1 - y\right) \cdot t_2\right)\\ \mathbf{if}\;t_1 \leq 0.9997523214764116:\\ \;\;\;\;\left(\frac{y + -1}{\frac{1}{y} - y} + t_0\right) + t_3\\ \mathbf{elif}\;t_1 \leq 1.0000146328459205:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 + \left(t_0 + \left(y + -1\right) \cdot t_2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma x (/ y (+ 1.0 y)) 1.0))
        (t_1 (/ (* (- 1.0 x) y) (+ 1.0 y)))
        (t_2 (/ y (- 1.0 (* y y))))
        (t_3 (fma (+ y -1.0) t_2 (* (- 1.0 y) t_2))))
   (if (<= t_1 0.9997523214764116)
     (+ (+ (/ (+ y -1.0) (- (/ 1.0 y) y)) t_0) t_3)
     (if (<= t_1 1.0000146328459205)
       (-
        (+ (+ x (/ x (* y y))) (/ 1.0 (pow y 3.0)))
        (+ (/ x (pow y 3.0)) (+ (/ 1.0 (* y y)) (/ (+ x -1.0) y))))
       (+ t_3 (+ t_0 (* (+ y -1.0) t_2)))))))

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 = fma(x, (y / (1.0 + y)), 1.0);
	double t_1 = ((1.0 - x) * y) / (1.0 + y);
	double t_2 = y / (1.0 - (y * y));
	double t_3 = fma((y + -1.0), t_2, ((1.0 - y) * t_2));
	double tmp;
	if (t_1 <= 0.9997523214764116) {
		tmp = (((y + -1.0) / ((1.0 / y) - y)) + t_0) + t_3;
	} else if (t_1 <= 1.0000146328459205) {
		tmp = ((x + (x / (y * y))) + (1.0 / pow(y, 3.0))) - ((x / pow(y, 3.0)) + ((1.0 / (y * y)) + ((x + -1.0) / y)));
	} else {
		tmp = t_3 + (t_0 + ((y + -1.0) * t_2));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	22.1
Target	0.2
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 0.999752321476411576

   1. Initial program 7.1

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

   2. Simplified 0.0

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

   3. Taylor expanded in x around 0 7.1

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

   4. Simplified 0.0

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

   5. Applied flip-+_binary64 0.2

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

   6. Applied associate-/r/_binary64 0.2

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

   7. Applied *-un-lft-identity_binary64 0.2

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

   8. Applied prod-diff_binary64 0.2

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

   9. Applied add-cbrt-cube_binary64 12.9

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

   10. Simplified 12.7

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

   11. Applied rem-cbrt-cube_binary64 0.0

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

   if 0.999752321476411576 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.00001463284592051

   1. Initial program 59.2

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

   2. Simplified 59.2

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

   3. Taylor expanded in y around inf 0.1

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

   4. Simplified 0.1

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

   if 1.00001463284592051 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 20.7

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

   2. Simplified 0.2

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

   3. Taylor expanded in x around 0 20.7

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

   4. Simplified 0.1

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

   5. Applied flip-+_binary64 0.8

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

   6. Applied associate-/r/_binary64 0.8

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

   7. Applied *-un-lft-identity_binary64 0.8

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

   8. Applied prod-diff_binary64 0.8

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

   9. Applied fma-udef_binary64 0.8

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

   10. Simplified 0.8

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.9997523214764116:\\ \;\;\;\;\left(\frac{y + -1}{\frac{1}{y} - y} + \mathsf{fma}\left(x, \frac{y}{1 + y}, 1\right)\right) + \mathsf{fma}\left(y + -1, \frac{y}{1 - y \cdot y}, \left(1 - y\right) \cdot \frac{y}{1 - y \cdot y}\right)\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0000146328459205:\\ \;\;\;\;\left(\left(x + \frac{x}{y \cdot y}\right) + \frac{1}{{y}^{3}}\right) - \left(\frac{x}{{y}^{3}} + \left(\frac{1}{y \cdot y} + \frac{x + -1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + -1, \frac{y}{1 - y \cdot y}, \left(1 - y\right) \cdot \frac{y}{1 - y \cdot y}\right) + \left(\mathsf{fma}\left(x, \frac{y}{1 + y}, 1\right) + \left(y + -1\right) \cdot \frac{y}{1 - y \cdot y}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022095 
(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))))