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

Average Error: 23.0 → 0.2

Time: 6.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\left(1 - x\right) \cdot y}{1 + y}\\ t_1 := \frac{y}{1 + y}\\ \mathbf{if}\;t_0 \leq 0.7185485297680201:\\ \;\;\;\;\mathsf{fma}\left(t_1, x, 1\right) - t_1\\ \mathbf{elif}\;t_0 \leq 1.0003928918252876:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{x}{y \cdot y}\\ t_3 := \frac{t_2 + \left(\frac{1}{y} + \left(x + -1\right)\right)}{y}\\ \left(\left(\frac{1}{{y}^{3}} + \left(x + t_2\right)\right) - t_3\right) + t_3 \cdot 0 \end{array}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 - y \cdot y}, y + -1, \mathsf{fma}\left(x, t_1, 1\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- 1.0 x) y) (+ 1.0 y))) (t_1 (/ y (+ 1.0 y))))
   (if (<= t_0 0.7185485297680201)
     (- (fma t_1 x 1.0) t_1)
     (if (<= t_0 1.0003928918252876)
       (let* ((t_2 (/ x (* y y))) (t_3 (/ (+ t_2 (+ (/ 1.0 y) (+ x -1.0))) y)))
         (+ (- (+ (/ 1.0 (pow y 3.0)) (+ x t_2)) t_3) (* t_3 0.0)))
       (fma (/ y (- 1.0 (* y y))) (+ y -1.0) (fma x t_1 1.0))))))

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 - x) * y) / (1.0 + y);
	double t_1 = y / (1.0 + y);
	double tmp;
	if (t_0 <= 0.7185485297680201) {
		tmp = fma(t_1, x, 1.0) - t_1;
	} else if (t_0 <= 1.0003928918252876) {
		double t_2 = x / (y * y);
		double t_3 = (t_2 + ((1.0 / y) + (x + -1.0))) / y;
		tmp = (((1.0 / pow(y, 3.0)) + (x + t_2)) - t_3) + (t_3 * 0.0);
	} else {
		tmp = fma((y / (1.0 - (y * y))), (y + -1.0), fma(x, t_1, 1.0));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	23.0
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)) < 0.71854852976802008

   1. Initial program 7.0

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

   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 7.0

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

   4. Simplified 0.0

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

   5. Applied *-un-lft-identity_binary64 0.0

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

   6. Applied *-un-lft-identity_binary64 0.0

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

   7. Applied times-frac_binary64 0.0

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

   8. Applied cancel-sign-sub-inv_binary64 0.0

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

   if 0.71854852976802008 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1)) < 1.0003928918252876

   1. Initial program 58.2

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

   2. Simplified 58.1

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

   3. Taylor expanded in y around inf 0.3

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

   4. Simplified 0.3

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

   5. Applied div-inv_binary64 0.3

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

   6. Applied *-un-lft-identity_binary64 0.3

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

   7. Applied times-frac_binary64 0.3

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

   8. Applied distribute-rgt-out_binary64 0.3

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

   9. Applied cube-mult_binary64 0.3

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

   10. Applied *-un-lft-identity_binary64 0.3

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

   11. Applied times-frac_binary64 0.3

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

   12. Applied distribute-lft-out_binary64 0.3

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

   13. Applied add-cube-cbrt_binary64 1.0

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

   14. Applied prod-diff_binary64 1.0

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

   15. Simplified 0.3

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

   16. Simplified 0.3

       \[\leadsto \left(\left(\frac{1}{{y}^{3}} + \left(x + \frac{x}{y \cdot y}\right)\right) - \frac{\frac{x}{y \cdot y} + \left(\frac{1}{y} + \left(x + -1\right)\right)}{y}\right) + \color{blue}{0 \cdot \frac{\frac{x}{y \cdot y} + \left(\frac{1}{y} + \left(x + -1\right)\right)}{y}} \]

   if 1.0003928918252876 < (/.f64 (*.f64 (-.f64 1 x) y) (+.f64 y 1))

   1. Initial program 21.0

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

   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 21.0

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

   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 0.4

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

   6. Applied associate-/r/_binary64 0.4

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

   7. Applied add-cube-cbrt_binary64 1.7

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

   8. Applied prod-diff_binary64 1.7

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

   9. Simplified 0.4

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

   10. Simplified 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 0.7185485297680201:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 + y}, x, 1\right) - \frac{y}{1 + y}\\ \mathbf{elif}\;\frac{\left(1 - x\right) \cdot y}{1 + y} \leq 1.0003928918252876:\\ \;\;\;\;\left(\left(\frac{1}{{y}^{3}} + \left(x + \frac{x}{y \cdot y}\right)\right) - \frac{\frac{x}{y \cdot y} + \left(\frac{1}{y} + \left(x + -1\right)\right)}{y}\right) + \frac{\frac{x}{y \cdot y} + \left(\frac{1}{y} + \left(x + -1\right)\right)}{y} \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{1 - y \cdot y}, y + -1, \mathsf{fma}\left(x, \frac{y}{1 + y}, 1\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))