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

Average Error: 7.4 → 1.7

Time: 4.4s

Precision: 64

Math

\[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.2718441977740784 \cdot 10^{277}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= -inf.0)) {
		VAR = ((fma((y / ((t * z) - x)), z, x) - (x / ((t * z) - x))) * (1.0 / (x + 1.0)));
	} else {
		double VAR_1;
		if ((((x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0)) <= 1.2718441977740784e+277)) {
			VAR_1 = ((x + (1.0 / (((t * z) - x) / ((y * z) - x)))) / (x + 1.0));
		} else {
			VAR_1 = ((x + (y / t)) / (x + 1.0));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.4
Target	0.3
Herbie	1.7

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

Derivation

1. Split input into 3 regimes

2. if (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < -inf.0

   1. Initial program 64.0

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied div-sub 64.0

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

   4. Applied associate-+r- 64.0

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

   5. Simplified 4.2

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

   7. Applied div-inv 4.2

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

   if -inf.0 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)) < 1.2718441977740784e+277

   1. Initial program 0.8

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]
   2. Using strategy rm

   3. Applied clear-num 0.9

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}}{x + 1}\]

   if 1.2718441977740784e+277 < (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0))

   1. Initial program 60.4

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1}\]

   2. Taylor expanded around inf 10.5

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
3. Recombined 3 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} = -\infty:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{y}{t \cdot z - x}, z, x\right) - \frac{x}{t \cdot z - x}\right) \cdot \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le 1.2718441977740784 \cdot 10^{277}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t \cdot z - x}{y \cdot z - x}}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (/ (+ x (- (/ y (- t (/ x z))) (/ x (- (* t z) x)))) (+ x 1))

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