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

Average Error: 7.8 → 0.9

Time: 12.8s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le +inf.0:\\ \;\;\;\;\frac{x + \left(y \cdot \left(z \cdot \frac{1}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

C

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

↓

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	7.8
Target	0.4
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 5.1

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

   3. Applied div-sub 5.1

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

   4. Simplified 0.9

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

   6. Applied div-inv 1.0

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

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

   1. Initial program 64.0

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

   2. Taylor expanded around inf 0.0

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \le +inf.0:\\ \;\;\;\;\frac{x + \left(y \cdot \left(z \cdot \frac{1}{t \cdot z - x}\right) - \frac{x}{t \cdot z - x}\right)}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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 1.0))

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))