Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 13.1 → 1.9

Time: 5.4s

Precision: binary64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.55850091344555163 \cdot 10^{110} \lor \neg \left(x \le 25947.6304069558137\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x <= -3.5585009134455516e+110) || !(x <= 25947.630406955814))) {
		VAR = (x / (z / ((double) (y + z))));
	} else {
		VAR = ((double) (x + (((double) (x * y)) / z)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	13.1
Target	3.3
Herbie	1.9

\[\frac{x}{\frac{z}{y + z}}\]

Derivation

1. Split input into 2 regimes

2. if x < -3.55850091344555163e110 or 25947.6304069558137 < x

   1. Initial program 27.5

      \[\frac{x \cdot \left(y + z\right)}{z}\]
   2. Using strategy rm

   3. Applied associate-/l* 0.1

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]

   if -3.55850091344555163e110 < x < 25947.6304069558137

   1. Initial program 5.7

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

   2. Taylor expanded around 0 2.8

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.55850091344555163 \cdot 10^{110} \lor \neg \left(x \le 25947.6304069558137\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))