Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 2.5

Time: 2.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -8.82644305233315512 \cdot 10^{-5} \lor \neg \left(z \le 4.9542007534756208 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((z <= -8.826443052333155e-05) || !(z <= 4.954200753475621e-114))) {
		VAR = (x / (z / (y + z)));
	} else {
		VAR = (1.0 / (z / (x * (y + z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	2.9
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if z < -8.826443052333155e-05 or 4.954200753475621e-114 < z

   1. Initial program 14.9

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

   3. Applied associate-/l* 0.3

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

   if -8.826443052333155e-05 < z < 4.954200753475621e-114

   1. Initial program 7.1

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

   3. Applied clear-num 7.2

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -8.82644305233315512 \cdot 10^{-5} \lor \neg \left(z \le 4.9542007534756208 \cdot 10^{-114}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(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))