Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.4 → 1.4

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1483102598056.9634 \lor \neg \left(x \le 4.575777435395605 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((x <= -1483102598056.9634) || !(x <= 4.575777435395605e-20))) {
		VAR = ((double) (x / ((double) (z / ((double) (y + z))))));
	} else {
		VAR = ((double) (((double) (((double) (x * y)) / z)) + x));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.4
Target	3.0
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if x < -1483102598056.9634 or 4.575777435395605e-20 < x

   1. Initial program 22.5

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

   3. Applied associate-/l* 0.1

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

   if -1483102598056.9634 < x < 4.575777435395605e-20

   1. Initial program 4.6

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

   2. Taylor expanded around 0 2.5

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1483102598056.9634 \lor \neg \left(x \le 4.575777435395605 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020174 
(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))