Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.2 → 2.5

Time: 2.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.4132173672170657 \cdot 10^{-241} \lor \neg \left(x \le 3.8615698792640332 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

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 <= -1.4132173672170657e-241) || !(x <= 3.861569879264033e-09))) {
		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	12.2
Target	3.1
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if x < -1.4132173672170657e-241 or 3.8615698792640332e-9 < x

   1. Initial program 15.2

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

   3. Applied associate-/l* 1.9

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

   if -1.4132173672170657e-241 < x < 3.8615698792640332e-9

   1. Initial program 6.4

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

   2. Taylor expanded around 0 3.6

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

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4132173672170657 \cdot 10^{-241} \lor \neg \left(x \le 3.8615698792640332 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020161 
(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))