Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 0.5

Time: 1.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -3.1422801519322542 \cdot 10^{85} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.10620055177297528 \cdot 10^{-72} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 7.1035923095445766 \cdot 10^{280}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

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 (((((x * (y + z)) / z) <= -inf.0) || !((((x * (y + z)) / z) <= -3.142280151932254e+85) || !((((x * (y + z)) / z) <= 1.1062005517729753e-72) || !(((x * (y + z)) / z) <= 7.103592309544577e+280))))) {
		VAR = fma((y / z), x, x);
	} else {
		VAR = ((x * (y + z)) / z);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.3
Target	3.1
Herbie	0.5

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -3.142280151932254e+85 < (/ (* x (+ y z)) z) < 1.1062005517729753e-72 or 7.103592309544577e+280 < (/ (* x (+ y z)) z)

   1. Initial program 20.2

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]

   if -inf.0 < (/ (* x (+ y z)) z) < -3.142280151932254e+85 or 1.1062005517729753e-72 < (/ (* x (+ y z)) z) < 7.103592309544577e+280

   1. Initial program 0.3

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -3.1422801519322542 \cdot 10^{85} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.10620055177297528 \cdot 10^{-72} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 7.1035923095445766 \cdot 10^{280}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(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))