Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.8 → 2.1

Time: 2.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -5.29259606783455435 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -7.234852918192252 \cdot 10^{43}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]

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 ((((x * (y + z)) / z) <= -5.292596067834554e+295)) {
		VAR = fma((x / z), y, x);
	} else {
		double VAR_1;
		if ((((x * (y + z)) / z) <= -7.234852918192252e+43)) {
			VAR_1 = ((x * (y + z)) / z);
		} else {
			VAR_1 = fma((y / z), x, x);
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.8
Target	3.2
Herbie	2.1

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (+ y z)) z) < -5.292596067834554e+295

   1. Initial program 59.1

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

   2. Taylor expanded around 0 19.7

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

   3. Simplified 2.1

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

   if -5.292596067834554e+295 < (/ (* x (+ y z)) z) < -7.234852918192252e+43

   1. Initial program 0.2

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

   if -7.234852918192252e+43 < (/ (* x (+ y z)) z)

   1. Initial program 10.6

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

   2. Simplified 2.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -5.29259606783455435 \cdot 10^{295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -7.234852918192252 \cdot 10^{43}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +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))