Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.5 → 1.4

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -6.75135214178766355 \cdot 10^{198}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -1.94806065160282452 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.85109499375252098 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.10698314773329783 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

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 ((((double) (((double) (x * ((double) (y + z)))) / z)) <= -6.751352141787664e+198)) {
		VAR = ((double) fma(((double) (x / z)), y, x));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * ((double) (y + z)))) / z)) <= -1.9480606516028245e-98)) {
			VAR_1 = ((double) (((double) (x * ((double) (y + z)))) / z));
		} else {
			double VAR_2;
			if ((((double) (((double) (x * ((double) (y + z)))) / z)) <= 2.851094993752521e-41)) {
				VAR_2 = ((double) fma(((double) (y / z)), x, x));
			} else {
				double VAR_3;
				if ((((double) (((double) (x * ((double) (y + z)))) / z)) <= 2.106983147733298e+305)) {
					VAR_3 = ((double) (((double) (x * ((double) (y + z)))) / z));
				} else {
					VAR_3 = ((double) fma(((double) (x / z)), y, x));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	2.9
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (+ y z)) z) < -6.751352141787664e+198 or 2.106983147733298e+305 < (/ (* x (+ y z)) z)

   1. Initial program 45.1

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

   2. Taylor expanded around 0 15.3

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

   3. Simplified 6.0

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

   if -6.751352141787664e+198 < (/ (* x (+ y z)) z) < -1.9480606516028245e-98 or 2.851094993752521e-41 < (/ (* x (+ y z)) z) < 2.106983147733298e+305

   1. Initial program 0.3

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

   if -1.9480606516028245e-98 < (/ (* x (+ y z)) z) < 2.851094993752521e-41

   1. Initial program 8.7

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

   2. Simplified 0.1

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -6.75135214178766355 \cdot 10^{198}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -1.94806065160282452 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.85109499375252098 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.10698314773329783 \cdot 10^{305}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 +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))