Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.5 → 0.6

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -5.13924030840783878 \cdot 10^{59}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 65413472161681629200:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.35854138642000028 \cdot 10^{266}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

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)) {
		VAR = fma((x / z), y, x);
	} else {
		double VAR_1;
		if ((((x * (y + z)) / z) <= -5.139240308407839e+59)) {
			VAR_1 = ((x * (y + z)) / z);
		} else {
			double VAR_2;
			if ((((x * (y + z)) / z) <= 6.541347216168163e+19)) {
				VAR_2 = fma((y / z), x, x);
			} else {
				double VAR_3;
				if ((((x * (y + z)) / z) <= 2.3585413864200003e+266)) {
					VAR_3 = ((x * (y + z)) / z);
				} else {
					VAR_3 = (x / (z / (y + z)));
				}
				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	2.7
Herbie	0.6

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

Derivation

1. Split input into 4 regimes

2. if (/ (* x (+ y z)) z) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded around 0 21.0

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

   3. Simplified 0.1

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

   if -inf.0 < (/ (* x (+ y z)) z) < -5.139240308407839e+59 or 6.541347216168163e+19 < (/ (* x (+ y z)) z) < 2.3585413864200003e+266

   1. Initial program 0.2

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

   if -5.139240308407839e+59 < (/ (* x (+ y z)) z) < 6.541347216168163e+19

   1. Initial program 5.7

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

   2. Simplified 0.4

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

   if 2.3585413864200003e+266 < (/ (* x (+ y z)) z)

   1. Initial program 46.9

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

   3. Applied associate-/l* 3.7

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y + z}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -5.13924030840783878 \cdot 10^{59}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 65413472161681629200:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.35854138642000028 \cdot 10^{266}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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))