Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.6 → 2.4

Time: 2.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.784404483351111 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 9.4322673509474408 \cdot 10^{246}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((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)) <= 1.784404483351111e+144)) {
		VAR = ((double) (x / ((double) (z / ((double) (y + z))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * ((double) (y + z)))) / z)) <= 9.43226735094744e+246)) {
			VAR_1 = ((double) (((double) (x * ((double) (y + z)))) / z));
		} else {
			VAR_1 = ((double) fma(((double) (y / z)), x, x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.6
Target	3.0
Herbie	2.4

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (+ y z)) z) < 1.784404483351111e+144

   1. Initial program 9.4

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

   3. Applied associate-/l* 2.3

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

   if 1.784404483351111e+144 < (/ (* x (+ y z)) z) < 9.43226735094744e+246

   1. Initial program 0.2

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

   if 9.43226735094744e+246 < (/ (* x (+ y z)) z)

   1. Initial program 45.6

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

   2. Simplified 4.2

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.784404483351111 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 9.4322673509474408 \cdot 10^{246}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))