Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.5 → 0.8

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -4.40375298531076147 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.52299297723786964 \cdot 10^{132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.57446106865930061 \cdot 10^{293}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

C

double code(double x, double y, double z) {
	return ((x * (y + z)) / z);
}

↓

double code(double x, double y, double z) {
	double temp;
	if ((((x * (y + z)) / z) <= -inf.0)) {
		temp = fma((y / z), x, x);
	} else {
		double temp_1;
		if ((((x * (y + z)) / z) <= -4.4037529853107615e+90)) {
			temp_1 = ((x * (y + z)) / z);
		} else {
			double temp_2;
			if ((((x * (y + z)) / z) <= 1.5229929772378696e+132)) {
				temp_2 = fma((y / z), x, x);
			} else {
				double temp_3;
				if ((((x * (y + z)) / z) <= 2.5744610686593006e+293)) {
					temp_3 = ((x * (y + z)) / z);
				} else {
					temp_3 = fma((x / z), y, x);
				}
				temp_2 = temp_3;
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.5
Target	3.0
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -4.4037529853107615e+90 < (/ (* x (+ y z)) z) < 1.5229929772378696e+132

   1. Initial program 11.3

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

   2. Simplified 0.7

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

   if -inf.0 < (/ (* x (+ y z)) z) < -4.4037529853107615e+90 or 1.5229929772378696e+132 < (/ (* x (+ y z)) z) < 2.5744610686593006e+293

   1. Initial program 0.2

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

   if 2.5744610686593006e+293 < (/ (* x (+ y z)) z)

   1. Initial program 57.6

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

   2. Taylor expanded around 0 20.5

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

   3. Simplified 3.3

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le -4.40375298531076147 \cdot 10^{90}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.52299297723786964 \cdot 10^{132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 2.57446106865930061 \cdot 10^{293}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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))