Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 2.4

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 3.53824064468242378 \cdot 10^{93}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.58984662664485094 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

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)) <= 3.538240644682424e+93)) {
		VAR = ((double) fma(((double) (y / z)), x, x));
	} else {
		double VAR_1;
		if ((((double) (((double) (x * ((double) (y + z)))) / z)) <= 1.589846626644851e+255)) {
			VAR_1 = ((double) (((double) (x * ((double) (y + z)))) / z));
		} else {
			VAR_1 = ((double) fma(((double) (x / z)), y, x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.3
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) < 3.538240644682424e+93

   1. Initial program 9.7

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

   2. Simplified 2.4

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

   if 3.538240644682424e+93 < (/ (* x (+ y z)) z) < 1.589846626644851e+255

   1. Initial program 0.2

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

   if 1.589846626644851e+255 < (/ (* x (+ y z)) z)

   1. Initial program 45.8

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

   2. Taylor expanded around 0 14.8

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

   3. Simplified 5.3

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

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 3.53824064468242378 \cdot 10^{93}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.58984662664485094 \cdot 10^{255}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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))