Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 13.1 → 2.2

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.69309411293973363 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.82587901953215762 \cdot 10^{299}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z) {
	double VAR;
	if (((((double) (x * ((double) (y + z)))) / z) <= 1.6930941129397336e-270)) {
		VAR = ((double) (x + ((double) ((x / ((double) (((double) cbrt(z)) * ((double) cbrt(z))))) * (y / ((double) cbrt(z)))))));
	} else {
		double VAR_1;
		if (((((double) (x * ((double) (y + z)))) / z) <= 1.8258790195321576e+299)) {
			VAR_1 = (((double) (x * ((double) (y + z)))) / z);
		} else {
			VAR_1 = ((double) (x + ((double) (x * (y / z)))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	13.1
Target	3.3
Herbie	2.2

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

Derivation

1. Split input into 3 regimes

2. if (/ (* x (+ y z)) z) < 1.69309411293973363e-270

   1. Initial program 14.6

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

   2. Simplified 3.2

      \[\leadsto \color{blue}{x + x \cdot \frac{y}{z}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 3.6

      \[\leadsto x + x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

   5. Applied *-un-lft-identity 3.6

      \[\leadsto x + x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

   6. Applied times-frac 3.6

      \[\leadsto x + x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)}\]

   7. Applied associate-*r* 3.6

      \[\leadsto x + \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}}\]

   8. Simplified 3.6

      \[\leadsto x + \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}}\]

   if 1.69309411293973363e-270 < (/ (* x (+ y z)) z) < 1.82587901953215762e299

   1. Initial program 0.4

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

   if 1.82587901953215762e299 < (/ (* x (+ y z)) z)

   1. Initial program 60.6

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

   2. Simplified 0.8

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

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.69309411293973363 \cdot 10^{-270}:\\ \;\;\;\;x + \frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \le 1.82587901953215762 \cdot 10^{299}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + x \cdot \frac{y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))