Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.2 → 0.2

Time: 3.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -inf.0 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -4405354812.0436611 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.554719158539295 \cdot 10^{41} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 4.81708140039638585 \cdot 10^{302}\right)\right)\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

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)) <= -inf.0) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= -4405354812.043661) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= 1.554719158539295e+41) || !(((double) (((double) (x * ((double) (y + z)))) / z)) <= 4.817081400396386e+302))))) {
		VAR = ((double) (x * ((double) (((double) (y + z)) / z))));
	} else {
		VAR = ((double) (((double) (x * ((double) (y + z)))) / z));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.0
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -4405354812.043661 < (/ (* x (+ y z)) z) < 1.554719158539295e+41 or 4.817081400396386e+302 < (/ (* x (+ y z)) z)

   1. Initial program 19.6

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

   3. Applied *-un-lft-identity 19.6

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

   4. Applied times-frac 0.2

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

   5. Simplified 0.2

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

   if -inf.0 < (/ (* x (+ y z)) z) < -4405354812.043661 or 1.554719158539295e+41 < (/ (* x (+ y z)) z) < 4.817081400396386e+302

   1. Initial program 0.2

      \[\frac{x \cdot \left(y + z\right)}{z}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -inf.0 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -4405354812.0436611 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.554719158539295 \cdot 10^{41} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 4.81708140039638585 \cdot 10^{302}\right)\right)\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020148 
(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))