Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 0.7

Time: 2.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -1.1242336700964007 \cdot 10^{282} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -969610033840.32202 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.48620450230642644 \cdot 10^{141} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 4.7595502063025706 \cdot 10^{303}\right)\right)\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

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)) <= -1.1242336700964007e+282) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= -969610033840.322) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= 1.4862045023064264e+141) || !(((double) (((double) (x * ((double) (y + z)))) / z)) <= 4.7595502063025706e+303))))) {
		VAR = ((double) (x * ((double) (((double) (y / z)) + 1.0))));
	} 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	12.3
Target	2.9
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -1.1242336700964007e282 or -969610033840.32202 < (/ (* x (+ y z)) z) < 1.48620450230642644e141 or 4.7595502063025706e303 < (/ (* x (+ y z)) z)

   1. Initial program 17.1

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

   3. Applied *-un-lft-identity 17.1

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

   4. Applied times-frac 0.9

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

   5. Simplified 0.9

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

   6. Taylor expanded around 0 0.9

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

   if -1.1242336700964007e282 < (/ (* x (+ y z)) z) < -969610033840.32202 or 1.48620450230642644e141 < (/ (* x (+ y z)) z) < 4.7595502063025706e303

   1. Initial program 0.2

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -1.1242336700964007 \cdot 10^{282} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -969610033840.32202 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 1.48620450230642644 \cdot 10^{141} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 4.7595502063025706 \cdot 10^{303}\right)\right)\right):\\ \;\;\;\;x \cdot \left(\frac{y}{z} + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(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))