Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 13.1 → 0.3

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -1.6477445116941071 \cdot 10^{56} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 8.0133830656333959 \cdot 10^{-67} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 7.426458603776121 \cdot 10^{296}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

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)) <= -inf.0) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= -1.6477445116941071e+56) || !((((double) (((double) (x * ((double) (y + z)))) / z)) <= 8.013383065633396e-67) || !(((double) (((double) (x * ((double) (y + z)))) / z)) <= 7.42645860377612e+296))))) {
		VAR = ((double) fma(((double) (y / z)), x, x));
	} 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	13.1
Target	2.7
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -1.6477445116941071e+56 < (/ (* x (+ y z)) z) < 8.013383065633396e-67 or 7.42645860377612e+296 < (/ (* x (+ y z)) z)

   1. Initial program 22.2

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

   2. Simplified 0.3

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

   if -inf.0 < (/ (* x (+ y z)) z) < -1.6477445116941071e+56 or 8.013383065633396e-67 < (/ (* x (+ y z)) z) < 7.42645860377612e+296

   1. Initial program 0.2

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -1.6477445116941071 \cdot 10^{56} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 8.0133830656333959 \cdot 10^{-67} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 7.426458603776121 \cdot 10^{296}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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))