Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.2 → 0.3

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -3.949073247736926 \cdot 10^{+55}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 6.0762204890108674 \cdot 10^{+293}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array}\]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ y z)) z) (- INFINITY))
   (/ x (/ z (+ y z)))
   (if (<= (/ (* x (+ y z)) z) -3.949073247736926e+55)
     (+ x (/ (* x y) z))
     (if (<= (/ (* x (+ y z)) z) 0.0)
       (/ x (/ z (+ y z)))
       (if (<= (/ (* x (+ y z)) z) 6.0762204890108674e+293)
         (+ x (/ (* x y) z))
         (* (+ y z) (/ x z)))))))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if (((x * (y + z)) / z) <= -((double) INFINITY)) {
		tmp = x / (z / (y + z));
	} else if (((x * (y + z)) / z) <= -3.949073247736926e+55) {
		tmp = x + ((x * y) / z);
	} else if (((x * (y + z)) / z) <= 0.0) {
		tmp = x / (z / (y + z));
	} else if (((x * (y + z)) / z) <= 6.0762204890108674e+293) {
		tmp = x + ((x * y) / z);
	} else {
		tmp = (y + z) * (x / z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.1
Herbie	0.3

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -inf.0 or -3.94907324773692607e55 < (/.f64 (*.f64 x (+.f64 y z)) z) < -0.0

   1. Initial program 20.1

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

   3. Applied associate-/l*_binary64_12493 0.2

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

   if -inf.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < -3.94907324773692607e55 or -0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 6.07622048901086744e293

   1. Initial program 0.4

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

   2. Taylor expanded around 0 0.3

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

   if 6.07622048901086744e293 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 59.8

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

   3. Applied associate-/l*_binary64_12493 1.4

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

   5. Applied associate-/r/_binary64_12494 1.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -3.949073247736926 \cdot 10^{+55}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 6.0762204890108674 \cdot 10^{+293}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot \frac{x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020303 
(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))