Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.2 → 0.8

Time: 2.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2.0130703549749886 \cdot 10^{+249}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -6.019602592663955 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.4021532079817052 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 6.841831142745054 \cdot 10^{+291}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (* x (+ y z)) z) -2.0130703549749886e+249)
   (* x (/ (+ y z) z))
   (if (<= (/ (* x (+ y z)) z) -6.019602592663955e-146)
     (/ (* x (+ y z)) z)
     (if (<= (/ (* x (+ y z)) z) 2.4021532079817052e+29)
       (/ x (/ z (+ y z)))
       (if (<= (/ (* x (+ y z)) z) 6.841831142745054e+291)
         (/ (* x (+ y z)) z)
         (* x (/ (+ y z) 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) <= -2.0130703549749886e+249) {
		tmp = x * ((y + z) / z);
	} else if (((x * (y + z)) / z) <= -6.019602592663955e-146) {
		tmp = (x * (y + z)) / z;
	} else if (((x * (y + z)) / z) <= 2.4021532079817052e+29) {
		tmp = x / (z / (y + z));
	} else if (((x * (y + z)) / z) <= 6.841831142745054e+291) {
		tmp = (x * (y + z)) / z;
	} else {
		tmp = x * ((y + z) / z);
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.1
Herbie	0.8

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -2.0130703549749886e249 or 6.8418311427450543e291 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 50.3

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

   3. Applied *-un-lft-identity_binary64_11872 50.3

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

   4. Applied times-frac_binary64_11878 3.7

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

   5. Simplified 3.7

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

   if -2.0130703549749886e249 < (/.f64 (*.f64 x (+.f64 y z)) z) < -6.0196025926639548e-146 or 2.40215320798170521e29 < (/.f64 (*.f64 x (+.f64 y z)) z) < 6.8418311427450543e291

   1. Initial program 0.2

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

   if -6.0196025926639548e-146 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2.40215320798170521e29

   1. Initial program 8.5

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

   3. Applied associate-/l*_binary64_11819 0.1

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq -2.0130703549749886 \cdot 10^{+249}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq -6.019602592663955 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2.4021532079817052 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{\frac{z}{y + z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 6.841831142745054 \cdot 10^{+291}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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))