Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.8 → 2.2

Time: 13.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -79155848252694193587582928945152:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 2.110775235893212719144090988535183713368 \cdot 10^{-216}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r22132911 = x;
        double r22132912 = y;
        double r22132913 = z;
        double r22132914 = r22132912 + r22132913;
        double r22132915 = r22132911 * r22132914;
        double r22132916 = r22132915 / r22132913;
        return r22132916;
}

↓

double f(double x, double y, double z) {
        double r22132917 = z;
        double r22132918 = -7.915584825269419e+31;
        bool r22132919 = r22132917 <= r22132918;
        double r22132920 = x;
        double r22132921 = y;
        double r22132922 = r22132917 + r22132921;
        double r22132923 = r22132917 / r22132922;
        double r22132924 = r22132920 / r22132923;
        double r22132925 = 2.1107752358932127e-216;
        bool r22132926 = r22132917 <= r22132925;
        double r22132927 = r22132920 * r22132921;
        double r22132928 = r22132927 / r22132917;
        double r22132929 = r22132920 + r22132928;
        double r22132930 = r22132926 ? r22132929 : r22132924;
        double r22132931 = r22132919 ? r22132924 : r22132930;
        return r22132931;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.8
Target	3.1
Herbie	2.2

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

Derivation

1. Split input into 2 regimes

2. if z < -7.915584825269419e+31 or 2.1107752358932127e-216 < z

   1. Initial program 15.0

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

   3. Applied associate-/l* 1.2

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

   if -7.915584825269419e+31 < z < 2.1107752358932127e-216

   1. Initial program 7.1

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

   2. Taylor expanded around 0 4.7

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -79155848252694193587582928945152:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 2.110775235893212719144090988535183713368 \cdot 10^{-216}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))