Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.8 → 2.2

Time: 11.4s

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 r30736783 = x;
        double r30736784 = y;
        double r30736785 = z;
        double r30736786 = r30736784 + r30736785;
        double r30736787 = r30736783 * r30736786;
        double r30736788 = r30736787 / r30736785;
        return r30736788;
}

↓

double f(double x, double y, double z) {
        double r30736789 = z;
        double r30736790 = -7.915584825269419e+31;
        bool r30736791 = r30736789 <= r30736790;
        double r30736792 = x;
        double r30736793 = y;
        double r30736794 = r30736789 + r30736793;
        double r30736795 = r30736789 / r30736794;
        double r30736796 = r30736792 / r30736795;
        double r30736797 = 2.1107752358932127e-216;
        bool r30736798 = r30736789 <= r30736797;
        double r30736799 = r30736792 * r30736793;
        double r30736800 = r30736799 / r30736789;
        double r30736801 = r30736792 + r30736800;
        double r30736802 = r30736798 ? r30736801 : r30736796;
        double r30736803 = r30736791 ? r30736796 : r30736802;
        return r30736803;
}

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))