Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.7 → 0.8

Time: 2.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -3.611697980994097954465201743191884507679 \cdot 10^{68} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 2237331398669139509528705024365953024 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 8.147235450072648293691296944251903929426 \cdot 10^{229}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r406846 = x;
        double r406847 = y;
        double r406848 = z;
        double r406849 = r406847 + r406848;
        double r406850 = r406846 * r406849;
        double r406851 = r406850 / r406848;
        return r406851;
}

↓

double f(double x, double y, double z) {
        double r406852 = x;
        double r406853 = y;
        double r406854 = z;
        double r406855 = r406853 + r406854;
        double r406856 = r406852 * r406855;
        double r406857 = r406856 / r406854;
        double r406858 = -inf.0;
        bool r406859 = r406857 <= r406858;
        double r406860 = -3.611697980994098e+68;
        bool r406861 = r406857 <= r406860;
        double r406862 = 2.2373313986691395e+36;
        bool r406863 = r406857 <= r406862;
        double r406864 = 8.147235450072648e+229;
        bool r406865 = r406857 <= r406864;
        double r406866 = !r406865;
        bool r406867 = r406863 || r406866;
        double r406868 = !r406867;
        bool r406869 = r406861 || r406868;
        double r406870 = !r406869;
        bool r406871 = r406859 || r406870;
        double r406872 = r406853 / r406854;
        double r406873 = fma(r406872, r406852, r406852);
        double r406874 = r406871 ? r406873 : r406857;
        return r406874;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.7
Target	3.2
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -3.611697980994098e+68 < (/ (* x (+ y z)) z) < 2.2373313986691395e+36 or 8.147235450072648e+229 < (/ (* x (+ y z)) z)

   1. Initial program 17.9

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

   2. Simplified 1.0

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

   if -inf.0 < (/ (* x (+ y z)) z) < -3.611697980994098e+68 or 2.2373313986691395e+36 < (/ (* x (+ y z)) z) < 8.147235450072648e+229

   1. Initial program 0.2

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -3.611697980994097954465201743191884507679 \cdot 10^{68} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 2237331398669139509528705024365953024 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 8.147235450072648293691296944251903929426 \cdot 10^{229}\right)\right)\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))