Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.2 → 2.5

Time: 2.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -8.290545694620129804415821640580454334178 \cdot 10^{-146} \lor \neg \left(x \le 1.08942122341774857321999335085630447302 \cdot 10^{-199}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r426104 = x;
        double r426105 = y;
        double r426106 = z;
        double r426107 = r426105 + r426106;
        double r426108 = r426104 * r426107;
        double r426109 = r426108 / r426106;
        return r426109;
}

↓

double f(double x, double y, double z) {
        double r426110 = x;
        double r426111 = -8.29054569462013e-146;
        bool r426112 = r426110 <= r426111;
        double r426113 = 1.0894212234177486e-199;
        bool r426114 = r426110 <= r426113;
        double r426115 = !r426114;
        bool r426116 = r426112 || r426115;
        double r426117 = y;
        double r426118 = z;
        double r426119 = r426117 / r426118;
        double r426120 = fma(r426119, r426110, r426110);
        double r426121 = r426110 / r426118;
        double r426122 = fma(r426121, r426117, r426110);
        double r426123 = r426116 ? r426120 : r426122;
        return r426123;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.2
Target	3.2
Herbie	2.5

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

Derivation

1. Split input into 2 regimes

2. if x < -8.29054569462013e-146 or 1.0894212234177486e-199 < x

   1. Initial program 13.3

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

   2. Simplified 1.8

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

   if -8.29054569462013e-146 < x < 1.0894212234177486e-199

   1. Initial program 9.1

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

   2. Taylor expanded around 0 4.9

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

   3. Simplified 4.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.290545694620129804415821640580454334178 \cdot 10^{-146} \lor \neg \left(x \le 1.08942122341774857321999335085630447302 \cdot 10^{-199}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 +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))