Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.7 → 1.7

Time: 1.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -6.9342098286425376 \cdot 10^{301} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -1.46715174269513036 \cdot 10^{-77}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r390631 = x;
        double r390632 = y;
        double r390633 = z;
        double r390634 = r390632 + r390633;
        double r390635 = r390631 * r390634;
        double r390636 = r390635 / r390633;
        return r390636;
}

↓

double f(double x, double y, double z) {
        double r390637 = x;
        double r390638 = y;
        double r390639 = z;
        double r390640 = r390638 + r390639;
        double r390641 = r390637 * r390640;
        double r390642 = r390641 / r390639;
        double r390643 = -6.934209828642538e+301;
        bool r390644 = r390642 <= r390643;
        double r390645 = -1.4671517426951304e-77;
        bool r390646 = r390642 <= r390645;
        double r390647 = !r390646;
        bool r390648 = r390644 || r390647;
        double r390649 = r390638 / r390639;
        double r390650 = fma(r390649, r390637, r390637);
        double r390651 = r390648 ? r390650 : r390642;
        return r390651;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.7
Target	3.1
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -6.934209828642538e+301 or -1.4671517426951304e-77 < (/ (* x (+ y z)) z)

   1. Initial program 17.3

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

   2. Simplified 2.2

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

   if -6.934209828642538e+301 < (/ (* x (+ y z)) z) < -1.4671517426951304e-77

   1. Initial program 0.3

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \le -6.9342098286425376 \cdot 10^{301} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -1.46715174269513036 \cdot 10^{-77}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 +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))