Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.7 → 0.8

Time: 2.6s

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 r420598 = x;
        double r420599 = y;
        double r420600 = z;
        double r420601 = r420599 + r420600;
        double r420602 = r420598 * r420601;
        double r420603 = r420602 / r420600;
        return r420603;
}

↓

double f(double x, double y, double z) {
        double r420604 = x;
        double r420605 = y;
        double r420606 = z;
        double r420607 = r420605 + r420606;
        double r420608 = r420604 * r420607;
        double r420609 = r420608 / r420606;
        double r420610 = -inf.0;
        bool r420611 = r420609 <= r420610;
        double r420612 = -3.611697980994098e+68;
        bool r420613 = r420609 <= r420612;
        double r420614 = 2.2373313986691395e+36;
        bool r420615 = r420609 <= r420614;
        double r420616 = 8.147235450072648e+229;
        bool r420617 = r420609 <= r420616;
        double r420618 = !r420617;
        bool r420619 = r420615 || r420618;
        double r420620 = !r420619;
        bool r420621 = r420613 || r420620;
        double r420622 = !r420621;
        bool r420623 = r420611 || r420622;
        double r420624 = r420605 / r420606;
        double r420625 = fma(r420624, r420604, r420604);
        double r420626 = r420623 ? r420625 : r420609;
        return r420626;
}

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))