Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.0 → 3.1

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.161724350185571 \cdot 10^{-40} \lor \neg \left(x \le 1.1452918085668837 \cdot 10^{-153}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r510455 = x;
        double r510456 = y;
        double r510457 = z;
        double r510458 = r510456 + r510457;
        double r510459 = r510455 * r510458;
        double r510460 = r510459 / r510457;
        return r510460;
}

↓

double f(double x, double y, double z) {
        double r510461 = x;
        double r510462 = -2.161724350185571e-40;
        bool r510463 = r510461 <= r510462;
        double r510464 = 1.1452918085668837e-153;
        bool r510465 = r510461 <= r510464;
        double r510466 = !r510465;
        bool r510467 = r510463 || r510466;
        double r510468 = y;
        double r510469 = z;
        double r510470 = r510468 / r510469;
        double r510471 = fma(r510470, r510461, r510461);
        double r510472 = 1.0;
        double r510473 = r510468 + r510469;
        double r510474 = r510461 * r510473;
        double r510475 = r510469 / r510474;
        double r510476 = r510472 / r510475;
        double r510477 = r510467 ? r510471 : r510476;
        return r510477;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.0
Target	3.0
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if x < -2.161724350185571e-40 or 1.1452918085668837e-153 < x

   1. Initial program 15.9

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

   2. Simplified 1.1

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

   if -2.161724350185571e-40 < x < 1.1452918085668837e-153

   1. Initial program 6.2

      \[\frac{x \cdot \left(y + z\right)}{z}\]
   2. Using strategy rm

   3. Applied clear-num 6.3

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.161724350185571 \cdot 10^{-40} \lor \neg \left(x \le 1.1452918085668837 \cdot 10^{-153}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{x \cdot \left(y + z\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))