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 r359271 = x;
        double r359272 = y;
        double r359273 = z;
        double r359274 = r359272 + r359273;
        double r359275 = r359271 * r359274;
        double r359276 = r359275 / r359273;
        return r359276;
}

↓

double f(double x, double y, double z) {
        double r359277 = x;
        double r359278 = y;
        double r359279 = z;
        double r359280 = r359278 + r359279;
        double r359281 = r359277 * r359280;
        double r359282 = r359281 / r359279;
        double r359283 = -inf.0;
        bool r359284 = r359282 <= r359283;
        double r359285 = -3.611697980994098e+68;
        bool r359286 = r359282 <= r359285;
        double r359287 = 2.2373313986691395e+36;
        bool r359288 = r359282 <= r359287;
        double r359289 = 8.147235450072648e+229;
        bool r359290 = r359282 <= r359289;
        double r359291 = !r359290;
        bool r359292 = r359288 || r359291;
        double r359293 = !r359292;
        bool r359294 = r359286 || r359293;
        double r359295 = !r359294;
        bool r359296 = r359284 || r359295;
        double r359297 = r359278 / r359279;
        double r359298 = fma(r359297, r359277, r359277);
        double r359299 = r359296 ? r359298 : r359282;
        return r359299;
}

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))