Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 0.9

Time: 13.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -6.314719169796707913625869261166949931185 \cdot 10^{133} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 260817532774.372833251953125\right) \land \frac{x \cdot \left(y + z\right)}{z} \le 5.286185397003034641152716277040577167977 \cdot 10^{280}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r278498 = x;
        double r278499 = y;
        double r278500 = z;
        double r278501 = r278499 + r278500;
        double r278502 = r278498 * r278501;
        double r278503 = r278502 / r278500;
        return r278503;
}

↓

double f(double x, double y, double z) {
        double r278504 = x;
        double r278505 = y;
        double r278506 = z;
        double r278507 = r278505 + r278506;
        double r278508 = r278504 * r278507;
        double r278509 = r278508 / r278506;
        double r278510 = -inf.0;
        bool r278511 = r278509 <= r278510;
        double r278512 = -6.314719169796708e+133;
        bool r278513 = r278509 <= r278512;
        double r278514 = 260817532774.37283;
        bool r278515 = r278509 <= r278514;
        double r278516 = !r278515;
        double r278517 = 5.286185397003035e+280;
        bool r278518 = r278509 <= r278517;
        bool r278519 = r278516 && r278518;
        bool r278520 = r278513 || r278519;
        double r278521 = !r278520;
        bool r278522 = r278511 || r278521;
        double r278523 = r278507 / r278506;
        double r278524 = r278504 * r278523;
        double r278525 = r278522 ? r278524 : r278509;
        return r278525;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.3
Target	3.2
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if (/ (* x (+ y z)) z) < -inf.0 or -6.314719169796708e+133 < (/ (* x (+ y z)) z) < 260817532774.37283 or 5.286185397003035e+280 < (/ (* x (+ y z)) z)

   1. Initial program 17.4

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

   3. Applied *-un-lft-identity 17.4

      \[\leadsto \frac{x \cdot \left(y + z\right)}{\color{blue}{1 \cdot z}}\]

   4. Applied times-frac 1.2

      \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{y + z}{z}}\]

   5. Simplified 1.2

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

   if -inf.0 < (/ (* x (+ y z)) z) < -6.314719169796708e+133 or 260817532774.37283 < (/ (* x (+ y z)) z) < 5.286185397003035e+280

   1. Initial program 0.2

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} = -\infty \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le -6.314719169796707913625869261166949931185 \cdot 10^{133} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \le 260817532774.372833251953125\right) \land \frac{x \cdot \left(y + z\right)}{z} \le 5.286185397003034641152716277040577167977 \cdot 10^{280}\right):\\ \;\;\;\;x \cdot \frac{y + z}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))