Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.3 → 0.9

Time: 13.4s

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 r297531 = x;
        double r297532 = y;
        double r297533 = z;
        double r297534 = r297532 + r297533;
        double r297535 = r297531 * r297534;
        double r297536 = r297535 / r297533;
        return r297536;
}

↓

double f(double x, double y, double z) {
        double r297537 = x;
        double r297538 = y;
        double r297539 = z;
        double r297540 = r297538 + r297539;
        double r297541 = r297537 * r297540;
        double r297542 = r297541 / r297539;
        double r297543 = -inf.0;
        bool r297544 = r297542 <= r297543;
        double r297545 = -6.314719169796708e+133;
        bool r297546 = r297542 <= r297545;
        double r297547 = 260817532774.37283;
        bool r297548 = r297542 <= r297547;
        double r297549 = !r297548;
        double r297550 = 5.286185397003035e+280;
        bool r297551 = r297542 <= r297550;
        bool r297552 = r297549 && r297551;
        bool r297553 = r297546 || r297552;
        double r297554 = !r297553;
        bool r297555 = r297544 || r297554;
        double r297556 = r297540 / r297539;
        double r297557 = r297537 * r297556;
        double r297558 = r297555 ? r297557 : r297542;
        return r297558;
}

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))