Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.1 → 1.7

Time: 10.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -66400270581518072926458347520:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 6.946459264278644888850392293793207714708 \cdot 10^{-106}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r25936138 = x;
        double r25936139 = y;
        double r25936140 = z;
        double r25936141 = r25936139 + r25936140;
        double r25936142 = r25936138 * r25936141;
        double r25936143 = r25936142 / r25936140;
        return r25936143;
}

↓

double f(double x, double y, double z) {
        double r25936144 = z;
        double r25936145 = -6.640027058151807e+28;
        bool r25936146 = r25936144 <= r25936145;
        double r25936147 = x;
        double r25936148 = y;
        double r25936149 = r25936144 + r25936148;
        double r25936150 = r25936144 / r25936149;
        double r25936151 = r25936147 / r25936150;
        double r25936152 = 6.946459264278645e-106;
        bool r25936153 = r25936144 <= r25936152;
        double r25936154 = r25936147 * r25936148;
        double r25936155 = r25936154 / r25936144;
        double r25936156 = r25936147 + r25936155;
        double r25936157 = r25936153 ? r25936156 : r25936151;
        double r25936158 = r25936146 ? r25936151 : r25936157;
        return r25936158;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.1
Target	3.1
Herbie	1.7

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

Derivation

1. Split input into 2 regimes

2. if z < -6.640027058151807e+28 or 6.946459264278645e-106 < z

   1. Initial program 15.4

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

   3. Applied associate-/l* 0.4

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

   if -6.640027058151807e+28 < z < 6.946459264278645e-106

   1. Initial program 6.7

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

   2. Taylor expanded around 0 3.9

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -66400270581518072926458347520:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \mathbf{elif}\;z \le 6.946459264278644888850392293793207714708 \cdot 10^{-106}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{z + y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

  :herbie-target
  (/ x (/ z (+ y z)))

  (/ (* x (+ y z)) z))