Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Average Error: 12.0 → 2.1

Time: 12.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -2.869128326247180131055491464033383801505 \cdot 10^{99} \lor \neg \left(z \le -1.442321906530695273017718499006101704898 \cdot 10^{-263}\right) \land \left(z \le 3.670705713904214514938236895518577498891 \cdot 10^{-228} \lor \neg \left(z \le 1.013639971791185277516860999161516023027 \cdot 10^{-81}\right)\right):\\ \;\;\;\;x \cdot \frac{z + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r286228 = x;
        double r286229 = y;
        double r286230 = z;
        double r286231 = r286229 + r286230;
        double r286232 = r286228 * r286231;
        double r286233 = r286232 / r286230;
        return r286233;
}

↓

double f(double x, double y, double z) {
        double r286234 = z;
        double r286235 = -2.86912832624718e+99;
        bool r286236 = r286234 <= r286235;
        double r286237 = -1.4423219065306953e-263;
        bool r286238 = r286234 <= r286237;
        double r286239 = !r286238;
        double r286240 = 3.6707057139042145e-228;
        bool r286241 = r286234 <= r286240;
        double r286242 = 1.0136399717911853e-81;
        bool r286243 = r286234 <= r286242;
        double r286244 = !r286243;
        bool r286245 = r286241 || r286244;
        bool r286246 = r286239 && r286245;
        bool r286247 = r286236 || r286246;
        double r286248 = x;
        double r286249 = y;
        double r286250 = r286234 + r286249;
        double r286251 = r286250 / r286234;
        double r286252 = r286248 * r286251;
        double r286253 = r286248 * r286249;
        double r286254 = r286253 / r286234;
        double r286255 = r286248 + r286254;
        double r286256 = r286247 ? r286252 : r286255;
        return r286256;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	12.0
Target	2.9
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if z < -2.86912832624718e+99 or -1.4423219065306953e-263 < z < 3.6707057139042145e-228 or 1.0136399717911853e-81 < z

   1. Initial program 15.8

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

   3. Applied *-un-lft-identity 15.8

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

   4. Applied times-frac 1.7

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

   5. Simplified 1.7

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

   6. Simplified 1.7

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

   if -2.86912832624718e+99 < z < -1.4423219065306953e-263 or 3.6707057139042145e-228 < z < 1.0136399717911853e-81

   1. Initial program 5.8

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

   2. Taylor expanded around 0 2.6

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

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.869128326247180131055491464033383801505 \cdot 10^{99} \lor \neg \left(z \le -1.442321906530695273017718499006101704898 \cdot 10^{-263}\right) \land \left(z \le 3.670705713904214514938236895518577498891 \cdot 10^{-228} \lor \neg \left(z \le 1.013639971791185277516860999161516023027 \cdot 10^{-81}\right)\right):\\ \;\;\;\;x \cdot \frac{z + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{x \cdot y}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))