Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 10.7s

Precision: 64

Math

\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

↓

\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]

C

double f(double x, double y, double z) {
        double r439335 = x;
        double r439336 = y;
        double r439337 = 0.5;
        double r439338 = r439336 + r439337;
        double r439339 = log(r439336);
        double r439340 = r439338 * r439339;
        double r439341 = r439335 - r439340;
        double r439342 = r439341 + r439336;
        double r439343 = z;
        double r439344 = r439342 - r439343;
        return r439344;
}

↓

double f(double x, double y, double z) {
        double r439345 = x;
        double r439346 = y;
        double r439347 = log(r439346);
        double r439348 = 0.5;
        double r439349 = r439346 + r439348;
        double r439350 = z;
        double r439351 = fma(r439347, r439349, r439350);
        double r439352 = r439351 - r439346;
        double r439353 = r439345 - r439352;
        return r439353;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y\]

Derivation

1. Initial program 0.1

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]

2. Simplified 0.1

   \[\leadsto \color{blue}{x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)}\]

3. Final simplification 0.1

   \[\leadsto x - \left(\mathsf{fma}\left(\log y, y + 0.5, z\right) - y\right)\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))