Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 13.5s

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 r503715 = x;
        double r503716 = y;
        double r503717 = 0.5;
        double r503718 = r503716 + r503717;
        double r503719 = log(r503716);
        double r503720 = r503718 * r503719;
        double r503721 = r503715 - r503720;
        double r503722 = r503721 + r503716;
        double r503723 = z;
        double r503724 = r503722 - r503723;
        return r503724;
}

↓

double f(double x, double y, double z) {
        double r503725 = x;
        double r503726 = y;
        double r503727 = log(r503726);
        double r503728 = 0.5;
        double r503729 = r503726 + r503728;
        double r503730 = z;
        double r503731 = fma(r503727, r503729, r503730);
        double r503732 = r503731 - r503726;
        double r503733 = r503725 - r503732;
        return r503733;
}

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 2020034 +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))