Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 8.2s

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 r357664 = x;
        double r357665 = y;
        double r357666 = 0.5;
        double r357667 = r357665 + r357666;
        double r357668 = log(r357665);
        double r357669 = r357667 * r357668;
        double r357670 = r357664 - r357669;
        double r357671 = r357670 + r357665;
        double r357672 = z;
        double r357673 = r357671 - r357672;
        return r357673;
}

↓

double f(double x, double y, double z) {
        double r357674 = x;
        double r357675 = y;
        double r357676 = log(r357675);
        double r357677 = 0.5;
        double r357678 = r357675 + r357677;
        double r357679 = z;
        double r357680 = fma(r357676, r357678, r357679);
        double r357681 = r357680 - r357675;
        double r357682 = r357674 - r357681;
        return r357682;
}

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 2020062 +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))