Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 9.2s

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 r336589 = x;
        double r336590 = y;
        double r336591 = 0.5;
        double r336592 = r336590 + r336591;
        double r336593 = log(r336590);
        double r336594 = r336592 * r336593;
        double r336595 = r336589 - r336594;
        double r336596 = r336595 + r336590;
        double r336597 = z;
        double r336598 = r336596 - r336597;
        return r336598;
}

↓

double f(double x, double y, double z) {
        double r336599 = x;
        double r336600 = y;
        double r336601 = log(r336600);
        double r336602 = 0.5;
        double r336603 = r336600 + r336602;
        double r336604 = z;
        double r336605 = fma(r336601, r336603, r336604);
        double r336606 = r336605 - r336600;
        double r336607 = r336599 - r336606;
        return r336607;
}

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 2020025 +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))