Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 4.0s

Precision: 64

Math

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

↓

\[\left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right) + \mathsf{fma}\left(\log y, y + 0.5, z - y\right) \cdot \left(\left(-1\right) + 1\right)\]

C

double f(double x, double y, double z) {
        double r328365 = x;
        double r328366 = y;
        double r328367 = 0.5;
        double r328368 = r328366 + r328367;
        double r328369 = log(r328366);
        double r328370 = r328368 * r328369;
        double r328371 = r328365 - r328370;
        double r328372 = r328371 + r328366;
        double r328373 = z;
        double r328374 = r328372 - r328373;
        return r328374;
}

↓

double f(double x, double y, double z) {
        double r328375 = x;
        double r328376 = y;
        double r328377 = log(r328376);
        double r328378 = 0.5;
        double r328379 = r328376 + r328378;
        double r328380 = z;
        double r328381 = r328380 - r328376;
        double r328382 = fma(r328377, r328379, r328381);
        double r328383 = r328375 - r328382;
        double r328384 = 1.0;
        double r328385 = -r328384;
        double r328386 = r328385 + r328384;
        double r328387 = r328382 * r328386;
        double r328388 = r328383 + r328387;
        return r328388;
}

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. Using strategy rm

4. Applied fma-udef 0.1

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

5. Applied associate--l+ 0.1

   \[\leadsto x - \color{blue}{\left(\log y \cdot \left(y + 0.5\right) + \left(z - y\right)\right)}\]
6. Using strategy rm

7. Applied add-cube-cbrt 0.9

   \[\leadsto x - \color{blue}{\left(\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\right) \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}}\]

8. Applied add-sqr-sqrt 32.8

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x}} - \left(\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\right) \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\]

9. Applied prod-diff 32.8

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \left(\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}, \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}, \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \left(\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\right)\right)}\]

10. Simplified 0.1

    \[\leadsto \color{blue}{\left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}, \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}, \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \left(\sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)} \cdot \sqrt[3]{\log y \cdot \left(y + 0.5\right) + \left(z - y\right)}\right)\right)\]

11. Simplified 0.1

    \[\leadsto \left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right) + \color{blue}{\mathsf{fma}\left(\log y, y + 0.5, z - y\right) \cdot \left(\left(-1\right) + 1\right)}\]

12. Final simplification 0.1

    \[\leadsto \left(x - \mathsf{fma}\left(\log y, y + 0.5, z - y\right)\right) + \mathsf{fma}\left(\log y, y + 0.5, z - y\right) \cdot \left(\left(-1\right) + 1\right)\]

Reproduce

herbie shell --seed 2020039 +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))