Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 53.1s

Precision: 64

Math

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

↓

\[\left(\left(x + y\right) - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

C

double f(double x, double y, double z) {
        double r263801 = x;
        double r263802 = y;
        double r263803 = 0.5;
        double r263804 = r263802 + r263803;
        double r263805 = log(r263802);
        double r263806 = r263804 * r263805;
        double r263807 = r263801 - r263806;
        double r263808 = r263807 + r263802;
        double r263809 = z;
        double r263810 = r263808 - r263809;
        return r263810;
}

↓

double f(double x, double y, double z) {
        double r263811 = x;
        double r263812 = y;
        double r263813 = r263811 + r263812;
        double r263814 = log(r263812);
        double r263815 = 0.5;
        double r263816 = r263812 + r263815;
        double r263817 = z;
        double r263818 = fma(r263814, r263816, r263817);
        double r263819 = r263813 - r263818;
        double r263820 = 0.0;
        double r263821 = fma(r263816, r263814, r263817);
        double r263822 = r263820 * r263821;
        double r263823 = r263819 + r263822;
        return r263823;
}

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

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

4. Applied add-cube-cbrt 0.9

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

5. Applied add-cube-cbrt 1.3

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

6. Applied prod-diff 1.3

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

7. Simplified 0.5

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

8. Simplified 0.5

   \[\leadsto \left({\left(\sqrt[3]{y + x}\right)}^{3} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + \color{blue}{0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)}\]

9. Taylor expanded around 0 0.1

   \[\leadsto \left(\color{blue}{\left(x + y\right)} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

10. Final simplification 0.1

    \[\leadsto \left(\left(x + y\right) - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

Reproduce

herbie shell --seed 2019303 +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))