Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.1

Time: 54.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r313149 = x;
        double r313150 = y;
        double r313151 = 0.5;
        double r313152 = r313150 + r313151;
        double r313153 = log(r313150);
        double r313154 = r313152 * r313153;
        double r313155 = r313149 - r313154;
        double r313156 = r313155 + r313150;
        double r313157 = z;
        double r313158 = r313156 - r313157;
        return r313158;
}

↓

double f(double x, double y, double z) {
        double r313159 = y;
        double r313160 = x;
        double r313161 = r313159 + r313160;
        double r313162 = log(r313159);
        double r313163 = 0.5;
        double r313164 = r313159 + r313163;
        double r313165 = z;
        double r313166 = fma(r313162, r313164, r313165);
        double r313167 = r313161 - r313166;
        double r313168 = 0.0;
        double r313169 = fma(r313164, r313162, r313165);
        double r313170 = r313168 * r313169;
        double r313171 = r313167 + r313170;
        return r313171;
}

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. Using strategy rm

10. Applied *-un-lft-identity 0.5

    \[\leadsto \left({\left(\sqrt[3]{\color{blue}{1 \cdot \left(y + x\right)}}\right)}^{3} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

11. Applied cbrt-prod 0.5

    \[\leadsto \left({\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{y + x}\right)}}^{3} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

12. Applied unpow-prod-down 0.5

    \[\leadsto \left(\color{blue}{{\left(\sqrt[3]{1}\right)}^{3} \cdot {\left(\sqrt[3]{y + x}\right)}^{3}} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

13. Simplified 0.5

    \[\leadsto \left(\color{blue}{1} \cdot {\left(\sqrt[3]{y + x}\right)}^{3} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

14. Simplified 0.1

    \[\leadsto \left(1 \cdot \color{blue}{\left(x + y\right)} - \mathsf{fma}\left(\log y, y + 0.5, z\right)\right) + 0 \cdot \mathsf{fma}\left(y + 0.5, \log y, z\right)\]

15. Final simplification 0.1

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

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))