Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.2

Time: 6.1s

Precision: 64

Math

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

↓

\[\left(\left(x - \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

C

double f(double x, double y, double z) {
        double r405264 = x;
        double r405265 = y;
        double r405266 = 0.5;
        double r405267 = r405265 + r405266;
        double r405268 = log(r405265);
        double r405269 = r405267 * r405268;
        double r405270 = r405264 - r405269;
        double r405271 = r405270 + r405265;
        double r405272 = z;
        double r405273 = r405271 - r405272;
        return r405273;
}

↓

double f(double x, double y, double z) {
        double r405274 = x;
        double r405275 = y;
        double r405276 = cbrt(r405275);
        double r405277 = r405276 * r405276;
        double r405278 = cbrt(r405277);
        double r405279 = log(r405278);
        double r405280 = 2.0;
        double r405281 = r405279 * r405280;
        double r405282 = 0.5;
        double r405283 = r405275 + r405282;
        double r405284 = r405281 * r405283;
        double r405285 = cbrt(r405276);
        double r405286 = log(r405285);
        double r405287 = r405286 * r405280;
        double r405288 = r405287 * r405283;
        double r405289 = r405284 + r405288;
        double r405290 = log(r405276);
        double r405291 = r405290 * r405283;
        double r405292 = r405289 + r405291;
        double r405293 = r405274 - r405292;
        double r405294 = r405293 + r405275;
        double r405295 = z;
        double r405296 = r405294 - r405295;
        return r405296;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.2

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

Derivation

1. Initial program 0.1

   \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.2

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

5. Applied distribute-lft-in 0.2

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

6. Simplified 0.2

   \[\leadsto \left(\left(x - \left(\color{blue}{\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(y + 0.5\right) \cdot \log \left(\sqrt[3]{y}\right)\right)\right) + y\right) - z\]

7. Simplified 0.2

   \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)}\right)\right) + y\right) - z\]
8. Using strategy rm

9. Applied add-cube-cbrt 0.2

   \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

10. Applied cbrt-prod 0.2

    \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

11. Applied log-prod 0.2

    \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \left(2 \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)}\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

12. Applied distribute-lft-in 0.2

    \[\leadsto \left(\left(x - \left(\left(y + 0.5\right) \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + 2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

13. Applied distribute-lft-in 0.2

    \[\leadsto \left(\left(x - \left(\color{blue}{\left(\left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\right) + \left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right)} + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

14. Simplified 0.2

    \[\leadsto \left(\left(x - \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right)} + \left(y + 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

15. Simplified 0.2

    \[\leadsto \left(\left(x - \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right) + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right)}\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

16. Final simplification 0.2

    \[\leadsto \left(\left(x - \left(\left(\left(\log \left(\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right) + \left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot 2\right) \cdot \left(y + 0.5\right)\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(y + 0.5\right)\right)\right) + y\right) - z\]

Reproduce

herbie shell --seed 2020065 
(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))