Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.2

Time: 22.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\left(x + \left(\left(y - \left(\left(-\log y\right) \cdot \frac{-1}{6}\right) \cdot \left(3 \cdot y + 1.5\right)\right) - \log \left(\sqrt{y}\right) \cdot \left(y + 0.5\right)\right)\right) - z\]

C

double f(double x, double y, double z) {
        double r292075 = x;
        double r292076 = y;
        double r292077 = 0.5;
        double r292078 = r292076 + r292077;
        double r292079 = log(r292076);
        double r292080 = r292078 * r292079;
        double r292081 = r292075 - r292080;
        double r292082 = r292081 + r292076;
        double r292083 = z;
        double r292084 = r292082 - r292083;
        return r292084;
}

↓

double f(double x, double y, double z) {
        double r292085 = x;
        double r292086 = y;
        double r292087 = log(r292086);
        double r292088 = -r292087;
        double r292089 = -0.16666666666666666;
        double r292090 = r292088 * r292089;
        double r292091 = 3.0;
        double r292092 = r292091 * r292086;
        double r292093 = 1.5;
        double r292094 = r292092 + r292093;
        double r292095 = r292090 * r292094;
        double r292096 = r292086 - r292095;
        double r292097 = sqrt(r292086);
        double r292098 = log(r292097);
        double r292099 = 0.5;
        double r292100 = r292086 + r292099;
        double r292101 = r292098 * r292100;
        double r292102 = r292096 - r292101;
        double r292103 = r292085 + r292102;
        double r292104 = z;
        double r292105 = r292103 - r292104;
        return r292105;
}

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 sub-neg 0.1

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

4. Applied associate-+l+ 0.1

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

5. Simplified 0.1

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

7. Applied add-sqr-sqrt 0.1

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

8. Applied log-prod 0.1

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

9. Applied distribute-rgt-in 0.1

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

10. Applied associate--r+ 0.1

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

11. Simplified 0.1

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

13. Applied add-cube-cbrt 0.1

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

14. Applied log-prod 0.1

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

15. Applied distribute-lft-in 0.1

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

16. Simplified 0.1

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

17. Taylor expanded around inf 0.2

    \[\leadsto \left(x + \left(\left(y - \color{blue}{\left(3 \cdot \left(y \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{6}}\right)\right) + 1.5 \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{6}}\right)\right)}\right) - \log \left(\sqrt{y}\right) \cdot \left(y + 0.5\right)\right)\right) - z\]

18. Simplified 0.2

    \[\leadsto \left(x + \left(\left(y - \color{blue}{\left(\left(-\log y\right) \cdot \frac{-1}{6}\right) \cdot \left(3 \cdot y + 1.5\right)}\right) - \log \left(\sqrt{y}\right) \cdot \left(y + 0.5\right)\right)\right) - z\]

19. Final simplification 0.2

    \[\leadsto \left(x + \left(\left(y - \left(\left(-\log y\right) \cdot \frac{-1}{6}\right) \cdot \left(3 \cdot y + 1.5\right)\right) - \log \left(\sqrt{y}\right) \cdot \left(y + 0.5\right)\right)\right) - z\]

Reproduce

herbie shell --seed 2019350 
(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))