Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 6.2 → 6.2

Time: 7.9s

Precision: 64

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

↓

\[\left(\left(x - 0.5\right) \cdot \log \left(e^{\log x \cdot \frac{1}{3}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

C

double f(double x, double y, double z) {
        double r494883 = x;
        double r494884 = 0.5;
        double r494885 = r494883 - r494884;
        double r494886 = log(r494883);
        double r494887 = r494885 * r494886;
        double r494888 = r494887 - r494883;
        double r494889 = 0.91893853320467;
        double r494890 = r494888 + r494889;
        double r494891 = y;
        double r494892 = 0.0007936500793651;
        double r494893 = r494891 + r494892;
        double r494894 = z;
        double r494895 = r494893 * r494894;
        double r494896 = 0.0027777777777778;
        double r494897 = r494895 - r494896;
        double r494898 = r494897 * r494894;
        double r494899 = 0.083333333333333;
        double r494900 = r494898 + r494899;
        double r494901 = r494900 / r494883;
        double r494902 = r494890 + r494901;
        return r494902;
}

↓

double f(double x, double y, double z) {
        double r494903 = x;
        double r494904 = 0.5;
        double r494905 = r494903 - r494904;
        double r494906 = log(r494903);
        double r494907 = 0.3333333333333333;
        double r494908 = r494906 * r494907;
        double r494909 = exp(r494908);
        double r494910 = cbrt(r494903);
        double r494911 = r494909 * r494910;
        double r494912 = log(r494911);
        double r494913 = r494905 * r494912;
        double r494914 = log(r494910);
        double r494915 = r494914 * r494905;
        double r494916 = r494915 - r494903;
        double r494917 = 0.91893853320467;
        double r494918 = r494916 + r494917;
        double r494919 = r494913 + r494918;
        double r494920 = y;
        double r494921 = 0.0007936500793651;
        double r494922 = r494920 + r494921;
        double r494923 = z;
        double r494924 = r494922 * r494923;
        double r494925 = 0.0027777777777778;
        double r494926 = r494924 - r494925;
        double r494927 = r494926 * r494923;
        double r494928 = 0.083333333333333;
        double r494929 = r494927 + r494928;
        double r494930 = r494929 / r494903;
        double r494931 = r494919 + r494930;
        return r494931;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	1.3
Herbie	6.2

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

1. Initial program 6.2

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
2. Using strategy rm

3. Applied add-cube-cbrt 6.2

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

4. Applied log-prod 6.2

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

5. Applied distribute-lft-in 6.2

   \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

6. Applied associate--l+ 6.2

   \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right)\right)} + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

7. Applied associate-+l+ 6.2

   \[\leadsto \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) - x\right) + 0.91893853320467001\right)\right)} + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

8. Simplified 6.2

   \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \color{blue}{\left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)}\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

9. Taylor expanded around inf 6.2

   \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\color{blue}{{\left(\frac{1}{x}\right)}^{\frac{-1}{3}}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
10. Using strategy rm

11. Applied add-exp-log 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{\frac{-1}{3}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

12. Applied rec-exp 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left({\color{blue}{\left(e^{-\log x}\right)}}^{\frac{-1}{3}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

13. Applied pow-exp 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\color{blue}{e^{\left(-\log x\right) \cdot \frac{-1}{3}}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

14. Simplified 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(e^{\color{blue}{\log x \cdot \frac{1}{3}}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

15. Final simplification 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(e^{\log x \cdot \frac{1}{3}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))