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

Average Error: 6.2 → 6.2

Time: 8.2s

Precision: 64

Math

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

↓

\[\left(\left(x - 0.5\right) \cdot \log \left(\left({\left({\left({x}^{\frac{1}{3}}\right)}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

C

double f(double x, double y, double z) {
        double r562756 = x;
        double r562757 = 0.5;
        double r562758 = r562756 - r562757;
        double r562759 = log(r562756);
        double r562760 = r562758 * r562759;
        double r562761 = r562760 - r562756;
        double r562762 = 0.91893853320467;
        double r562763 = r562761 + r562762;
        double r562764 = y;
        double r562765 = 0.0007936500793651;
        double r562766 = r562764 + r562765;
        double r562767 = z;
        double r562768 = r562766 * r562767;
        double r562769 = 0.0027777777777778;
        double r562770 = r562768 - r562769;
        double r562771 = r562770 * r562767;
        double r562772 = 0.083333333333333;
        double r562773 = r562771 + r562772;
        double r562774 = r562773 / r562756;
        double r562775 = r562763 + r562774;
        return r562775;
}

↓

double f(double x, double y, double z) {
        double r562776 = x;
        double r562777 = 0.5;
        double r562778 = r562776 - r562777;
        double r562779 = 0.3333333333333333;
        double r562780 = pow(r562776, r562779);
        double r562781 = 0.6666666666666666;
        double r562782 = cbrt(r562781);
        double r562783 = r562782 * r562782;
        double r562784 = pow(r562780, r562783);
        double r562785 = pow(r562784, r562782);
        double r562786 = cbrt(r562776);
        double r562787 = pow(r562786, r562779);
        double r562788 = r562785 * r562787;
        double r562789 = 1.0;
        double r562790 = r562789 * r562780;
        double r562791 = r562788 * r562790;
        double r562792 = log(r562791);
        double r562793 = r562778 * r562792;
        double r562794 = log(r562786);
        double r562795 = r562794 * r562778;
        double r562796 = r562795 - r562776;
        double r562797 = 0.91893853320467;
        double r562798 = r562796 + r562797;
        double r562799 = r562793 + r562798;
        double r562800 = y;
        double r562801 = 0.0007936500793651;
        double r562802 = r562800 + r562801;
        double r562803 = z;
        double r562804 = r562802 * r562803;
        double r562805 = 0.0027777777777778;
        double r562806 = r562804 - r562805;
        double r562807 = r562806 * r562803;
        double r562808 = 0.083333333333333;
        double r562809 = r562807 + r562808;
        double r562810 = r562809 / r562776;
        double r562811 = r562799 + r562810;
        return r562811;
}

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. Using strategy rm

10. Applied *-un-lft-identity 6.2

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

11. Applied cbrt-prod 6.2

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

12. Simplified 6.2

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

13. Simplified 6.2

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

15. Applied add-cube-cbrt 6.2

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

16. Simplified 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\left(\color{blue}{{\left({x}^{\frac{1}{3}}\right)}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

17. Simplified 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\left({\left({x}^{\frac{1}{3}}\right)}^{\frac{2}{3}} \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
18. Using strategy rm

19. Applied add-cube-cbrt 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\left({\left({x}^{\frac{1}{3}}\right)}^{\color{blue}{\left(\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right) \cdot \sqrt[3]{\frac{2}{3}}\right)}} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

20. Applied pow-unpow 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\left(\color{blue}{{\left({\left({x}^{\frac{1}{3}}\right)}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)}} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

21. Final simplification 6.2

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\left({\left({\left({x}^{\frac{1}{3}}\right)}^{\left(\sqrt[3]{\frac{2}{3}} \cdot \sqrt[3]{\frac{2}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{2}{3}}\right)} \cdot {\left(\sqrt[3]{x}\right)}^{\frac{1}{3}}\right) \cdot \left(1 \cdot {x}^{\frac{1}{3}}\right)\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.91893853320467001\right)\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

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)))