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

Average Error: 6.3 → 6.4

Time: 19.6s

Precision: 64

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

Math

\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

↓

\[\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right) - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

C

double f(double x, double y, double z) {
        double r674772 = x;
        double r674773 = 0.5;
        double r674774 = r674772 - r674773;
        double r674775 = log(r674772);
        double r674776 = r674774 * r674775;
        double r674777 = r674776 - r674772;
        double r674778 = 0.91893853320467;
        double r674779 = r674777 + r674778;
        double r674780 = y;
        double r674781 = 0.0007936500793651;
        double r674782 = r674780 + r674781;
        double r674783 = z;
        double r674784 = r674782 * r674783;
        double r674785 = 0.0027777777777778;
        double r674786 = r674784 - r674785;
        double r674787 = r674786 * r674783;
        double r674788 = 0.083333333333333;
        double r674789 = r674787 + r674788;
        double r674790 = r674789 / r674772;
        double r674791 = r674779 + r674790;
        return r674791;
}

↓

double f(double x, double y, double z) {
        double r674792 = x;
        double r674793 = 0.5;
        double r674794 = r674792 - r674793;
        double r674795 = sqrt(r674792);
        double r674796 = log(r674795);
        double r674797 = sqrt(r674795);
        double r674798 = log(r674797);
        double r674799 = r674796 + r674798;
        double r674800 = r674794 * r674799;
        double r674801 = 2.0;
        double r674802 = cbrt(r674797);
        double r674803 = log(r674802);
        double r674804 = r674801 * r674803;
        double r674805 = r674794 * r674804;
        double r674806 = r674794 * r674803;
        double r674807 = r674805 + r674806;
        double r674808 = r674807 - r674792;
        double r674809 = r674800 + r674808;
        double r674810 = 0.91893853320467;
        double r674811 = r674809 + r674810;
        double r674812 = y;
        double r674813 = 0.0007936500793651;
        double r674814 = r674812 + r674813;
        double r674815 = z;
        double r674816 = r674814 * r674815;
        double r674817 = 0.0027777777777778;
        double r674818 = r674816 - r674817;
        double r674819 = r674818 * r674815;
        double r674820 = 0.083333333333333;
        double r674821 = r674819 + r674820;
        double r674822 = r674821 / r674792;
        double r674823 = r674811 + r674822;
        return r674823;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.3
Target	1.2
Herbie	6.4

\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

1. Initial program 6.3

   \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
2. Using strategy rm

3. Applied add-sqr-sqrt 6.3

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

4. Applied log-prod 6.3

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{x}\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

5. Applied distribute-lft-in 6.3

   \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

6. Applied associate--l+ 6.4

   \[\leadsto \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)\right)} + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
7. Using strategy rm

8. Applied add-sqr-sqrt 6.4

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

9. Applied sqrt-prod 6.4

   \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\sqrt{\sqrt{x}} \cdot \sqrt{\sqrt{x}}\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

10. Applied log-prod 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt{\sqrt{x}}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

11. Applied distribute-lft-in 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right)\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

12. Applied associate--l+ 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right) - x\right)\right)}\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

13. Applied associate-+r+ 6.4

    \[\leadsto \left(\color{blue}{\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right) - x\right)\right)} + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

14. Simplified 6.4

    \[\leadsto \left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right)} + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{\sqrt{x}}\right) - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
15. Using strategy rm

16. Applied add-cube-cbrt 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt{\sqrt{x}}} \cdot \sqrt[3]{\sqrt{\sqrt{x}}}\right) \cdot \sqrt[3]{\sqrt{\sqrt{x}}}\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

17. Applied log-prod 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{\sqrt{x}}} \cdot \sqrt[3]{\sqrt{\sqrt{x}}}\right) + \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

18. Applied distribute-lft-in 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}} \cdot \sqrt[3]{\sqrt{\sqrt{x}}}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right)} - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

19. Simplified 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right) - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

20. Final simplification 6.4

    \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt{x}\right) + \log \left(\sqrt{\sqrt{x}}\right)\right) + \left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt{\sqrt{x}}}\right)\right) - x\right)\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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)))