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

Average Error: 5.9 → 5.9

Time: 9.0s

Precision: 64

Math

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

↓

\[\left(\left(x - 0.5\right) \cdot \log \left(e^{\log \left({x}^{\frac{1}{3}}\right)} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

C

double f(double x, double y, double z) {
        double r442762 = x;
        double r442763 = 0.5;
        double r442764 = r442762 - r442763;
        double r442765 = log(r442762);
        double r442766 = r442764 * r442765;
        double r442767 = r442766 - r442762;
        double r442768 = 0.91893853320467;
        double r442769 = r442767 + r442768;
        double r442770 = y;
        double r442771 = 0.0007936500793651;
        double r442772 = r442770 + r442771;
        double r442773 = z;
        double r442774 = r442772 * r442773;
        double r442775 = 0.0027777777777778;
        double r442776 = r442774 - r442775;
        double r442777 = r442776 * r442773;
        double r442778 = 0.083333333333333;
        double r442779 = r442777 + r442778;
        double r442780 = r442779 / r442762;
        double r442781 = r442769 + r442780;
        return r442781;
}

↓

double f(double x, double y, double z) {
        double r442782 = x;
        double r442783 = 0.5;
        double r442784 = r442782 - r442783;
        double r442785 = 0.3333333333333333;
        double r442786 = pow(r442782, r442785);
        double r442787 = log(r442786);
        double r442788 = exp(r442787);
        double r442789 = cbrt(r442782);
        double r442790 = r442788 * r442789;
        double r442791 = log(r442790);
        double r442792 = r442784 * r442791;
        double r442793 = log(r442789);
        double r442794 = r442793 * r442784;
        double r442795 = r442794 - r442782;
        double r442796 = 0.91893853320467;
        double r442797 = r442795 + r442796;
        double r442798 = r442792 + r442797;
        double r442799 = y;
        double r442800 = 0.0007936500793651;
        double r442801 = r442799 + r442800;
        double r442802 = z;
        double r442803 = r442801 * r442802;
        double r442804 = 0.0027777777777778;
        double r442805 = r442803 - r442804;
        double r442806 = r442805 * r442802;
        double r442807 = 0.083333333333333;
        double r442808 = r442806 + r442807;
        double r442809 = r442808 / r442782;
        double r442810 = r442798 + r442809;
        return r442810;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.0
Herbie	5.9

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

Derivation

1. Initial program 5.9

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

3. Applied add-cube-cbrt 5.9

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

4. Applied log-prod 6.0

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

5. Applied distribute-lft-in 6.0

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

6. Applied associate--l+ 6.0

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

7. Applied associate-+l+ 6.0

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

8. Simplified 6.0

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

10. Applied pow1/3 5.9

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\color{blue}{{x}^{\frac{1}{3}}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
11. Using strategy rm

12. Applied add-exp-log 5.9

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left({\color{blue}{\left(e^{\log x}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

13. Applied pow-exp 6.0

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\color{blue}{e^{\log x \cdot \frac{1}{3}}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

14. Simplified 5.9

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(e^{\color{blue}{\log \left({x}^{\frac{1}{3}}\right)}} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

15. Final simplification 5.9

    \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(e^{\log \left({x}^{\frac{1}{3}}\right)} \cdot \sqrt[3]{x}\right) + \left(\left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

Reproduce

herbie shell --seed 2020002 
(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)))