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

Average Error: 6.1 → 4.3

Time: 7.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 217404447530713645000:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot 1\right) \cdot \left(1 \cdot x + \left(-0.5\right)\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot 1\right) \cdot \left(1 \cdot x + \left(-0.5\right)\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, 1, 0.5\right), \log x, \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r396761 = x;
        double r396762 = 0.5;
        double r396763 = r396761 - r396762;
        double r396764 = log(r396761);
        double r396765 = r396763 * r396764;
        double r396766 = r396765 - r396761;
        double r396767 = 0.91893853320467;
        double r396768 = r396766 + r396767;
        double r396769 = y;
        double r396770 = 0.0007936500793651;
        double r396771 = r396769 + r396770;
        double r396772 = z;
        double r396773 = r396771 * r396772;
        double r396774 = 0.0027777777777778;
        double r396775 = r396773 - r396774;
        double r396776 = r396775 * r396772;
        double r396777 = 0.083333333333333;
        double r396778 = r396776 + r396777;
        double r396779 = r396778 / r396761;
        double r396780 = r396768 + r396779;
        return r396780;
}

↓

double f(double x, double y, double z) {
        double r396781 = x;
        double r396782 = 2.1740444753071365e+20;
        bool r396783 = r396781 <= r396782;
        double r396784 = 2.0;
        double r396785 = cbrt(r396781);
        double r396786 = log(r396785);
        double r396787 = r396784 * r396786;
        double r396788 = 1.0;
        double r396789 = r396787 * r396788;
        double r396790 = r396788 * r396781;
        double r396791 = 0.5;
        double r396792 = -r396791;
        double r396793 = r396790 + r396792;
        double r396794 = r396789 * r396793;
        double r396795 = r396786 * r396788;
        double r396796 = r396795 * r396793;
        double r396797 = r396794 + r396796;
        double r396798 = fma(r396792, r396788, r396791);
        double r396799 = log(r396781);
        double r396800 = y;
        double r396801 = 0.0007936500793651;
        double r396802 = r396800 + r396801;
        double r396803 = z;
        double r396804 = r396802 * r396803;
        double r396805 = 0.0027777777777778;
        double r396806 = r396804 - r396805;
        double r396807 = r396806 * r396803;
        double r396808 = 0.083333333333333;
        double r396809 = r396807 + r396808;
        double r396810 = r396809 / r396781;
        double r396811 = 0.91893853320467;
        double r396812 = r396781 - r396811;
        double r396813 = r396810 - r396812;
        double r396814 = fma(r396798, r396799, r396813);
        double r396815 = r396797 + r396814;
        double r396816 = pow(r396803, r396784);
        double r396817 = r396816 / r396781;
        double r396818 = r396801 * r396817;
        double r396819 = r396788 / r396781;
        double r396820 = log(r396819);
        double r396821 = fma(r396820, r396781, r396781);
        double r396822 = r396818 - r396821;
        double r396823 = fma(r396817, r396800, r396822);
        double r396824 = r396783 ? r396815 : r396823;
        return r396824;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.1
Target	1.2
Herbie	4.3

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

Derivation

1. Split input into 2 regimes

2. if x < 2.1740444753071365e+20

   1. Initial program 0.2

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

   2. Simplified 0.2

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

   4. Applied fma-udef 0.2

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

   6. Applied *-un-lft-identity 0.2

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

   7. Applied add-sqr-sqrt 0.2

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

   8. Applied prod-diff 0.2

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

   9. Applied distribute-rgt-in 0.2

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

   10. Applied associate-+l+ 0.2

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

   11. Simplified 0.2

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

   13. Applied add-cube-cbrt 0.2

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

   14. Applied log-prod 0.2

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

   15. Applied distribute-lft-in 0.2

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

   16. Simplified 0.2

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

   17. Simplified 0.2

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

   if 2.1740444753071365e+20 < x

   1. Initial program 10.8

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

   2. Simplified 10.7

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

   3. Taylor expanded around inf 10.9

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

   4. Simplified 7.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 217404447530713645000:\\ \;\;\;\;\left(\left(\left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) \cdot 1\right) \cdot \left(1 \cdot x + \left(-0.5\right)\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot 1\right) \cdot \left(1 \cdot x + \left(-0.5\right)\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(-0.5, 1, 0.5\right), \log x, \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x} - \left(x - 0.91893853320467001\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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)))