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

Average Error: 6.0 → 3.9

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 561024664316101460000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\left(1 \cdot {x}^{\frac{1}{3}}\right) \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r421655 = x;
        double r421656 = 0.5;
        double r421657 = r421655 - r421656;
        double r421658 = log(r421655);
        double r421659 = r421657 * r421658;
        double r421660 = r421659 - r421655;
        double r421661 = 0.91893853320467;
        double r421662 = r421660 + r421661;
        double r421663 = y;
        double r421664 = 0.0007936500793651;
        double r421665 = r421663 + r421664;
        double r421666 = z;
        double r421667 = r421665 * r421666;
        double r421668 = 0.0027777777777778;
        double r421669 = r421667 - r421668;
        double r421670 = r421669 * r421666;
        double r421671 = 0.083333333333333;
        double r421672 = r421670 + r421671;
        double r421673 = r421672 / r421655;
        double r421674 = r421662 + r421673;
        return r421674;
}

↓

double f(double x, double y, double z) {
        double r421675 = x;
        double r421676 = 5.6102466431610146e+20;
        bool r421677 = r421675 <= r421676;
        double r421678 = 0.5;
        double r421679 = r421675 - r421678;
        double r421680 = 1.0;
        double r421681 = 0.3333333333333333;
        double r421682 = pow(r421675, r421681);
        double r421683 = r421680 * r421682;
        double r421684 = cbrt(r421675);
        double r421685 = r421683 * r421684;
        double r421686 = log(r421685);
        double r421687 = r421679 * r421686;
        double r421688 = log(r421684);
        double r421689 = r421688 * r421679;
        double r421690 = r421689 - r421675;
        double r421691 = r421687 + r421690;
        double r421692 = 0.91893853320467;
        double r421693 = r421691 + r421692;
        double r421694 = y;
        double r421695 = 0.0007936500793651;
        double r421696 = r421694 + r421695;
        double r421697 = z;
        double r421698 = r421696 * r421697;
        double r421699 = 0.0027777777777778;
        double r421700 = r421698 - r421699;
        double r421701 = r421700 * r421697;
        double r421702 = 0.083333333333333;
        double r421703 = r421701 + r421702;
        double r421704 = r421703 / r421675;
        double r421705 = r421693 + r421704;
        double r421706 = 2.0;
        double r421707 = pow(r421697, r421706);
        double r421708 = r421707 / r421675;
        double r421709 = r421695 * r421708;
        double r421710 = r421680 / r421675;
        double r421711 = log(r421710);
        double r421712 = fma(r421711, r421675, r421675);
        double r421713 = r421709 - r421712;
        double r421714 = fma(r421708, r421694, r421713);
        double r421715 = r421677 ? r421705 : r421714;
        return r421715;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.3
Herbie	3.9

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

Derivation

1. Split input into 2 regimes

2. if x < 5.6102466431610146e+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. Using strategy rm

   3. Applied add-cube-cbrt 0.2

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

   4. Applied log-prod 0.2

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

   5. Applied distribute-lft-in 0.2

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

   6. Applied associate--l+ 0.2

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

   7. Simplified 0.2

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

   9. Applied *-un-lft-identity 0.2

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

   10. Applied cbrt-prod 0.2

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

   11. Simplified 0.2

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

   12. Simplified 0.2

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

   if 5.6102466431610146e+20 < x

   1. Initial program 10.4

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

   2. Simplified 10.4

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

   3. Taylor expanded around inf 10.5

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

   4. Simplified 6.8

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

4. Final simplification 3.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 561024664316101460000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\left(1 \cdot {x}^{\frac{1}{3}}\right) \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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)))