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

Average Error: 5.8 → 0.3

Time: 27.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 21506604914.314178466796875:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt[3]{\left(x - 0.5\right) \cdot \log x}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right) + \left(\left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r18239982 = x;
        double r18239983 = 0.5;
        double r18239984 = r18239982 - r18239983;
        double r18239985 = log(r18239982);
        double r18239986 = r18239984 * r18239985;
        double r18239987 = r18239986 - r18239982;
        double r18239988 = 0.91893853320467;
        double r18239989 = r18239987 + r18239988;
        double r18239990 = y;
        double r18239991 = 0.0007936500793651;
        double r18239992 = r18239990 + r18239991;
        double r18239993 = z;
        double r18239994 = r18239992 * r18239993;
        double r18239995 = 0.0027777777777778;
        double r18239996 = r18239994 - r18239995;
        double r18239997 = r18239996 * r18239993;
        double r18239998 = 0.083333333333333;
        double r18239999 = r18239997 + r18239998;
        double r18240000 = r18239999 / r18239982;
        double r18240001 = r18239989 + r18240000;
        return r18240001;
}

↓

double f(double x, double y, double z) {
        double r18240002 = x;
        double r18240003 = 21506604914.31418;
        bool r18240004 = r18240002 <= r18240003;
        double r18240005 = z;
        double r18240006 = y;
        double r18240007 = 0.0007936500793651;
        double r18240008 = r18240006 + r18240007;
        double r18240009 = r18240005 * r18240008;
        double r18240010 = 0.0027777777777778;
        double r18240011 = r18240009 - r18240010;
        double r18240012 = r18240005 * r18240011;
        double r18240013 = 0.083333333333333;
        double r18240014 = r18240012 + r18240013;
        double r18240015 = r18240014 / r18240002;
        double r18240016 = 0.91893853320467;
        double r18240017 = 0.5;
        double r18240018 = r18240002 - r18240017;
        double r18240019 = log(r18240002);
        double r18240020 = r18240018 * r18240019;
        double r18240021 = cbrt(r18240020);
        double r18240022 = r18240021 * r18240021;
        double r18240023 = r18240021 * r18240022;
        double r18240024 = r18240023 - r18240002;
        double r18240025 = r18240016 + r18240024;
        double r18240026 = r18240015 + r18240025;
        double r18240027 = r18240002 / r18240005;
        double r18240028 = r18240005 / r18240027;
        double r18240029 = r18240028 * r18240008;
        double r18240030 = r18240005 / r18240002;
        double r18240031 = r18240030 * r18240010;
        double r18240032 = r18240029 - r18240031;
        double r18240033 = 1.0;
        double r18240034 = r18240033 / r18240002;
        double r18240035 = -0.3333333333333333;
        double r18240036 = pow(r18240034, r18240035);
        double r18240037 = log(r18240036);
        double r18240038 = r18240037 * r18240018;
        double r18240039 = r18240038 - r18240002;
        double r18240040 = r18240039 + r18240016;
        double r18240041 = cbrt(r18240002);
        double r18240042 = r18240041 * r18240041;
        double r18240043 = log(r18240042);
        double r18240044 = r18240043 * r18240018;
        double r18240045 = r18240040 + r18240044;
        double r18240046 = r18240032 + r18240045;
        double r18240047 = r18240004 ? r18240026 : r18240046;
        return r18240047;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.2
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < 21506604914.31418

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.2

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

   if 21506604914.31418 < x

   1. Initial program 10.2

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

   3. Applied add-cube-cbrt 10.2

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

   4. Applied log-prod 10.2

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

   5. Applied distribute-rgt-in 10.2

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

   6. Applied associate--l+ 10.2

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

   7. Applied associate-+l+ 10.2

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

   8. Taylor expanded around inf 10.1

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

   9. Taylor expanded around inf 10.2

      \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\]

   10. Simplified 0.4

       \[\leadsto \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 21506604914.314178466796875:\\ \;\;\;\;\frac{z \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \left(\sqrt[3]{\left(x - 0.5\right) \cdot \log x} \cdot \sqrt[3]{\left(x - 0.5\right) \cdot \log x}\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z}{x} \cdot 0.002777777777777800001512975569539776188321\right) + \left(\left(\left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :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)))