Average Error: 6.0 → 3.7
Time: 30.8s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.176380466602636012645528110948300751537 \cdot 10^{89}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{1}{12.00000000000004796163466380676254630089 \cdot x + x \cdot \left(z \cdot 0.4000000000000064059868520871532382443547 - {z}^{2} \cdot 0.1009522780952416126654114236771420110017\right)}\\ \end{array}\]
\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 1.176380466602636012645528110948300751537 \cdot 10^{89}:\\
\;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\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}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{1}{12.00000000000004796163466380676254630089 \cdot x + x \cdot \left(z \cdot 0.4000000000000064059868520871532382443547 - {z}^{2} \cdot 0.1009522780952416126654114236771420110017\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r289092 = x;
        double r289093 = 0.5;
        double r289094 = r289092 - r289093;
        double r289095 = log(r289092);
        double r289096 = r289094 * r289095;
        double r289097 = r289096 - r289092;
        double r289098 = 0.91893853320467;
        double r289099 = r289097 + r289098;
        double r289100 = y;
        double r289101 = 0.0007936500793651;
        double r289102 = r289100 + r289101;
        double r289103 = z;
        double r289104 = r289102 * r289103;
        double r289105 = 0.0027777777777778;
        double r289106 = r289104 - r289105;
        double r289107 = r289106 * r289103;
        double r289108 = 0.083333333333333;
        double r289109 = r289107 + r289108;
        double r289110 = r289109 / r289092;
        double r289111 = r289099 + r289110;
        return r289111;
}
↓
double f(double x, double y, double z) {
        double r289112 = x;
        double r289113 = 1.176380466602636e+89;
        bool r289114 = r289112 <= r289113;
        double r289115 = 0.5;
        double r289116 = r289112 - r289115;
        double r289117 = 2.0;
        double r289118 = cbrt(r289112);
        double r289119 = log(r289118);
        double r289120 = r289117 * r289119;
        double r289121 = r289116 * r289120;
        double r289122 = r289119 * r289116;
        double r289123 = r289121 + r289122;
        double r289124 = r289123 - r289112;
        double r289125 = 0.91893853320467;
        double r289126 = r289124 + r289125;
        double r289127 = y;
        double r289128 = 0.0007936500793651;
        double r289129 = r289127 + r289128;
        double r289130 = z;
        double r289131 = r289129 * r289130;
        double r289132 = 0.0027777777777778;
        double r289133 = r289131 - r289132;
        double r289134 = r289133 * r289130;
        double r289135 = 0.083333333333333;
        double r289136 = r289134 + r289135;
        double r289137 = r289136 / r289112;
        double r289138 = r289126 + r289137;
        double r289139 = log(r289112);
        double r289140 = r289116 * r289139;
        double r289141 = r289140 - r289112;
        double r289142 = r289141 + r289125;
        double r289143 = 1.0;
        double r289144 = 12.000000000000048;
        double r289145 = r289144 * r289112;
        double r289146 = 0.4000000000000064;
        double r289147 = r289130 * r289146;
        double r289148 = pow(r289130, r289117);
        double r289149 = 0.10095227809524161;
        double r289150 = r289148 * r289149;
        double r289151 = r289147 - r289150;
        double r289152 = r289112 * r289151;
        double r289153 = r289145 + r289152;
        double r289154 = r289143 / r289153;
        double r289155 = r289142 + r289154;
        double r289156 = r289114 ? r289138 : r289155;
        return r289156;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.0
Target1.2
Herbie3.7
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.176380466602636e+89

    1. Initial program 0.9

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

    3. Applied add-cube-cbrt0.9

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

    4. Applied log-prod1.0

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

    5. Applied distribute-lft-in1.0

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

    6. Simplified1.0

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

    7. Simplified1.0

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

    if 1.176380466602636e+89 < x

    1. Initial program 12.3

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

    3. Applied clear-num12.3

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

    4. Taylor expanded around 0 12.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{1}{\color{blue}{\left(0.4000000000000064059868520871532382443547 \cdot \left(x \cdot z\right) + 12.00000000000004796163466380676254630089 \cdot x\right) - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)}}\]

    5. Simplified7.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{1}{\color{blue}{12.00000000000004796163466380676254630089 \cdot x + x \cdot \left(z \cdot 0.4000000000000064059868520871532382443547 - {z}^{2} \cdot 0.1009522780952416126654114236771420110017\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.176380466602636012645528110948300751537 \cdot 10^{89}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{1}{12.00000000000004796163466380676254630089 \cdot x + x \cdot \left(z \cdot 0.4000000000000064059868520871532382443547 - {z}^{2} \cdot 0.1009522780952416126654114236771420110017\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))