Average Error: 6.0 → 4.1
Time: 8.0s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 15119356296086663000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\mathsf{fma}\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778, z, 0.0833333333333329956\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]
\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 15119356296086663000:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\mathsf{fma}\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778, z, 0.0833333333333329956\right)}{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}
double f(double x, double y, double z) {
        double r475132 = x;
        double r475133 = 0.5;
        double r475134 = r475132 - r475133;
        double r475135 = log(r475132);
        double r475136 = r475134 * r475135;
        double r475137 = r475136 - r475132;
        double r475138 = 0.91893853320467;
        double r475139 = r475137 + r475138;
        double r475140 = y;
        double r475141 = 0.0007936500793651;
        double r475142 = r475140 + r475141;
        double r475143 = z;
        double r475144 = r475142 * r475143;
        double r475145 = 0.0027777777777778;
        double r475146 = r475144 - r475145;
        double r475147 = r475146 * r475143;
        double r475148 = 0.083333333333333;
        double r475149 = r475147 + r475148;
        double r475150 = r475149 / r475132;
        double r475151 = r475139 + r475150;
        return r475151;
}
↓
double f(double x, double y, double z) {
        double r475152 = x;
        double r475153 = 1.5119356296086663e+19;
        bool r475154 = r475152 <= r475153;
        double r475155 = 0.5;
        double r475156 = r475152 - r475155;
        double r475157 = log(r475152);
        double r475158 = r475156 * r475157;
        double r475159 = r475158 - r475152;
        double r475160 = 0.91893853320467;
        double r475161 = r475159 + r475160;
        double r475162 = y;
        double r475163 = 0.0007936500793651;
        double r475164 = r475162 + r475163;
        double r475165 = z;
        double r475166 = r475164 * r475165;
        double r475167 = 0.0027777777777778;
        double r475168 = r475166 - r475167;
        double r475169 = 0.083333333333333;
        double r475170 = fma(r475168, r475165, r475169);
        double r475171 = r475170 / r475152;
        double r475172 = r475161 + r475171;
        double r475173 = 2.0;
        double r475174 = pow(r475165, r475173);
        double r475175 = r475174 / r475152;
        double r475176 = r475163 * r475175;
        double r475177 = 1.0;
        double r475178 = r475177 / r475152;
        double r475179 = log(r475178);
        double r475180 = fma(r475179, r475152, r475152);
        double r475181 = r475176 - r475180;
        double r475182 = fma(r475175, r475162, r475181);
        double r475183 = r475154 ? r475172 : r475182;
        return r475183;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.0
Target1.3
Herbie4.1
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 1.5119356296086663e+19

    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 *-un-lft-identity0.2

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

    4. Applied associate-/r*0.2

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

    5. Simplified0.2

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

    if 1.5119356296086663e+19 < x

    1. Initial program 10.6

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

    2. Simplified10.5

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

    3. Taylor expanded around inf 10.6

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

    4. Simplified7.2

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

  4. Final simplification4.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 15119356296086663000:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\mathsf{fma}\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778, z, 0.0833333333333329956\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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)))