Average Error: 6.0 → 3.5
Time: 7.8s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.75269981231680683 \cdot 10^{274}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{elif}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 2.33178853399688958 \cdot 10^{260}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}} - \left(x - 0.91893853320467001\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.400000000000006406 \cdot x, z, 12.000000000000048 \cdot x - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.91893853320467001\right)\right)\\ \end{array}\]
\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}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.75269981231680683 \cdot 10^{274}:\\
\;\;\;\;\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)\\

\mathbf{elif}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 2.33178853399688958 \cdot 10^{260}:\\
\;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}} - \left(x - 0.91893853320467001\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.400000000000006406 \cdot x, z, 12.000000000000048 \cdot x - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.91893853320467001\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r426087 = x;
        double r426088 = 0.5;
        double r426089 = r426087 - r426088;
        double r426090 = log(r426087);
        double r426091 = r426089 * r426090;
        double r426092 = r426091 - r426087;
        double r426093 = 0.91893853320467;
        double r426094 = r426092 + r426093;
        double r426095 = y;
        double r426096 = 0.0007936500793651;
        double r426097 = r426095 + r426096;
        double r426098 = z;
        double r426099 = r426097 * r426098;
        double r426100 = 0.0027777777777778;
        double r426101 = r426099 - r426100;
        double r426102 = r426101 * r426098;
        double r426103 = 0.083333333333333;
        double r426104 = r426102 + r426103;
        double r426105 = r426104 / r426087;
        double r426106 = r426094 + r426105;
        return r426106;
}
↓
double f(double x, double y, double z) {
        double r426107 = y;
        double r426108 = 0.0007936500793651;
        double r426109 = r426107 + r426108;
        double r426110 = z;
        double r426111 = r426109 * r426110;
        double r426112 = 0.0027777777777778;
        double r426113 = r426111 - r426112;
        double r426114 = r426113 * r426110;
        double r426115 = -4.752699812316807e+274;
        bool r426116 = r426114 <= r426115;
        double r426117 = 2.0;
        double r426118 = pow(r426110, r426117);
        double r426119 = x;
        double r426120 = r426118 / r426119;
        double r426121 = r426108 * r426120;
        double r426122 = 1.0;
        double r426123 = r426122 / r426119;
        double r426124 = log(r426123);
        double r426125 = fma(r426124, r426119, r426119);
        double r426126 = r426121 - r426125;
        double r426127 = fma(r426120, r426107, r426126);
        double r426128 = 2.3317885339968896e+260;
        bool r426129 = r426114 <= r426128;
        double r426130 = log(r426119);
        double r426131 = 0.5;
        double r426132 = r426119 - r426131;
        double r426133 = 0.083333333333333;
        double r426134 = r426114 + r426133;
        double r426135 = r426119 / r426134;
        double r426136 = r426122 / r426135;
        double r426137 = 0.91893853320467;
        double r426138 = r426119 - r426137;
        double r426139 = r426136 - r426138;
        double r426140 = fma(r426130, r426132, r426139);
        double r426141 = 0.4000000000000064;
        double r426142 = r426141 * r426119;
        double r426143 = 12.000000000000048;
        double r426144 = r426143 * r426119;
        double r426145 = 0.10095227809524161;
        double r426146 = r426119 * r426118;
        double r426147 = r426145 * r426146;
        double r426148 = r426144 - r426147;
        double r426149 = fma(r426142, r426110, r426148);
        double r426150 = r426122 / r426149;
        double r426151 = r426150 - r426138;
        double r426152 = fma(r426130, r426132, r426151);
        double r426153 = r426129 ? r426140 : r426152;
        double r426154 = r426116 ? r426127 : r426153;
        return r426154;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.0
Target1.3
Herbie3.5
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 3 regimes

  2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -4.752699812316807e+274

    1. Initial program 55.3

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

    2. Simplified55.3

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

    3. Taylor expanded around inf 55.3

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

    4. Simplified17.0

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

    if -4.752699812316807e+274 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 2.3317885339968896e+260

    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. Simplified0.2

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

    4. Applied clear-num0.3

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

    if 2.3317885339968896e+260 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

    1. Initial program 48.1

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

    2. Simplified48.1

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

    4. Applied clear-num48.1

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

    5. Taylor expanded around 0 45.9

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\color{blue}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}} - \left(x - 0.91893853320467001\right)\right)\]

    6. Simplified32.0

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\color{blue}{\mathsf{fma}\left(0.400000000000006406 \cdot x, z, 12.000000000000048 \cdot x - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)\right)}} - \left(x - 0.91893853320467001\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification3.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le -4.75269981231680683 \cdot 10^{274}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{elif}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 2.33178853399688958 \cdot 10^{260}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\frac{x}{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}} - \left(x - 0.91893853320467001\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.400000000000006406 \cdot x, z, 12.000000000000048 \cdot x - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.91893853320467001\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))