Average Error: 30.1 → 1.1
Time: 13.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.7503197233736622 \cdot 10^{25} \lor \neg \left(z \le 14972530473052320000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.7503197233736622 \cdot 10^{25} \lor \neg \left(z \le 14972530473052320000\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r329760 = x;
        double r329761 = y;
        double r329762 = z;
        double r329763 = 3.13060547623;
        double r329764 = r329762 * r329763;
        double r329765 = 11.1667541262;
        double r329766 = r329764 + r329765;
        double r329767 = r329766 * r329762;
        double r329768 = t;
        double r329769 = r329767 + r329768;
        double r329770 = r329769 * r329762;
        double r329771 = a;
        double r329772 = r329770 + r329771;
        double r329773 = r329772 * r329762;
        double r329774 = b;
        double r329775 = r329773 + r329774;
        double r329776 = r329761 * r329775;
        double r329777 = 15.234687407;
        double r329778 = r329762 + r329777;
        double r329779 = r329778 * r329762;
        double r329780 = 31.4690115749;
        double r329781 = r329779 + r329780;
        double r329782 = r329781 * r329762;
        double r329783 = 11.9400905721;
        double r329784 = r329782 + r329783;
        double r329785 = r329784 * r329762;
        double r329786 = 0.607771387771;
        double r329787 = r329785 + r329786;
        double r329788 = r329776 / r329787;
        double r329789 = r329760 + r329788;
        return r329789;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r329790 = z;
        double r329791 = -1.7503197233736622e+25;
        bool r329792 = r329790 <= r329791;
        double r329793 = 1.497253047305232e+19;
        bool r329794 = r329790 <= r329793;
        double r329795 = !r329794;
        bool r329796 = r329792 || r329795;
        double r329797 = t;
        double r329798 = 2.0;
        double r329799 = pow(r329790, r329798);
        double r329800 = r329797 / r329799;
        double r329801 = y;
        double r329802 = 3.13060547623;
        double r329803 = x;
        double r329804 = fma(r329802, r329801, r329803);
        double r329805 = fma(r329800, r329801, r329804);
        double r329806 = r329790 * r329802;
        double r329807 = 11.1667541262;
        double r329808 = r329806 + r329807;
        double r329809 = r329808 * r329790;
        double r329810 = r329809 + r329797;
        double r329811 = r329810 * r329790;
        double r329812 = a;
        double r329813 = r329811 + r329812;
        double r329814 = r329813 * r329790;
        double r329815 = b;
        double r329816 = r329814 + r329815;
        double r329817 = r329801 * r329816;
        double r329818 = 15.234687407;
        double r329819 = r329790 + r329818;
        double r329820 = r329819 * r329790;
        double r329821 = 31.4690115749;
        double r329822 = r329820 + r329821;
        double r329823 = r329822 * r329790;
        double r329824 = 11.9400905721;
        double r329825 = r329823 + r329824;
        double r329826 = r329825 * r329790;
        double r329827 = 0.607771387771;
        double r329828 = r329826 + r329827;
        double r329829 = r329817 / r329828;
        double r329830 = r329803 + r329829;
        double r329831 = r329796 ? r329805 : r329830;
        return r329831;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original30.1
Target0.9
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.7503197233736622e+25 or 1.497253047305232e+19 < z

    1. Initial program 58.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Simplified56.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)}\]

    3. Taylor expanded around inf 8.7

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.13060547622999996 \cdot y\right)}\]

    4. Simplified1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)}\]

    if -1.7503197233736622e+25 < z < 1.497253047305232e+19

    1. Initial program 0.6

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.7503197233736622 \cdot 10^{25} \lor \neg \left(z \le 14972530473052320000\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{{z}^{2}}, y, \mathsf{fma}\left(3.13060547622999996, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))