Average Error: 29.3 → 1.1
Time: 5.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.7622742072428377 \cdot 10^{61} \lor \neg \left(z \le 4.1810546146577023 \cdot 10^{44}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + t \cdot \frac{1}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -5.7622742072428377 \cdot 10^{61} \lor \neg \left(z \le 4.1810546146577023 \cdot 10^{44}\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + t \cdot \frac{1}{{z}^{2}}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r349878 = x;
        double r349879 = y;
        double r349880 = z;
        double r349881 = 3.13060547623;
        double r349882 = r349880 * r349881;
        double r349883 = 11.1667541262;
        double r349884 = r349882 + r349883;
        double r349885 = r349884 * r349880;
        double r349886 = t;
        double r349887 = r349885 + r349886;
        double r349888 = r349887 * r349880;
        double r349889 = a;
        double r349890 = r349888 + r349889;
        double r349891 = r349890 * r349880;
        double r349892 = b;
        double r349893 = r349891 + r349892;
        double r349894 = r349879 * r349893;
        double r349895 = 15.234687407;
        double r349896 = r349880 + r349895;
        double r349897 = r349896 * r349880;
        double r349898 = 31.4690115749;
        double r349899 = r349897 + r349898;
        double r349900 = r349899 * r349880;
        double r349901 = 11.9400905721;
        double r349902 = r349900 + r349901;
        double r349903 = r349902 * r349880;
        double r349904 = 0.607771387771;
        double r349905 = r349903 + r349904;
        double r349906 = r349894 / r349905;
        double r349907 = r349878 + r349906;
        return r349907;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r349908 = z;
        double r349909 = -5.762274207242838e+61;
        bool r349910 = r349908 <= r349909;
        double r349911 = 4.181054614657702e+44;
        bool r349912 = r349908 <= r349911;
        double r349913 = !r349912;
        bool r349914 = r349910 || r349913;
        double r349915 = y;
        double r349916 = 3.13060547623;
        double r349917 = t;
        double r349918 = 1.0;
        double r349919 = 2.0;
        double r349920 = pow(r349908, r349919);
        double r349921 = r349918 / r349920;
        double r349922 = r349917 * r349921;
        double r349923 = r349916 + r349922;
        double r349924 = x;
        double r349925 = fma(r349915, r349923, r349924);
        double r349926 = 15.234687407;
        double r349927 = r349908 + r349926;
        double r349928 = 31.4690115749;
        double r349929 = fma(r349927, r349908, r349928);
        double r349930 = 11.9400905721;
        double r349931 = fma(r349929, r349908, r349930);
        double r349932 = 0.607771387771;
        double r349933 = fma(r349931, r349908, r349932);
        double r349934 = r349915 / r349933;
        double r349935 = 11.1667541262;
        double r349936 = fma(r349908, r349916, r349935);
        double r349937 = fma(r349936, r349908, r349917);
        double r349938 = a;
        double r349939 = fma(r349937, r349908, r349938);
        double r349940 = b;
        double r349941 = fma(r349939, r349908, r349940);
        double r349942 = fma(r349934, r349941, r349924);
        double r349943 = r349914 ? r349925 : r349942;
        return r349943;
}

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

Original29.3
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 < -5.762274207242838e+61 or 4.181054614657702e+44 < z

    1. Initial program 61.8

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

    2. Simplified60.3

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

    3. Taylor expanded around inf 8.4

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

    4. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied div-inv0.8

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \color{blue}{t \cdot \frac{1}{{z}^{2}}}, x\right)\]

    if -5.762274207242838e+61 < z < 4.181054614657702e+44

    1. Initial program 2.6

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

    2. Simplified1.4

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.7622742072428377 \cdot 10^{61} \lor \neg \left(z \le 4.1810546146577023 \cdot 10^{44}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + t \cdot \frac{1}{{z}^{2}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +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))))