Average Error: 29.7 → 4.5
Time: 5.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -6.13124396009091248 \cdot 10^{46} \lor \neg \left(z \le 6.275897102199269 \cdot 10^{44}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]
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 -6.13124396009091248 \cdot 10^{46} \lor \neg \left(z \le 6.275897102199269 \cdot 10^{44}\right):\\
\;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\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 r345805 = x;
        double r345806 = y;
        double r345807 = z;
        double r345808 = 3.13060547623;
        double r345809 = r345807 * r345808;
        double r345810 = 11.1667541262;
        double r345811 = r345809 + r345810;
        double r345812 = r345811 * r345807;
        double r345813 = t;
        double r345814 = r345812 + r345813;
        double r345815 = r345814 * r345807;
        double r345816 = a;
        double r345817 = r345815 + r345816;
        double r345818 = r345817 * r345807;
        double r345819 = b;
        double r345820 = r345818 + r345819;
        double r345821 = r345806 * r345820;
        double r345822 = 15.234687407;
        double r345823 = r345807 + r345822;
        double r345824 = r345823 * r345807;
        double r345825 = 31.4690115749;
        double r345826 = r345824 + r345825;
        double r345827 = r345826 * r345807;
        double r345828 = 11.9400905721;
        double r345829 = r345827 + r345828;
        double r345830 = r345829 * r345807;
        double r345831 = 0.607771387771;
        double r345832 = r345830 + r345831;
        double r345833 = r345821 / r345832;
        double r345834 = r345805 + r345833;
        return r345834;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r345835 = z;
        double r345836 = -6.131243960090912e+46;
        bool r345837 = r345835 <= r345836;
        double r345838 = 6.275897102199269e+44;
        bool r345839 = r345835 <= r345838;
        double r345840 = !r345839;
        bool r345841 = r345837 || r345840;
        double r345842 = x;
        double r345843 = 3.13060547623;
        double r345844 = y;
        double r345845 = r345843 * r345844;
        double r345846 = t;
        double r345847 = r345846 * r345844;
        double r345848 = 2.0;
        double r345849 = pow(r345835, r345848);
        double r345850 = r345847 / r345849;
        double r345851 = r345845 + r345850;
        double r345852 = 36.527041698806414;
        double r345853 = r345844 / r345835;
        double r345854 = r345852 * r345853;
        double r345855 = r345851 - r345854;
        double r345856 = r345842 + r345855;
        double r345857 = r345835 * r345843;
        double r345858 = 11.1667541262;
        double r345859 = r345857 + r345858;
        double r345860 = r345859 * r345835;
        double r345861 = r345860 + r345846;
        double r345862 = r345861 * r345835;
        double r345863 = a;
        double r345864 = r345862 + r345863;
        double r345865 = r345864 * r345835;
        double r345866 = b;
        double r345867 = r345865 + r345866;
        double r345868 = 15.234687407;
        double r345869 = r345835 + r345868;
        double r345870 = r345869 * r345835;
        double r345871 = 31.4690115749;
        double r345872 = r345870 + r345871;
        double r345873 = r345872 * r345835;
        double r345874 = 11.9400905721;
        double r345875 = r345873 + r345874;
        double r345876 = r345875 * r345835;
        double r345877 = 0.607771387771;
        double r345878 = r345876 + r345877;
        double r345879 = r345867 / r345878;
        double r345880 = r345844 * r345879;
        double r345881 = r345842 + r345880;
        double r345882 = r345841 ? r345856 : r345881;
        return r345882;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target1.0
Herbie4.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -6.131243960090912e+46 or 6.275897102199269e+44 < z

    1. Initial program 60.7

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

    2. Taylor expanded around inf 8.7

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    if -6.131243960090912e+46 < z < 6.275897102199269e+44

    1. Initial program 2.0

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

    3. Applied *-un-lft-identity2.0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004\right)}}\]

    4. Applied times-frac0.7

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\]

    5. Simplified0.7

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

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -6.13124396009091248 \cdot 10^{46} \lor \neg \left(z \le 6.275897102199269 \cdot 10^{44}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))