Average Error: 29.4 → 4.7
Time: 5.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.0056051312042543 \cdot 10^{25} \lor \neg \left(z \le 1.2468316103113146 \cdot 10^{24}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]
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 -3.0056051312042543 \cdot 10^{25} \lor \neg \left(z \le 1.2468316103113146 \cdot 10^{24}\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 r390946 = x;
        double r390947 = y;
        double r390948 = z;
        double r390949 = 3.13060547623;
        double r390950 = r390948 * r390949;
        double r390951 = 11.1667541262;
        double r390952 = r390950 + r390951;
        double r390953 = r390952 * r390948;
        double r390954 = t;
        double r390955 = r390953 + r390954;
        double r390956 = r390955 * r390948;
        double r390957 = a;
        double r390958 = r390956 + r390957;
        double r390959 = r390958 * r390948;
        double r390960 = b;
        double r390961 = r390959 + r390960;
        double r390962 = r390947 * r390961;
        double r390963 = 15.234687407;
        double r390964 = r390948 + r390963;
        double r390965 = r390964 * r390948;
        double r390966 = 31.4690115749;
        double r390967 = r390965 + r390966;
        double r390968 = r390967 * r390948;
        double r390969 = 11.9400905721;
        double r390970 = r390968 + r390969;
        double r390971 = r390970 * r390948;
        double r390972 = 0.607771387771;
        double r390973 = r390971 + r390972;
        double r390974 = r390962 / r390973;
        double r390975 = r390946 + r390974;
        return r390975;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r390976 = z;
        double r390977 = -3.0056051312042543e+25;
        bool r390978 = r390976 <= r390977;
        double r390979 = 1.2468316103113146e+24;
        bool r390980 = r390976 <= r390979;
        double r390981 = !r390980;
        bool r390982 = r390978 || r390981;
        double r390983 = x;
        double r390984 = 3.13060547623;
        double r390985 = y;
        double r390986 = r390984 * r390985;
        double r390987 = t;
        double r390988 = r390987 * r390985;
        double r390989 = 2.0;
        double r390990 = pow(r390976, r390989);
        double r390991 = r390988 / r390990;
        double r390992 = r390986 + r390991;
        double r390993 = 36.527041698806414;
        double r390994 = r390985 / r390976;
        double r390995 = r390993 * r390994;
        double r390996 = r390992 - r390995;
        double r390997 = r390983 + r390996;
        double r390998 = r390976 * r390984;
        double r390999 = 11.1667541262;
        double r391000 = r390998 + r390999;
        double r391001 = r391000 * r390976;
        double r391002 = r391001 + r390987;
        double r391003 = r391002 * r390976;
        double r391004 = a;
        double r391005 = r391003 + r391004;
        double r391006 = r391005 * r390976;
        double r391007 = b;
        double r391008 = r391006 + r391007;
        double r391009 = 15.234687407;
        double r391010 = r390976 + r391009;
        double r391011 = r391010 * r390976;
        double r391012 = 31.4690115749;
        double r391013 = r391011 + r391012;
        double r391014 = r391013 * r390976;
        double r391015 = 11.9400905721;
        double r391016 = r391014 + r391015;
        double r391017 = r391016 * r390976;
        double r391018 = 0.607771387771;
        double r391019 = r391017 + r391018;
        double r391020 = r391008 / r391019;
        double r391021 = r390985 * r391020;
        double r391022 = r390983 + r391021;
        double r391023 = r390982 ? r390997 : r391022;
        return r391023;
}

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

Original29.4
Target1.0
Herbie4.7
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.0056051312042543e+25 or 1.2468316103113146e+24 < z

    1. Initial program 58.1

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

    2. Taylor expanded around inf 9.0

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

    if -3.0056051312042543e+25 < z < 1.2468316103113146e+24

    1. Initial program 0.9

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

    3. Applied *-un-lft-identity0.9

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

    4. Applied times-frac0.5

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

    5. Simplified0.5

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

  4. Final simplification4.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.0056051312042543 \cdot 10^{25} \lor \neg \left(z \le 1.2468316103113146 \cdot 10^{24}\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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))))