Average Error: 30.1 → 4.5
Time: 16.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.39389244567685115 \cdot 10^{38} \lor \neg \left(z \le 1751680100761613370000\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]
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 -3.39389244567685115 \cdot 10^{38} \lor \neg \left(z \le 1751680100761613370000\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 + \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}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r476060 = x;
        double r476061 = y;
        double r476062 = z;
        double r476063 = 3.13060547623;
        double r476064 = r476062 * r476063;
        double r476065 = 11.1667541262;
        double r476066 = r476064 + r476065;
        double r476067 = r476066 * r476062;
        double r476068 = t;
        double r476069 = r476067 + r476068;
        double r476070 = r476069 * r476062;
        double r476071 = a;
        double r476072 = r476070 + r476071;
        double r476073 = r476072 * r476062;
        double r476074 = b;
        double r476075 = r476073 + r476074;
        double r476076 = r476061 * r476075;
        double r476077 = 15.234687407;
        double r476078 = r476062 + r476077;
        double r476079 = r476078 * r476062;
        double r476080 = 31.4690115749;
        double r476081 = r476079 + r476080;
        double r476082 = r476081 * r476062;
        double r476083 = 11.9400905721;
        double r476084 = r476082 + r476083;
        double r476085 = r476084 * r476062;
        double r476086 = 0.607771387771;
        double r476087 = r476085 + r476086;
        double r476088 = r476076 / r476087;
        double r476089 = r476060 + r476088;
        return r476089;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r476090 = z;
        double r476091 = -3.393892445676851e+38;
        bool r476092 = r476090 <= r476091;
        double r476093 = 1.7516801007616134e+21;
        bool r476094 = r476090 <= r476093;
        double r476095 = !r476094;
        bool r476096 = r476092 || r476095;
        double r476097 = x;
        double r476098 = 3.13060547623;
        double r476099 = y;
        double r476100 = r476098 * r476099;
        double r476101 = t;
        double r476102 = r476101 * r476099;
        double r476103 = 2.0;
        double r476104 = pow(r476090, r476103);
        double r476105 = r476102 / r476104;
        double r476106 = r476100 + r476105;
        double r476107 = 36.527041698806414;
        double r476108 = r476099 / r476090;
        double r476109 = r476107 * r476108;
        double r476110 = r476106 - r476109;
        double r476111 = r476097 + r476110;
        double r476112 = 15.234687407;
        double r476113 = r476090 + r476112;
        double r476114 = r476113 * r476090;
        double r476115 = 31.4690115749;
        double r476116 = r476114 + r476115;
        double r476117 = r476116 * r476090;
        double r476118 = 11.9400905721;
        double r476119 = r476117 + r476118;
        double r476120 = r476119 * r476090;
        double r476121 = 0.607771387771;
        double r476122 = r476120 + r476121;
        double r476123 = r476090 * r476098;
        double r476124 = 11.1667541262;
        double r476125 = r476123 + r476124;
        double r476126 = r476125 * r476090;
        double r476127 = r476126 + r476101;
        double r476128 = r476127 * r476090;
        double r476129 = a;
        double r476130 = r476128 + r476129;
        double r476131 = r476130 * r476090;
        double r476132 = b;
        double r476133 = r476131 + r476132;
        double r476134 = r476122 / r476133;
        double r476135 = r476099 / r476134;
        double r476136 = r476097 + r476135;
        double r476137 = r476096 ? r476111 : r476136;
        return r476137;
}

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

Original30.1
Target0.9
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 < -3.393892445676851e+38 or 1.7516801007616134e+21 < z

    1. Initial program 59.3

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

    2. Taylor expanded around inf 8.5

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

    if -3.393892445676851e+38 < z < 1.7516801007616134e+21

    1. Initial program 1.2

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

    3. Applied associate-/l*0.6

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

  4. Final simplification4.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.39389244567685115 \cdot 10^{38} \lor \neg \left(z \le 1751680100761613370000\right):\\ \;\;\;\;x + \left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))