Average Error: 29.5 → 4.8
Time: 6.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -17052070997538.8926 \lor \neg \left(z \le 130676711668149.234\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]
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 -17052070997538.8926 \lor \neg \left(z \le 130676711668149.234\right):\\
\;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 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 r336139 = x;
        double r336140 = y;
        double r336141 = z;
        double r336142 = 3.13060547623;
        double r336143 = r336141 * r336142;
        double r336144 = 11.1667541262;
        double r336145 = r336143 + r336144;
        double r336146 = r336145 * r336141;
        double r336147 = t;
        double r336148 = r336146 + r336147;
        double r336149 = r336148 * r336141;
        double r336150 = a;
        double r336151 = r336149 + r336150;
        double r336152 = r336151 * r336141;
        double r336153 = b;
        double r336154 = r336152 + r336153;
        double r336155 = r336140 * r336154;
        double r336156 = 15.234687407;
        double r336157 = r336141 + r336156;
        double r336158 = r336157 * r336141;
        double r336159 = 31.4690115749;
        double r336160 = r336158 + r336159;
        double r336161 = r336160 * r336141;
        double r336162 = 11.9400905721;
        double r336163 = r336161 + r336162;
        double r336164 = r336163 * r336141;
        double r336165 = 0.607771387771;
        double r336166 = r336164 + r336165;
        double r336167 = r336155 / r336166;
        double r336168 = r336139 + r336167;
        return r336168;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r336169 = z;
        double r336170 = -17052070997538.893;
        bool r336171 = r336169 <= r336170;
        double r336172 = 130676711668149.23;
        bool r336173 = r336169 <= r336172;
        double r336174 = !r336173;
        bool r336175 = r336171 || r336174;
        double r336176 = x;
        double r336177 = y;
        double r336178 = 3.13060547623;
        double r336179 = t;
        double r336180 = r336179 * r336177;
        double r336181 = 2.0;
        double r336182 = pow(r336169, r336181);
        double r336183 = r336180 / r336182;
        double r336184 = 36.527041698806414;
        double r336185 = r336177 / r336169;
        double r336186 = r336184 * r336185;
        double r336187 = r336183 - r336186;
        double r336188 = fma(r336177, r336178, r336187);
        double r336189 = r336176 + r336188;
        double r336190 = r336169 * r336178;
        double r336191 = 11.1667541262;
        double r336192 = r336190 + r336191;
        double r336193 = r336192 * r336169;
        double r336194 = r336193 + r336179;
        double r336195 = r336194 * r336169;
        double r336196 = a;
        double r336197 = r336195 + r336196;
        double r336198 = r336197 * r336169;
        double r336199 = b;
        double r336200 = r336198 + r336199;
        double r336201 = 15.234687407;
        double r336202 = r336169 + r336201;
        double r336203 = r336202 * r336169;
        double r336204 = 31.4690115749;
        double r336205 = r336203 + r336204;
        double r336206 = r336205 * r336169;
        double r336207 = 11.9400905721;
        double r336208 = r336206 + r336207;
        double r336209 = r336208 * r336169;
        double r336210 = 0.607771387771;
        double r336211 = r336209 + r336210;
        double r336212 = r336200 / r336211;
        double r336213 = r336177 * r336212;
        double r336214 = r336176 + r336213;
        double r336215 = r336175 ? r336189 : r336214;
        return r336215;
}

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

Original29.5
Target1.1
Herbie4.8
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -17052070997538.893 or 130676711668149.23 < z

    1. Initial program 56.8

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

    2. Taylor expanded around inf 9.0

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

    3. Simplified9.0

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

    if -17052070997538.893 < z < 130676711668149.23

    1. Initial program 0.3

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

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

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

    4. Applied times-frac0.3

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

    5. Simplified0.3

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

  4. Final simplification4.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -17052070997538.8926 \lor \neg \left(z \le 130676711668149.234\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))