Average Error: 29.4 → 1.0
Time: 21.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}
↓
\begin{array}{l}
\mathbf{if}\;z \le -3240660210645078190185871572992:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{elif}\;z \le 587734457247325707726001012736:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r12354128 = x;
        double r12354129 = y;
        double r12354130 = z;
        double r12354131 = 3.13060547623;
        double r12354132 = r12354130 * r12354131;
        double r12354133 = 11.1667541262;
        double r12354134 = r12354132 + r12354133;
        double r12354135 = r12354134 * r12354130;
        double r12354136 = t;
        double r12354137 = r12354135 + r12354136;
        double r12354138 = r12354137 * r12354130;
        double r12354139 = a;
        double r12354140 = r12354138 + r12354139;
        double r12354141 = r12354140 * r12354130;
        double r12354142 = b;
        double r12354143 = r12354141 + r12354142;
        double r12354144 = r12354129 * r12354143;
        double r12354145 = 15.234687407;
        double r12354146 = r12354130 + r12354145;
        double r12354147 = r12354146 * r12354130;
        double r12354148 = 31.4690115749;
        double r12354149 = r12354147 + r12354148;
        double r12354150 = r12354149 * r12354130;
        double r12354151 = 11.9400905721;
        double r12354152 = r12354150 + r12354151;
        double r12354153 = r12354152 * r12354130;
        double r12354154 = 0.607771387771;
        double r12354155 = r12354153 + r12354154;
        double r12354156 = r12354144 / r12354155;
        double r12354157 = r12354128 + r12354156;
        return r12354157;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r12354158 = z;
        double r12354159 = -3.240660210645078e+30;
        bool r12354160 = r12354158 <= r12354159;
        double r12354161 = y;
        double r12354162 = t;
        double r12354163 = r12354158 * r12354158;
        double r12354164 = r12354162 / r12354163;
        double r12354165 = 3.13060547623;
        double r12354166 = r12354164 + r12354165;
        double r12354167 = x;
        double r12354168 = fma(r12354161, r12354166, r12354167);
        double r12354169 = 5.877344572473257e+29;
        bool r12354170 = r12354158 <= r12354169;
        double r12354171 = 1.0;
        double r12354172 = 15.234687407;
        double r12354173 = r12354158 + r12354172;
        double r12354174 = 31.4690115749;
        double r12354175 = fma(r12354158, r12354173, r12354174);
        double r12354176 = 11.9400905721;
        double r12354177 = fma(r12354158, r12354175, r12354176);
        double r12354178 = 0.607771387771;
        double r12354179 = fma(r12354158, r12354177, r12354178);
        double r12354180 = r12354171 / r12354179;
        double r12354181 = 11.1667541262;
        double r12354182 = fma(r12354165, r12354158, r12354181);
        double r12354183 = fma(r12354182, r12354158, r12354162);
        double r12354184 = a;
        double r12354185 = fma(r12354158, r12354183, r12354184);
        double r12354186 = b;
        double r12354187 = fma(r12354185, r12354158, r12354186);
        double r12354188 = r12354180 * r12354187;
        double r12354189 = fma(r12354161, r12354188, r12354167);
        double r12354190 = r12354170 ? r12354189 : r12354168;
        double r12354191 = r12354160 ? r12354168 : r12354190;
        return r12354191;
}

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

Original29.4
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.240660210645078e+30 or 5.877344572473257e+29 < z

    1. Initial program 58.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Simplified55.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)}\]

    3. Taylor expanded around inf 9.2

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right)}\]

    4. Simplified1.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)}\]

    if -3.240660210645078e+30 < z < 5.877344572473257e+29

    1. Initial program 1.1

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    2. Simplified0.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)}\]
    3. Using strategy rm

    4. Applied div-inv0.6

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}}, x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (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 1.0)))))

  (+ 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))))