Average Error: 29.2 → 1.1
Time: 29.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -87077245091641527145418274318303940308890000:\\ \;\;\;\;x + \left(\frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880641416057187598198652267456\right) + 3.130605476229999961645944495103321969509 \cdot y\right)\\ \mathbf{elif}\;z \le 177470294739602374656:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\frac{\left({\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)}^{3} + {31.46901157490000144889563671313226222992}^{3}\right) \cdot z}{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + \left(31.46901157490000144889563671313226222992 \cdot 31.46901157490000144889563671313226222992 - \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot 31.46901157490000144889563671313226222992\right)} + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{{z}^{2}} + \left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right)\right)\\ \end{array}\]
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 -87077245091641527145418274318303940308890000:\\
\;\;\;\;x + \left(\frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880641416057187598198652267456\right) + 3.130605476229999961645944495103321969509 \cdot y\right)\\

\mathbf{elif}\;z \le 177470294739602374656:\\
\;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\frac{\left({\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)}^{3} + {31.46901157490000144889563671313226222992}^{3}\right) \cdot z}{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + \left(31.46901157490000144889563671313226222992 \cdot 31.46901157490000144889563671313226222992 - \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot 31.46901157490000144889563671313226222992\right)} + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\

\mathbf{else}:\\
\;\;\;\;x + y \cdot \left(\frac{t}{{z}^{2}} + \left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r265087 = x;
        double r265088 = y;
        double r265089 = z;
        double r265090 = 3.13060547623;
        double r265091 = r265089 * r265090;
        double r265092 = 11.1667541262;
        double r265093 = r265091 + r265092;
        double r265094 = r265093 * r265089;
        double r265095 = t;
        double r265096 = r265094 + r265095;
        double r265097 = r265096 * r265089;
        double r265098 = a;
        double r265099 = r265097 + r265098;
        double r265100 = r265099 * r265089;
        double r265101 = b;
        double r265102 = r265100 + r265101;
        double r265103 = r265088 * r265102;
        double r265104 = 15.234687407;
        double r265105 = r265089 + r265104;
        double r265106 = r265105 * r265089;
        double r265107 = 31.4690115749;
        double r265108 = r265106 + r265107;
        double r265109 = r265108 * r265089;
        double r265110 = 11.9400905721;
        double r265111 = r265109 + r265110;
        double r265112 = r265111 * r265089;
        double r265113 = 0.607771387771;
        double r265114 = r265112 + r265113;
        double r265115 = r265103 / r265114;
        double r265116 = r265087 + r265115;
        return r265116;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r265117 = z;
        double r265118 = -8.707724509164153e+43;
        bool r265119 = r265117 <= r265118;
        double r265120 = x;
        double r265121 = y;
        double r265122 = r265121 / r265117;
        double r265123 = t;
        double r265124 = r265123 / r265117;
        double r265125 = 36.527041698806414;
        double r265126 = r265124 - r265125;
        double r265127 = r265122 * r265126;
        double r265128 = 3.13060547623;
        double r265129 = r265128 * r265121;
        double r265130 = r265127 + r265129;
        double r265131 = r265120 + r265130;
        double r265132 = 1.7747029473960237e+20;
        bool r265133 = r265117 <= r265132;
        double r265134 = r265117 * r265128;
        double r265135 = 11.1667541262;
        double r265136 = r265134 + r265135;
        double r265137 = r265136 * r265117;
        double r265138 = r265137 + r265123;
        double r265139 = r265138 * r265117;
        double r265140 = a;
        double r265141 = r265139 + r265140;
        double r265142 = r265141 * r265117;
        double r265143 = b;
        double r265144 = r265142 + r265143;
        double r265145 = 15.234687407;
        double r265146 = r265117 + r265145;
        double r265147 = r265146 * r265117;
        double r265148 = 3.0;
        double r265149 = pow(r265147, r265148);
        double r265150 = 31.4690115749;
        double r265151 = pow(r265150, r265148);
        double r265152 = r265149 + r265151;
        double r265153 = r265152 * r265117;
        double r265154 = r265147 * r265147;
        double r265155 = r265150 * r265150;
        double r265156 = r265147 * r265150;
        double r265157 = r265155 - r265156;
        double r265158 = r265154 + r265157;
        double r265159 = r265153 / r265158;
        double r265160 = 11.9400905721;
        double r265161 = r265159 + r265160;
        double r265162 = r265161 * r265117;
        double r265163 = 0.607771387771;
        double r265164 = r265162 + r265163;
        double r265165 = r265144 / r265164;
        double r265166 = r265121 * r265165;
        double r265167 = r265120 + r265166;
        double r265168 = 2.0;
        double r265169 = pow(r265117, r265168);
        double r265170 = r265123 / r265169;
        double r265171 = r265125 / r265117;
        double r265172 = r265128 - r265171;
        double r265173 = r265170 + r265172;
        double r265174 = r265121 * r265173;
        double r265175 = r265120 + r265174;
        double r265176 = r265133 ? r265167 : r265175;
        double r265177 = r265119 ? r265131 : r265176;
        return r265177;
}

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

Original29.2
Target0.9
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -8.707724509164153e+43

    1. Initial program 60.5

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

    3. Applied add-cube-cbrt60.5

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(\sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227} \cdot \sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\right) \cdot \sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}}}\]

    4. Applied times-frac58.5

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227} \cdot \sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}} \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt[3]{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}}}\]

    5. Taylor expanded around inf 9.0

      \[\leadsto x + \color{blue}{\left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]

    6. Simplified1.3

      \[\leadsto x + \color{blue}{\left(\frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880641416057187598198652267456\right) + 3.130605476229999961645944495103321969509 \cdot y\right)}\]

    if -8.707724509164153e+43 < z < 1.7747029473960237e+20

    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. Using strategy rm

    3. Applied *-un-lft-identity1.1

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227\right)}}\]

    4. Applied times-frac0.5

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}}\]

    5. Simplified0.5

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

    7. Applied flip3-+0.5

      \[\leadsto x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\color{blue}{\frac{{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)}^{3} + {31.46901157490000144889563671313226222992}^{3}}{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + \left(31.46901157490000144889563671313226222992 \cdot 31.46901157490000144889563671313226222992 - \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot 31.46901157490000144889563671313226222992\right)}} \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    8. Applied associate-*l/0.5

      \[\leadsto x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\color{blue}{\frac{\left({\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)}^{3} + {31.46901157490000144889563671313226222992}^{3}\right) \cdot z}{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + \left(31.46901157490000144889563671313226222992 \cdot 31.46901157490000144889563671313226222992 - \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot 31.46901157490000144889563671313226222992\right)}} + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

    if 1.7747029473960237e+20 < z

    1. Initial program 58.1

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

    3. Applied *-un-lft-identity58.1

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227\right)}}\]

    4. Applied times-frac55.3

      \[\leadsto x + \color{blue}{\frac{y}{1} \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}}\]

    5. Simplified55.3

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

    6. Taylor expanded around inf 2.1

      \[\leadsto x + y \cdot \color{blue}{\left(\left(\frac{t}{{z}^{2}} + 3.130605476229999961645944495103321969509\right) - 36.52704169880641416057187598198652267456 \cdot \frac{1}{z}\right)}\]

    7. Simplified2.1

      \[\leadsto x + y \cdot \color{blue}{\left(\frac{t}{{z}^{2}} + \left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -87077245091641527145418274318303940308890000:\\ \;\;\;\;x + \left(\frac{y}{z} \cdot \left(\frac{t}{z} - 36.52704169880641416057187598198652267456\right) + 3.130605476229999961645944495103321969509 \cdot y\right)\\ \mathbf{elif}\;z \le 177470294739602374656:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\frac{\left({\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right)}^{3} + {31.46901157490000144889563671313226222992}^{3}\right) \cdot z}{\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + \left(31.46901157490000144889563671313226222992 \cdot 31.46901157490000144889563671313226222992 - \left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) \cdot 31.46901157490000144889563671313226222992\right)} + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\frac{t}{{z}^{2}} + \left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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))))