Average Error: 30.2 → 1.1
Time: 32.2s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.807302090294030027556419953822894801612 \cdot 10^{70}:\\ \;\;\;\;x + y \cdot \left(\left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}\right)\\ \mathbf{elif}\;z \le 548437100344291305854952900722667028480:\\ \;\;\;\;x + y \cdot \left(\left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right) \cdot \frac{1}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}\right)\\ \end{array}\]
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 -3.807302090294030027556419953822894801612 \cdot 10^{70}:\\
\;\;\;\;x + y \cdot \left(\left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}\right)\\

\mathbf{elif}\;z \le 548437100344291305854952900722667028480:\\
\;\;\;\;x + y \cdot \left(\left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right) \cdot \frac{1}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r26592142 = x;
        double r26592143 = y;
        double r26592144 = z;
        double r26592145 = 3.13060547623;
        double r26592146 = r26592144 * r26592145;
        double r26592147 = 11.1667541262;
        double r26592148 = r26592146 + r26592147;
        double r26592149 = r26592148 * r26592144;
        double r26592150 = t;
        double r26592151 = r26592149 + r26592150;
        double r26592152 = r26592151 * r26592144;
        double r26592153 = a;
        double r26592154 = r26592152 + r26592153;
        double r26592155 = r26592154 * r26592144;
        double r26592156 = b;
        double r26592157 = r26592155 + r26592156;
        double r26592158 = r26592143 * r26592157;
        double r26592159 = 15.234687407;
        double r26592160 = r26592144 + r26592159;
        double r26592161 = r26592160 * r26592144;
        double r26592162 = 31.4690115749;
        double r26592163 = r26592161 + r26592162;
        double r26592164 = r26592163 * r26592144;
        double r26592165 = 11.9400905721;
        double r26592166 = r26592164 + r26592165;
        double r26592167 = r26592166 * r26592144;
        double r26592168 = 0.607771387771;
        double r26592169 = r26592167 + r26592168;
        double r26592170 = r26592158 / r26592169;
        double r26592171 = r26592142 + r26592170;
        return r26592171;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r26592172 = z;
        double r26592173 = -3.80730209029403e+70;
        bool r26592174 = r26592172 <= r26592173;
        double r26592175 = x;
        double r26592176 = y;
        double r26592177 = t;
        double r26592178 = r26592172 * r26592172;
        double r26592179 = r26592177 / r26592178;
        double r26592180 = 3.13060547623;
        double r26592181 = r26592179 + r26592180;
        double r26592182 = 36.527041698806414;
        double r26592183 = r26592182 / r26592172;
        double r26592184 = r26592181 - r26592183;
        double r26592185 = r26592176 * r26592184;
        double r26592186 = r26592175 + r26592185;
        double r26592187 = 5.484371003442913e+38;
        bool r26592188 = r26592172 <= r26592187;
        double r26592189 = r26592172 * r26592180;
        double r26592190 = 11.1667541262;
        double r26592191 = r26592189 + r26592190;
        double r26592192 = r26592191 * r26592172;
        double r26592193 = r26592192 + r26592177;
        double r26592194 = r26592193 * r26592172;
        double r26592195 = a;
        double r26592196 = r26592194 + r26592195;
        double r26592197 = r26592196 * r26592172;
        double r26592198 = b;
        double r26592199 = r26592197 + r26592198;
        double r26592200 = 1.0;
        double r26592201 = 15.234687407;
        double r26592202 = r26592172 + r26592201;
        double r26592203 = r26592202 * r26592172;
        double r26592204 = 31.4690115749;
        double r26592205 = r26592203 + r26592204;
        double r26592206 = r26592205 * r26592172;
        double r26592207 = 11.9400905721;
        double r26592208 = r26592206 + r26592207;
        double r26592209 = r26592208 * r26592172;
        double r26592210 = 0.607771387771;
        double r26592211 = r26592209 + r26592210;
        double r26592212 = r26592200 / r26592211;
        double r26592213 = r26592199 * r26592212;
        double r26592214 = r26592176 * r26592213;
        double r26592215 = r26592175 + r26592214;
        double r26592216 = r26592188 ? r26592215 : r26592186;
        double r26592217 = r26592174 ? r26592186 : r26592216;
        return r26592217;
}

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

Original30.2
Target0.8
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 2 regimes

  2. if z < -3.80730209029403e+70 or 5.484371003442913e+38 < z

    1. Initial program 61.7

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

    3. Applied *-un-lft-identity61.7

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

    4. Applied times-frac60.5

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

    5. Simplified60.5

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

    7. Applied div-inv60.5

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

    8. Taylor expanded around inf 0.7

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

    9. Simplified0.7

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

    if -3.80730209029403e+70 < z < 5.484371003442913e+38

    1. Initial program 2.9

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

    3. Applied *-un-lft-identity2.9

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

    4. Applied times-frac1.3

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

    5. Simplified1.3

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

    7. Applied div-inv1.3

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.807302090294030027556419953822894801612 \cdot 10^{70}:\\ \;\;\;\;x + y \cdot \left(\left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}\right)\\ \mathbf{elif}\;z \le 548437100344291305854952900722667028480:\\ \;\;\;\;x + y \cdot \left(\left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right) \cdot \frac{1}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\left(\frac{t}{z \cdot z} + 3.130605476229999961645944495103321969509\right) - \frac{36.52704169880641416057187598198652267456}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))