Average Error: 30.2 → 1.1
Time: 41.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.807302090294030027556419953822894801612 \cdot 10^{70}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 548437100344291305854952900722667028480:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \end{array}\]
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 -3.807302090294030027556419953822894801612 \cdot 10^{70}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{\frac{t}{z}}{z} - \frac{36.52704169880641416057187598198652267456}{z}\right) + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{elif}\;z \le 548437100344291305854952900722667028480:\\
\;\;\;\;\mathsf{fma}\left(y, \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, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r18145158 = x;
        double r18145159 = y;
        double r18145160 = z;
        double r18145161 = 3.13060547623;
        double r18145162 = r18145160 * r18145161;
        double r18145163 = 11.1667541262;
        double r18145164 = r18145162 + r18145163;
        double r18145165 = r18145164 * r18145160;
        double r18145166 = t;
        double r18145167 = r18145165 + r18145166;
        double r18145168 = r18145167 * r18145160;
        double r18145169 = a;
        double r18145170 = r18145168 + r18145169;
        double r18145171 = r18145170 * r18145160;
        double r18145172 = b;
        double r18145173 = r18145171 + r18145172;
        double r18145174 = r18145159 * r18145173;
        double r18145175 = 15.234687407;
        double r18145176 = r18145160 + r18145175;
        double r18145177 = r18145176 * r18145160;
        double r18145178 = 31.4690115749;
        double r18145179 = r18145177 + r18145178;
        double r18145180 = r18145179 * r18145160;
        double r18145181 = 11.9400905721;
        double r18145182 = r18145180 + r18145181;
        double r18145183 = r18145182 * r18145160;
        double r18145184 = 0.607771387771;
        double r18145185 = r18145183 + r18145184;
        double r18145186 = r18145174 / r18145185;
        double r18145187 = r18145158 + r18145186;
        return r18145187;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r18145188 = z;
        double r18145189 = -3.80730209029403e+70;
        bool r18145190 = r18145188 <= r18145189;
        double r18145191 = y;
        double r18145192 = t;
        double r18145193 = r18145192 / r18145188;
        double r18145194 = r18145193 / r18145188;
        double r18145195 = 36.527041698806414;
        double r18145196 = r18145195 / r18145188;
        double r18145197 = r18145194 - r18145196;
        double r18145198 = 3.13060547623;
        double r18145199 = r18145197 + r18145198;
        double r18145200 = x;
        double r18145201 = fma(r18145191, r18145199, r18145200);
        double r18145202 = 5.484371003442913e+38;
        bool r18145203 = r18145188 <= r18145202;
        double r18145204 = 11.1667541262;
        double r18145205 = fma(r18145198, r18145188, r18145204);
        double r18145206 = fma(r18145205, r18145188, r18145192);
        double r18145207 = a;
        double r18145208 = fma(r18145188, r18145206, r18145207);
        double r18145209 = b;
        double r18145210 = fma(r18145208, r18145188, r18145209);
        double r18145211 = 1.0;
        double r18145212 = 15.234687407;
        double r18145213 = r18145212 + r18145188;
        double r18145214 = 31.4690115749;
        double r18145215 = fma(r18145188, r18145213, r18145214);
        double r18145216 = 11.9400905721;
        double r18145217 = fma(r18145188, r18145215, r18145216);
        double r18145218 = 0.607771387771;
        double r18145219 = fma(r18145188, r18145217, r18145218);
        double r18145220 = r18145211 / r18145219;
        double r18145221 = r18145210 * r18145220;
        double r18145222 = fma(r18145191, r18145221, r18145200);
        double r18145223 = r18145203 ? r18145222 : r18145201;
        double r18145224 = r18145190 ? r18145201 : r18145223;
        return r18145224;
}

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

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. Simplified60.5

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

    3. Taylor expanded around inf 0.7

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

    4. Simplified0.7

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

    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. Simplified1.3

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

    4. Applied div-inv1.3

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

  4. Final simplification1.1

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

Reproduce

herbie shell --seed 2019174 +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))))