Average Error: 28.9 → 1.0
Time: 28.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000 \lor \neg \left(z \le 854771490603964610616754176\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{z}, y, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.2346874069999991263557603815570473671 + z, z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}}, x\right)\\ \end{array}\]
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 -11377676592436386882634106867972293791840000 \lor \neg \left(z \le 854771490603964610616754176\right):\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{z}, y, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.2346874069999991263557603815570473671 + z, z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r278217 = x;
        double r278218 = y;
        double r278219 = z;
        double r278220 = 3.13060547623;
        double r278221 = r278219 * r278220;
        double r278222 = 11.1667541262;
        double r278223 = r278221 + r278222;
        double r278224 = r278223 * r278219;
        double r278225 = t;
        double r278226 = r278224 + r278225;
        double r278227 = r278226 * r278219;
        double r278228 = a;
        double r278229 = r278227 + r278228;
        double r278230 = r278229 * r278219;
        double r278231 = b;
        double r278232 = r278230 + r278231;
        double r278233 = r278218 * r278232;
        double r278234 = 15.234687407;
        double r278235 = r278219 + r278234;
        double r278236 = r278235 * r278219;
        double r278237 = 31.4690115749;
        double r278238 = r278236 + r278237;
        double r278239 = r278238 * r278219;
        double r278240 = 11.9400905721;
        double r278241 = r278239 + r278240;
        double r278242 = r278241 * r278219;
        double r278243 = 0.607771387771;
        double r278244 = r278242 + r278243;
        double r278245 = r278233 / r278244;
        double r278246 = r278217 + r278245;
        return r278246;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r278247 = z;
        double r278248 = -1.1377676592436387e+43;
        bool r278249 = r278247 <= r278248;
        double r278250 = 8.547714906039646e+26;
        bool r278251 = r278247 <= r278250;
        double r278252 = !r278251;
        bool r278253 = r278249 || r278252;
        double r278254 = t;
        double r278255 = r278254 / r278247;
        double r278256 = r278255 / r278247;
        double r278257 = y;
        double r278258 = 3.13060547623;
        double r278259 = x;
        double r278260 = fma(r278258, r278257, r278259);
        double r278261 = fma(r278256, r278257, r278260);
        double r278262 = 1.0;
        double r278263 = 15.234687407;
        double r278264 = r278263 + r278247;
        double r278265 = 31.4690115749;
        double r278266 = fma(r278264, r278247, r278265);
        double r278267 = 11.9400905721;
        double r278268 = fma(r278247, r278266, r278267);
        double r278269 = 0.607771387771;
        double r278270 = fma(r278247, r278268, r278269);
        double r278271 = 11.1667541262;
        double r278272 = fma(r278258, r278247, r278271);
        double r278273 = fma(r278272, r278247, r278254);
        double r278274 = a;
        double r278275 = fma(r278247, r278273, r278274);
        double r278276 = b;
        double r278277 = fma(r278275, r278247, r278276);
        double r278278 = r278270 / r278277;
        double r278279 = r278262 / r278278;
        double r278280 = fma(r278257, r278279, r278259);
        double r278281 = r278253 ? r278261 : r278280;
        return r278281;
}

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

Original28.9
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 < -1.1377676592436387e+43 or 8.547714906039646e+26 < z

    1. Initial program 59.2

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

    2. Simplified56.4

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

    4. Applied clear-num56.4

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

    5. Simplified56.4

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

    6. Taylor expanded around inf 9.0

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

    7. Simplified1.5

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

    if -1.1377676592436387e+43 < z < 8.547714906039646e+26

    1. Initial program 1.3

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

    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 clear-num0.6

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

    5. Simplified0.6

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -11377676592436386882634106867972293791840000 \lor \neg \left(z \le 854771490603964610616754176\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{t}{z}}{z}, y, \mathsf{fma}\left(3.130605476229999961645944495103321969509, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.2346874069999991263557603815570473671 + z, z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))