Average Error: 29.4 → 1.0
Time: 23.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -9.17536549406960862736530076972690071175 \cdot 10^{51} \lor \neg \left(z \le 31413547445392871784448\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\sqrt[3]{\frac{t}{z}}}{z} \cdot \left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \end{array}\]
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 -9.17536549406960862736530076972690071175 \cdot 10^{51} \lor \neg \left(z \le 31413547445392871784448\right):\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\sqrt[3]{\frac{t}{z}}}{z} \cdot \left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right) + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r258189 = x;
        double r258190 = y;
        double r258191 = z;
        double r258192 = 3.13060547623;
        double r258193 = r258191 * r258192;
        double r258194 = 11.1667541262;
        double r258195 = r258193 + r258194;
        double r258196 = r258195 * r258191;
        double r258197 = t;
        double r258198 = r258196 + r258197;
        double r258199 = r258198 * r258191;
        double r258200 = a;
        double r258201 = r258199 + r258200;
        double r258202 = r258201 * r258191;
        double r258203 = b;
        double r258204 = r258202 + r258203;
        double r258205 = r258190 * r258204;
        double r258206 = 15.234687407;
        double r258207 = r258191 + r258206;
        double r258208 = r258207 * r258191;
        double r258209 = 31.4690115749;
        double r258210 = r258208 + r258209;
        double r258211 = r258210 * r258191;
        double r258212 = 11.9400905721;
        double r258213 = r258211 + r258212;
        double r258214 = r258213 * r258191;
        double r258215 = 0.607771387771;
        double r258216 = r258214 + r258215;
        double r258217 = r258205 / r258216;
        double r258218 = r258189 + r258217;
        return r258218;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r258219 = z;
        double r258220 = -9.175365494069609e+51;
        bool r258221 = r258219 <= r258220;
        double r258222 = 3.141354744539287e+22;
        bool r258223 = r258219 <= r258222;
        double r258224 = !r258223;
        bool r258225 = r258221 || r258224;
        double r258226 = y;
        double r258227 = t;
        double r258228 = r258227 / r258219;
        double r258229 = cbrt(r258228);
        double r258230 = r258229 / r258219;
        double r258231 = r258229 * r258229;
        double r258232 = r258230 * r258231;
        double r258233 = 3.13060547623;
        double r258234 = r258232 + r258233;
        double r258235 = x;
        double r258236 = fma(r258226, r258234, r258235);
        double r258237 = 11.1667541262;
        double r258238 = fma(r258233, r258219, r258237);
        double r258239 = fma(r258238, r258219, r258227);
        double r258240 = a;
        double r258241 = fma(r258219, r258239, r258240);
        double r258242 = b;
        double r258243 = fma(r258241, r258219, r258242);
        double r258244 = 15.234687407;
        double r258245 = r258244 + r258219;
        double r258246 = 31.4690115749;
        double r258247 = fma(r258219, r258245, r258246);
        double r258248 = 11.9400905721;
        double r258249 = fma(r258219, r258247, r258248);
        double r258250 = 0.607771387771;
        double r258251 = fma(r258219, r258249, r258250);
        double r258252 = r258243 / r258251;
        double r258253 = fma(r258226, r258252, r258235);
        double r258254 = r258225 ? r258236 : r258253;
        return r258254;
}

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
Target1.0
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 < -9.175365494069609e+51 or 3.141354744539287e+22 < z

    1. Initial program 59.9

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 9.2

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

    4. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\frac{t}{z}}{z}, x\right)}\]
    5. Using strategy rm

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

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

    7. Applied add-cube-cbrt1.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{\color{blue}{\left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right) \cdot \sqrt[3]{\frac{t}{z}}}}{1 \cdot z}, x\right)\]

    8. Applied times-frac1.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{\frac{\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}}{1} \cdot \frac{\sqrt[3]{\frac{t}{z}}}{z}}, x\right)\]

    9. Simplified1.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{\left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right)} \cdot \frac{\sqrt[3]{\frac{t}{z}}}{z}, x\right)\]

    if -9.175365494069609e+51 < z < 3.141354744539287e+22

    1. Initial program 1.7

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

    2. Simplified0.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -9.17536549406960862736530076972690071175 \cdot 10^{51} \lor \neg \left(z \le 31413547445392871784448\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\sqrt[3]{\frac{t}{z}}}{z} \cdot \left(\sqrt[3]{\frac{t}{z}} \cdot \sqrt[3]{\frac{t}{z}}\right) + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.2346874069999991263557603815570473671 + z, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +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))))