Average Error: 29.2 → 1.2
Time: 22.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -284409547868809266809657528464945600004100 \lor \neg \left(z \le 2055517233392067584\right):\\ \;\;\;\;x + \mathsf{fma}\left(-\frac{y}{z}, 36.52704169880641416057187598198652267456, y \cdot \left(3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}}\\ \end{array}\]
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 -284409547868809266809657528464945600004100 \lor \neg \left(z \le 2055517233392067584\right):\\
\;\;\;\;x + \mathsf{fma}\left(-\frac{y}{z}, 36.52704169880641416057187598198652267456, y \cdot \left(3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r258236 = x;
        double r258237 = y;
        double r258238 = z;
        double r258239 = 3.13060547623;
        double r258240 = r258238 * r258239;
        double r258241 = 11.1667541262;
        double r258242 = r258240 + r258241;
        double r258243 = r258242 * r258238;
        double r258244 = t;
        double r258245 = r258243 + r258244;
        double r258246 = r258245 * r258238;
        double r258247 = a;
        double r258248 = r258246 + r258247;
        double r258249 = r258248 * r258238;
        double r258250 = b;
        double r258251 = r258249 + r258250;
        double r258252 = r258237 * r258251;
        double r258253 = 15.234687407;
        double r258254 = r258238 + r258253;
        double r258255 = r258254 * r258238;
        double r258256 = 31.4690115749;
        double r258257 = r258255 + r258256;
        double r258258 = r258257 * r258238;
        double r258259 = 11.9400905721;
        double r258260 = r258258 + r258259;
        double r258261 = r258260 * r258238;
        double r258262 = 0.607771387771;
        double r258263 = r258261 + r258262;
        double r258264 = r258252 / r258263;
        double r258265 = r258236 + r258264;
        return r258265;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r258266 = z;
        double r258267 = -2.8440954786880927e+41;
        bool r258268 = r258266 <= r258267;
        double r258269 = 2.0555172333920676e+18;
        bool r258270 = r258266 <= r258269;
        double r258271 = !r258270;
        bool r258272 = r258268 || r258271;
        double r258273 = x;
        double r258274 = y;
        double r258275 = r258274 / r258266;
        double r258276 = -r258275;
        double r258277 = 36.527041698806414;
        double r258278 = 3.13060547623;
        double r258279 = t;
        double r258280 = 2.0;
        double r258281 = pow(r258266, r258280);
        double r258282 = r258279 / r258281;
        double r258283 = r258278 + r258282;
        double r258284 = r258274 * r258283;
        double r258285 = fma(r258276, r258277, r258284);
        double r258286 = r258273 + r258285;
        double r258287 = 15.234687407;
        double r258288 = r258266 + r258287;
        double r258289 = 31.4690115749;
        double r258290 = fma(r258288, r258266, r258289);
        double r258291 = 11.9400905721;
        double r258292 = fma(r258290, r258266, r258291);
        double r258293 = 0.607771387771;
        double r258294 = fma(r258292, r258266, r258293);
        double r258295 = 11.1667541262;
        double r258296 = fma(r258266, r258278, r258295);
        double r258297 = fma(r258296, r258266, r258279);
        double r258298 = a;
        double r258299 = fma(r258297, r258266, r258298);
        double r258300 = b;
        double r258301 = fma(r258299, r258266, r258300);
        double r258302 = r258294 / r258301;
        double r258303 = r258274 / r258302;
        double r258304 = r258273 + r258303;
        double r258305 = r258272 ? r258286 : r258304;
        return r258305;
}

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

Original29.2
Target1.0
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.8440954786880927e+41 or 2.0555172333920676e+18 < z

    1. Initial program 58.6

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

    3. Applied associate-/l*55.6

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

    4. Simplified55.6

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

    5. Taylor expanded around inf 8.8

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

    6. Simplified1.8

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

    if -2.8440954786880927e+41 < z < 2.0555172333920676e+18

    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 associate-/l*0.6

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

    4. Simplified0.6

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

  4. Final simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -284409547868809266809657528464945600004100 \lor \neg \left(z \le 2055517233392067584\right):\\ \;\;\;\;x + \mathsf{fma}\left(-\frac{y}{z}, 36.52704169880641416057187598198652267456, y \cdot \left(3.130605476229999961645944495103321969509 + \frac{t}{{z}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.2346874069999991263557603815570473671, z, 31.46901157490000144889563671313226222992\right), z, 11.94009057210000079862766142468899488449\right), z, 0.6077713877710000378584709324059076607227\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.130605476229999961645944495103321969509, 11.16675412620000074070958362426608800888\right), z, t\right), z, a\right), z, b\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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 -6.4993449962526318e53) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.0669654369142868e59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004) (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004))))