Average Error: 29.2 → 4.4
Time: 5.8s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.962175411905014607618718304664775132748 \cdot 10^{71} \lor \neg \left(z \le 3.872588552071848293825397508348311355608 \cdot 10^{49}\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]
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 -2.962175411905014607618718304664775132748 \cdot 10^{71} \lor \neg \left(z \le 3.872588552071848293825397508348311355608 \cdot 10^{49}\right):\\
\;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + 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}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r368255 = x;
        double r368256 = y;
        double r368257 = z;
        double r368258 = 3.13060547623;
        double r368259 = r368257 * r368258;
        double r368260 = 11.1667541262;
        double r368261 = r368259 + r368260;
        double r368262 = r368261 * r368257;
        double r368263 = t;
        double r368264 = r368262 + r368263;
        double r368265 = r368264 * r368257;
        double r368266 = a;
        double r368267 = r368265 + r368266;
        double r368268 = r368267 * r368257;
        double r368269 = b;
        double r368270 = r368268 + r368269;
        double r368271 = r368256 * r368270;
        double r368272 = 15.234687407;
        double r368273 = r368257 + r368272;
        double r368274 = r368273 * r368257;
        double r368275 = 31.4690115749;
        double r368276 = r368274 + r368275;
        double r368277 = r368276 * r368257;
        double r368278 = 11.9400905721;
        double r368279 = r368277 + r368278;
        double r368280 = r368279 * r368257;
        double r368281 = 0.607771387771;
        double r368282 = r368280 + r368281;
        double r368283 = r368271 / r368282;
        double r368284 = r368255 + r368283;
        return r368284;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r368285 = z;
        double r368286 = -2.9621754119050146e+71;
        bool r368287 = r368285 <= r368286;
        double r368288 = 3.8725885520718483e+49;
        bool r368289 = r368285 <= r368288;
        double r368290 = !r368289;
        bool r368291 = r368287 || r368290;
        double r368292 = x;
        double r368293 = 3.13060547623;
        double r368294 = y;
        double r368295 = r368293 * r368294;
        double r368296 = t;
        double r368297 = r368296 * r368294;
        double r368298 = 2.0;
        double r368299 = pow(r368285, r368298);
        double r368300 = r368297 / r368299;
        double r368301 = r368295 + r368300;
        double r368302 = 36.527041698806414;
        double r368303 = r368294 / r368285;
        double r368304 = r368302 * r368303;
        double r368305 = r368301 - r368304;
        double r368306 = r368292 + r368305;
        double r368307 = r368285 * r368293;
        double r368308 = 11.1667541262;
        double r368309 = r368307 + r368308;
        double r368310 = r368309 * r368285;
        double r368311 = r368310 + r368296;
        double r368312 = r368311 * r368285;
        double r368313 = a;
        double r368314 = r368312 + r368313;
        double r368315 = r368314 * r368285;
        double r368316 = b;
        double r368317 = r368315 + r368316;
        double r368318 = 15.234687407;
        double r368319 = r368285 + r368318;
        double r368320 = r368319 * r368285;
        double r368321 = 31.4690115749;
        double r368322 = r368320 + r368321;
        double r368323 = r368322 * r368285;
        double r368324 = 11.9400905721;
        double r368325 = r368323 + r368324;
        double r368326 = r368325 * r368285;
        double r368327 = 0.607771387771;
        double r368328 = r368326 + r368327;
        double r368329 = r368317 / r368328;
        double r368330 = r368294 * r368329;
        double r368331 = r368292 + r368330;
        double r368332 = r368291 ? r368306 : r368331;
        return r368332;
}

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

Original29.2
Target0.9
Herbie4.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.9621754119050146e+71 or 3.8725885520718483e+49 < z

    1. Initial program 62.3

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

    2. Taylor expanded around inf 8.4

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

    if -2.9621754119050146e+71 < z < 3.8725885520718483e+49

    1. Initial program 3.7

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

    3. Applied *-un-lft-identity3.7

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

    4. Applied times-frac1.4

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

    5. Simplified1.4

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.962175411905014607618718304664775132748 \cdot 10^{71} \lor \neg \left(z \le 3.872588552071848293825397508348311355608 \cdot 10^{49}\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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 -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (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)))))

  (+ 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))))