Average Error: 29.3 → 1.3
Time: 38.7s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.697774109576783041065302598687410151924 \cdot 10^{70}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z}{\frac{y}{z}}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 95577221718968741136415336631237087330300:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z}{\frac{y}{z}}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \end{array}\]
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 -3.697774109576783041065302598687410151924 \cdot 10^{70}:\\
\;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z}{\frac{y}{z}}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\

\mathbf{elif}\;z \le 95577221718968741136415336631237087330300:\\
\;\;\;\;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}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r72158283 = x;
        double r72158284 = y;
        double r72158285 = z;
        double r72158286 = 3.13060547623;
        double r72158287 = r72158285 * r72158286;
        double r72158288 = 11.1667541262;
        double r72158289 = r72158287 + r72158288;
        double r72158290 = r72158289 * r72158285;
        double r72158291 = t;
        double r72158292 = r72158290 + r72158291;
        double r72158293 = r72158292 * r72158285;
        double r72158294 = a;
        double r72158295 = r72158293 + r72158294;
        double r72158296 = r72158295 * r72158285;
        double r72158297 = b;
        double r72158298 = r72158296 + r72158297;
        double r72158299 = r72158284 * r72158298;
        double r72158300 = 15.234687407;
        double r72158301 = r72158285 + r72158300;
        double r72158302 = r72158301 * r72158285;
        double r72158303 = 31.4690115749;
        double r72158304 = r72158302 + r72158303;
        double r72158305 = r72158304 * r72158285;
        double r72158306 = 11.9400905721;
        double r72158307 = r72158305 + r72158306;
        double r72158308 = r72158307 * r72158285;
        double r72158309 = 0.607771387771;
        double r72158310 = r72158308 + r72158309;
        double r72158311 = r72158299 / r72158310;
        double r72158312 = r72158283 + r72158311;
        return r72158312;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r72158313 = z;
        double r72158314 = -3.697774109576783e+70;
        bool r72158315 = r72158313 <= r72158314;
        double r72158316 = x;
        double r72158317 = 3.13060547623;
        double r72158318 = y;
        double r72158319 = r72158317 * r72158318;
        double r72158320 = t;
        double r72158321 = r72158318 / r72158313;
        double r72158322 = r72158313 / r72158321;
        double r72158323 = r72158320 / r72158322;
        double r72158324 = r72158319 + r72158323;
        double r72158325 = 36.527041698806414;
        double r72158326 = r72158325 * r72158321;
        double r72158327 = r72158324 - r72158326;
        double r72158328 = r72158316 + r72158327;
        double r72158329 = 9.557722171896874e+40;
        bool r72158330 = r72158313 <= r72158329;
        double r72158331 = r72158313 * r72158317;
        double r72158332 = 11.1667541262;
        double r72158333 = r72158331 + r72158332;
        double r72158334 = r72158333 * r72158313;
        double r72158335 = r72158334 + r72158320;
        double r72158336 = r72158335 * r72158313;
        double r72158337 = a;
        double r72158338 = r72158336 + r72158337;
        double r72158339 = r72158338 * r72158313;
        double r72158340 = b;
        double r72158341 = r72158339 + r72158340;
        double r72158342 = 15.234687407;
        double r72158343 = r72158313 + r72158342;
        double r72158344 = r72158343 * r72158313;
        double r72158345 = 31.4690115749;
        double r72158346 = r72158344 + r72158345;
        double r72158347 = r72158346 * r72158313;
        double r72158348 = 11.9400905721;
        double r72158349 = r72158347 + r72158348;
        double r72158350 = r72158349 * r72158313;
        double r72158351 = 0.607771387771;
        double r72158352 = r72158350 + r72158351;
        double r72158353 = r72158341 / r72158352;
        double r72158354 = r72158318 * r72158353;
        double r72158355 = r72158316 + r72158354;
        double r72158356 = r72158330 ? r72158355 : r72158328;
        double r72158357 = r72158315 ? r72158328 : r72158356;
        return r72158357;
}

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

Original29.3
Target1.2
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.697774109576783e+70 or 9.557722171896874e+40 < z

    1. Initial program 61.6

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

    2. Taylor expanded around inf 8.8

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

    3. Simplified1.2

      \[\leadsto x + \color{blue}{\left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z \cdot z}{y}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]
    4. Using strategy rm

    5. Applied associate-/l*1.2

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

    if -3.697774109576783e+70 < z < 9.557722171896874e+40

    1. Initial program 3.4

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

    3. Applied *-un-lft-identity3.4

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

    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 simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.697774109576783041065302598687410151924 \cdot 10^{70}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z}{\frac{y}{z}}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{elif}\;z \le 95577221718968741136415336631237087330300:\\ \;\;\;\;x + y \cdot \frac{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t}{\frac{z}{\frac{y}{z}}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019173 
(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))))