Average Error: 29.3 → 4.2
Time: 5.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.47363773233135832562413998511827618689 \cdot 10^{49} \lor \neg \left(z \le 28286845129981610611744656683406721024\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]
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 -3.47363773233135832562413998511827618689 \cdot 10^{49} \lor \neg \left(z \le 28286845129981610611744656683406721024\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 + \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}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r502398 = x;
        double r502399 = y;
        double r502400 = z;
        double r502401 = 3.13060547623;
        double r502402 = r502400 * r502401;
        double r502403 = 11.1667541262;
        double r502404 = r502402 + r502403;
        double r502405 = r502404 * r502400;
        double r502406 = t;
        double r502407 = r502405 + r502406;
        double r502408 = r502407 * r502400;
        double r502409 = a;
        double r502410 = r502408 + r502409;
        double r502411 = r502410 * r502400;
        double r502412 = b;
        double r502413 = r502411 + r502412;
        double r502414 = r502399 * r502413;
        double r502415 = 15.234687407;
        double r502416 = r502400 + r502415;
        double r502417 = r502416 * r502400;
        double r502418 = 31.4690115749;
        double r502419 = r502417 + r502418;
        double r502420 = r502419 * r502400;
        double r502421 = 11.9400905721;
        double r502422 = r502420 + r502421;
        double r502423 = r502422 * r502400;
        double r502424 = 0.607771387771;
        double r502425 = r502423 + r502424;
        double r502426 = r502414 / r502425;
        double r502427 = r502398 + r502426;
        return r502427;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r502428 = z;
        double r502429 = -3.4736377323313583e+49;
        bool r502430 = r502428 <= r502429;
        double r502431 = 2.828684512998161e+37;
        bool r502432 = r502428 <= r502431;
        double r502433 = !r502432;
        bool r502434 = r502430 || r502433;
        double r502435 = x;
        double r502436 = 3.13060547623;
        double r502437 = y;
        double r502438 = r502436 * r502437;
        double r502439 = t;
        double r502440 = r502439 * r502437;
        double r502441 = 2.0;
        double r502442 = pow(r502428, r502441);
        double r502443 = r502440 / r502442;
        double r502444 = r502438 + r502443;
        double r502445 = 36.527041698806414;
        double r502446 = r502437 / r502428;
        double r502447 = r502445 * r502446;
        double r502448 = r502444 - r502447;
        double r502449 = r502435 + r502448;
        double r502450 = 15.234687407;
        double r502451 = r502428 + r502450;
        double r502452 = r502451 * r502428;
        double r502453 = 31.4690115749;
        double r502454 = r502452 + r502453;
        double r502455 = r502454 * r502428;
        double r502456 = 11.9400905721;
        double r502457 = r502455 + r502456;
        double r502458 = r502457 * r502428;
        double r502459 = 0.607771387771;
        double r502460 = r502458 + r502459;
        double r502461 = r502428 * r502436;
        double r502462 = 11.1667541262;
        double r502463 = r502461 + r502462;
        double r502464 = r502463 * r502428;
        double r502465 = r502464 + r502439;
        double r502466 = r502465 * r502428;
        double r502467 = a;
        double r502468 = r502466 + r502467;
        double r502469 = r502468 * r502428;
        double r502470 = b;
        double r502471 = r502469 + r502470;
        double r502472 = r502460 / r502471;
        double r502473 = r502437 / r502472;
        double r502474 = r502435 + r502473;
        double r502475 = r502434 ? r502449 : r502474;
        return r502475;
}

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
Target1.0
Herbie4.2
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.4736377323313583e+49 or 2.828684512998161e+37 < z

    1. Initial program 60.6

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

    2. Taylor expanded around inf 8.2

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

    if -3.4736377323313583e+49 < z < 2.828684512998161e+37

    1. Initial program 2.1

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

    3. Applied associate-/l*0.8

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

  4. Final simplification4.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.47363773233135832562413998511827618689 \cdot 10^{49} \lor \neg \left(z \le 28286845129981610611744656683406721024\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(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))))