Average Error: 29.6 → 4.4
Time: 6.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.978483443952488632012241591332400099399 \cdot 10^{48} \lor \neg \left(z \le 8.559962113309013051011544276033877945173 \cdot 10^{55}\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]
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 -3.978483443952488632012241591332400099399 \cdot 10^{48} \lor \neg \left(z \le 8.559962113309013051011544276033877945173 \cdot 10^{55}\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 r397415 = x;
        double r397416 = y;
        double r397417 = z;
        double r397418 = 3.13060547623;
        double r397419 = r397417 * r397418;
        double r397420 = 11.1667541262;
        double r397421 = r397419 + r397420;
        double r397422 = r397421 * r397417;
        double r397423 = t;
        double r397424 = r397422 + r397423;
        double r397425 = r397424 * r397417;
        double r397426 = a;
        double r397427 = r397425 + r397426;
        double r397428 = r397427 * r397417;
        double r397429 = b;
        double r397430 = r397428 + r397429;
        double r397431 = r397416 * r397430;
        double r397432 = 15.234687407;
        double r397433 = r397417 + r397432;
        double r397434 = r397433 * r397417;
        double r397435 = 31.4690115749;
        double r397436 = r397434 + r397435;
        double r397437 = r397436 * r397417;
        double r397438 = 11.9400905721;
        double r397439 = r397437 + r397438;
        double r397440 = r397439 * r397417;
        double r397441 = 0.607771387771;
        double r397442 = r397440 + r397441;
        double r397443 = r397431 / r397442;
        double r397444 = r397415 + r397443;
        return r397444;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r397445 = z;
        double r397446 = -3.9784834439524886e+48;
        bool r397447 = r397445 <= r397446;
        double r397448 = 8.559962113309013e+55;
        bool r397449 = r397445 <= r397448;
        double r397450 = !r397449;
        bool r397451 = r397447 || r397450;
        double r397452 = x;
        double r397453 = 3.13060547623;
        double r397454 = y;
        double r397455 = r397453 * r397454;
        double r397456 = t;
        double r397457 = r397456 * r397454;
        double r397458 = 2.0;
        double r397459 = pow(r397445, r397458);
        double r397460 = r397457 / r397459;
        double r397461 = r397455 + r397460;
        double r397462 = 36.527041698806414;
        double r397463 = r397454 / r397445;
        double r397464 = r397462 * r397463;
        double r397465 = r397461 - r397464;
        double r397466 = r397452 + r397465;
        double r397467 = 15.234687407;
        double r397468 = r397445 + r397467;
        double r397469 = r397468 * r397445;
        double r397470 = 31.4690115749;
        double r397471 = r397469 + r397470;
        double r397472 = r397471 * r397445;
        double r397473 = 11.9400905721;
        double r397474 = r397472 + r397473;
        double r397475 = r397474 * r397445;
        double r397476 = 0.607771387771;
        double r397477 = r397475 + r397476;
        double r397478 = r397445 * r397453;
        double r397479 = 11.1667541262;
        double r397480 = r397478 + r397479;
        double r397481 = r397480 * r397445;
        double r397482 = r397481 + r397456;
        double r397483 = r397482 * r397445;
        double r397484 = a;
        double r397485 = r397483 + r397484;
        double r397486 = r397485 * r397445;
        double r397487 = b;
        double r397488 = r397486 + r397487;
        double r397489 = r397477 / r397488;
        double r397490 = r397454 / r397489;
        double r397491 = r397452 + r397490;
        double r397492 = r397451 ? r397466 : r397491;
        return r397492;
}

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

Original29.6
Target1.0
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 < -3.9784834439524886e+48 or 8.559962113309013e+55 < 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 8.4

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

    if -3.9784834439524886e+48 < z < 8.559962113309013e+55

    1. Initial program 2.7

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

    3. Applied associate-/l*1.0

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

  4. Final simplification4.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.978483443952488632012241591332400099399 \cdot 10^{48} \lor \neg \left(z \le 8.559962113309013051011544276033877945173 \cdot 10^{55}\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))