Average Error: 28.9 → 4.6
Time: 15.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -309142137143483328 \lor \neg \left(z \le 1423778970433199129219338790265421824\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z} \cdot \sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z}\right) \cdot \sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z} + a\right) \cdot z + b}}\\ \end{array}\]
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 -309142137143483328 \lor \neg \left(z \le 1423778970433199129219338790265421824\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(\sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z} \cdot \sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z}\right) \cdot \sqrt[3]{\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 r404448 = x;
        double r404449 = y;
        double r404450 = z;
        double r404451 = 3.13060547623;
        double r404452 = r404450 * r404451;
        double r404453 = 11.1667541262;
        double r404454 = r404452 + r404453;
        double r404455 = r404454 * r404450;
        double r404456 = t;
        double r404457 = r404455 + r404456;
        double r404458 = r404457 * r404450;
        double r404459 = a;
        double r404460 = r404458 + r404459;
        double r404461 = r404460 * r404450;
        double r404462 = b;
        double r404463 = r404461 + r404462;
        double r404464 = r404449 * r404463;
        double r404465 = 15.234687407;
        double r404466 = r404450 + r404465;
        double r404467 = r404466 * r404450;
        double r404468 = 31.4690115749;
        double r404469 = r404467 + r404468;
        double r404470 = r404469 * r404450;
        double r404471 = 11.9400905721;
        double r404472 = r404470 + r404471;
        double r404473 = r404472 * r404450;
        double r404474 = 0.607771387771;
        double r404475 = r404473 + r404474;
        double r404476 = r404464 / r404475;
        double r404477 = r404448 + r404476;
        return r404477;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r404478 = z;
        double r404479 = -3.091421371434833e+17;
        bool r404480 = r404478 <= r404479;
        double r404481 = 1.423778970433199e+36;
        bool r404482 = r404478 <= r404481;
        double r404483 = !r404482;
        bool r404484 = r404480 || r404483;
        double r404485 = x;
        double r404486 = 3.13060547623;
        double r404487 = y;
        double r404488 = r404486 * r404487;
        double r404489 = t;
        double r404490 = r404489 * r404487;
        double r404491 = 2.0;
        double r404492 = pow(r404478, r404491);
        double r404493 = r404490 / r404492;
        double r404494 = r404488 + r404493;
        double r404495 = 36.527041698806414;
        double r404496 = r404487 / r404478;
        double r404497 = r404495 * r404496;
        double r404498 = r404494 - r404497;
        double r404499 = r404485 + r404498;
        double r404500 = 15.234687407;
        double r404501 = r404478 + r404500;
        double r404502 = r404501 * r404478;
        double r404503 = 31.4690115749;
        double r404504 = r404502 + r404503;
        double r404505 = r404504 * r404478;
        double r404506 = 11.9400905721;
        double r404507 = r404505 + r404506;
        double r404508 = r404507 * r404478;
        double r404509 = 0.607771387771;
        double r404510 = r404508 + r404509;
        double r404511 = r404478 * r404486;
        double r404512 = 11.1667541262;
        double r404513 = r404511 + r404512;
        double r404514 = r404513 * r404478;
        double r404515 = r404514 + r404489;
        double r404516 = r404515 * r404478;
        double r404517 = cbrt(r404516);
        double r404518 = r404517 * r404517;
        double r404519 = r404518 * r404517;
        double r404520 = a;
        double r404521 = r404519 + r404520;
        double r404522 = r404521 * r404478;
        double r404523 = b;
        double r404524 = r404522 + r404523;
        double r404525 = r404510 / r404524;
        double r404526 = r404487 / r404525;
        double r404527 = r404485 + r404526;
        double r404528 = r404484 ? r404499 : r404527;
        return r404528;
}

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

Original28.9
Target1.0
Herbie4.6
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.091421371434833e+17 or 1.423778970433199e+36 < z

    1. Initial program 58.5

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

    2. Taylor expanded around inf 8.9

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

    if -3.091421371434833e+17 < z < 1.423778970433199e+36

    1. Initial program 1.0

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

    3. Applied associate-/l*0.5

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

    5. Applied add-cube-cbrt0.6

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

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -309142137143483328 \lor \neg \left(z \le 1423778970433199129219338790265421824\right):\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z} \cdot \sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z}\right) \cdot \sqrt[3]{\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z} + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(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))))