Average Error: 29.6 → 1.3
Time: 23.0s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.375116000296380313236516536594876606522 \cdot 10^{48}:\\ \;\;\;\;\left(\left(\frac{y}{\frac{z \cdot z}{t}} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 2.567100374691052605767061754234624146341 \cdot 10^{70}:\\ \;\;\;\;x + \frac{z \cdot \left(a + \left(t + \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) \cdot z\right) \cdot z\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{\frac{z \cdot z}{t}} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]
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 -1.375116000296380313236516536594876606522 \cdot 10^{48}:\\
\;\;\;\;\left(\left(\frac{y}{\frac{z \cdot z}{t}} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\

\mathbf{elif}\;z \le 2.567100374691052605767061754234624146341 \cdot 10^{70}:\\
\;\;\;\;x + \frac{z \cdot \left(a + \left(t + \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) \cdot z\right) \cdot z\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z} \cdot y\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r20129728 = x;
        double r20129729 = y;
        double r20129730 = z;
        double r20129731 = 3.13060547623;
        double r20129732 = r20129730 * r20129731;
        double r20129733 = 11.1667541262;
        double r20129734 = r20129732 + r20129733;
        double r20129735 = r20129734 * r20129730;
        double r20129736 = t;
        double r20129737 = r20129735 + r20129736;
        double r20129738 = r20129737 * r20129730;
        double r20129739 = a;
        double r20129740 = r20129738 + r20129739;
        double r20129741 = r20129740 * r20129730;
        double r20129742 = b;
        double r20129743 = r20129741 + r20129742;
        double r20129744 = r20129729 * r20129743;
        double r20129745 = 15.234687407;
        double r20129746 = r20129730 + r20129745;
        double r20129747 = r20129746 * r20129730;
        double r20129748 = 31.4690115749;
        double r20129749 = r20129747 + r20129748;
        double r20129750 = r20129749 * r20129730;
        double r20129751 = 11.9400905721;
        double r20129752 = r20129750 + r20129751;
        double r20129753 = r20129752 * r20129730;
        double r20129754 = 0.607771387771;
        double r20129755 = r20129753 + r20129754;
        double r20129756 = r20129744 / r20129755;
        double r20129757 = r20129728 + r20129756;
        return r20129757;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r20129758 = z;
        double r20129759 = -1.3751160002963803e+48;
        bool r20129760 = r20129758 <= r20129759;
        double r20129761 = y;
        double r20129762 = r20129758 * r20129758;
        double r20129763 = t;
        double r20129764 = r20129762 / r20129763;
        double r20129765 = r20129761 / r20129764;
        double r20129766 = 3.13060547623;
        double r20129767 = r20129766 * r20129761;
        double r20129768 = r20129765 + r20129767;
        double r20129769 = 36.527041698806414;
        double r20129770 = r20129758 / r20129761;
        double r20129771 = r20129769 / r20129770;
        double r20129772 = r20129768 - r20129771;
        double r20129773 = x;
        double r20129774 = r20129772 + r20129773;
        double r20129775 = 2.5671003746910526e+70;
        bool r20129776 = r20129758 <= r20129775;
        double r20129777 = a;
        double r20129778 = 11.1667541262;
        double r20129779 = r20129766 * r20129758;
        double r20129780 = r20129778 + r20129779;
        double r20129781 = r20129780 * r20129758;
        double r20129782 = r20129763 + r20129781;
        double r20129783 = r20129782 * r20129758;
        double r20129784 = r20129777 + r20129783;
        double r20129785 = r20129758 * r20129784;
        double r20129786 = b;
        double r20129787 = r20129785 + r20129786;
        double r20129788 = 0.607771387771;
        double r20129789 = 31.4690115749;
        double r20129790 = 15.234687407;
        double r20129791 = r20129758 + r20129790;
        double r20129792 = r20129791 * r20129758;
        double r20129793 = r20129789 + r20129792;
        double r20129794 = r20129758 * r20129793;
        double r20129795 = 11.9400905721;
        double r20129796 = r20129794 + r20129795;
        double r20129797 = r20129796 * r20129758;
        double r20129798 = r20129788 + r20129797;
        double r20129799 = r20129787 / r20129798;
        double r20129800 = r20129799 * r20129761;
        double r20129801 = r20129773 + r20129800;
        double r20129802 = r20129776 ? r20129801 : r20129774;
        double r20129803 = r20129760 ? r20129774 : r20129802;
        return r20129803;
}

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
Target1.2
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 < -1.3751160002963803e+48 or 2.5671003746910526e+70 < z

    1. Initial program 62.2

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

    2. Taylor expanded around inf 9.0

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

    3. Simplified1.0

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

    if -1.3751160002963803e+48 < z < 2.5671003746910526e+70

    1. Initial program 3.8

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

    3. Applied *-un-lft-identity3.8

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

    4. Applied times-frac1.6

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

    5. Simplified1.6

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.375116000296380313236516536594876606522 \cdot 10^{48}:\\ \;\;\;\;\left(\left(\frac{y}{\frac{z \cdot z}{t}} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \mathbf{elif}\;z \le 2.567100374691052605767061754234624146341 \cdot 10^{70}:\\ \;\;\;\;x + \frac{z \cdot \left(a + \left(t + \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) \cdot z\right) \cdot z\right) + b}{0.6077713877710000378584709324059076607227 + \left(z \cdot \left(31.46901157490000144889563671313226222992 + \left(z + 15.2346874069999991263557603815570473671\right) \cdot z\right) + 11.94009057210000079862766142468899488449\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{y}{\frac{z \cdot z}{t}} + 3.130605476229999961645944495103321969509 \cdot y\right) - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))