Average Error: 29.4 → 1.0
Time: 21.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)\\ \end{array}\]
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 -3240660210645078190185871572992:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\

\mathbf{elif}\;z \le 587734457247325707726001012736:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r12531785 = x;
        double r12531786 = y;
        double r12531787 = z;
        double r12531788 = 3.13060547623;
        double r12531789 = r12531787 * r12531788;
        double r12531790 = 11.1667541262;
        double r12531791 = r12531789 + r12531790;
        double r12531792 = r12531791 * r12531787;
        double r12531793 = t;
        double r12531794 = r12531792 + r12531793;
        double r12531795 = r12531794 * r12531787;
        double r12531796 = a;
        double r12531797 = r12531795 + r12531796;
        double r12531798 = r12531797 * r12531787;
        double r12531799 = b;
        double r12531800 = r12531798 + r12531799;
        double r12531801 = r12531786 * r12531800;
        double r12531802 = 15.234687407;
        double r12531803 = r12531787 + r12531802;
        double r12531804 = r12531803 * r12531787;
        double r12531805 = 31.4690115749;
        double r12531806 = r12531804 + r12531805;
        double r12531807 = r12531806 * r12531787;
        double r12531808 = 11.9400905721;
        double r12531809 = r12531807 + r12531808;
        double r12531810 = r12531809 * r12531787;
        double r12531811 = 0.607771387771;
        double r12531812 = r12531810 + r12531811;
        double r12531813 = r12531801 / r12531812;
        double r12531814 = r12531785 + r12531813;
        return r12531814;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r12531815 = z;
        double r12531816 = -3.240660210645078e+30;
        bool r12531817 = r12531815 <= r12531816;
        double r12531818 = y;
        double r12531819 = t;
        double r12531820 = r12531819 / r12531815;
        double r12531821 = r12531820 / r12531815;
        double r12531822 = 3.13060547623;
        double r12531823 = r12531821 + r12531822;
        double r12531824 = x;
        double r12531825 = fma(r12531818, r12531823, r12531824);
        double r12531826 = 5.877344572473257e+29;
        bool r12531827 = r12531815 <= r12531826;
        double r12531828 = 1.0;
        double r12531829 = 15.234687407;
        double r12531830 = r12531815 + r12531829;
        double r12531831 = 31.4690115749;
        double r12531832 = fma(r12531815, r12531830, r12531831);
        double r12531833 = 11.9400905721;
        double r12531834 = fma(r12531815, r12531832, r12531833);
        double r12531835 = 0.607771387771;
        double r12531836 = fma(r12531815, r12531834, r12531835);
        double r12531837 = r12531828 / r12531836;
        double r12531838 = 11.1667541262;
        double r12531839 = fma(r12531822, r12531815, r12531838);
        double r12531840 = fma(r12531839, r12531815, r12531819);
        double r12531841 = a;
        double r12531842 = fma(r12531815, r12531840, r12531841);
        double r12531843 = b;
        double r12531844 = fma(r12531842, r12531815, r12531843);
        double r12531845 = r12531837 * r12531844;
        double r12531846 = fma(r12531818, r12531845, r12531824);
        double r12531847 = r12531815 * r12531815;
        double r12531848 = r12531819 / r12531847;
        double r12531849 = r12531822 + r12531848;
        double r12531850 = fma(r12531818, r12531849, r12531824);
        double r12531851 = r12531827 ? r12531846 : r12531850;
        double r12531852 = r12531817 ? r12531825 : r12531851;
        return r12531852;
}

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

Target

Original29.4
Target0.8
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -3.240660210645078e+30

    1. Initial program 58.8

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

    2. Simplified56.0

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

    3. Taylor expanded around inf 8.2

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

    4. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)}\]
    5. Using strategy rm

    6. Applied associate-/r*1.3

      \[\leadsto \mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \color{blue}{\frac{\frac{t}{z}}{z}}, x\right)\]

    if -3.240660210645078e+30 < z < 5.877344572473257e+29

    1. Initial program 1.1

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

    2. Simplified0.5

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

    4. Applied div-inv0.6

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

    if 5.877344572473257e+29 < z

    1. Initial program 58.9

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

    2. Simplified55.9

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

    3. Taylor expanded around inf 10.1

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

    4. Simplified1.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3240660210645078190185871572992:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.130605476229999961645944495103321969509, x\right)\\ \mathbf{elif}\;z \le 587734457247325707726001012736:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.2346874069999991263557603815570473671, 31.46901157490000144889563671313226222992\right), 11.94009057210000079862766142468899488449\right), 0.6077713877710000378584709324059076607227\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.130605476229999961645944495103321969509, z, 11.16675412620000074070958362426608800888\right), z, t\right), a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.130605476229999961645944495103321969509 + \frac{t}{z \cdot z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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))))