Average Error: 29.7 → 4.6
Time: 5.6s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.24583959860649812 \cdot 10^{53} \lor \neg \left(z \le 1.6181471283118757 \cdot 10^{58}\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.24583959860649812 \cdot 10^{53} \lor \neg \left(z \le 1.6181471283118757 \cdot 10^{58}\right):\\
\;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r296731 = x;
        double r296732 = y;
        double r296733 = z;
        double r296734 = 3.13060547623;
        double r296735 = r296733 * r296734;
        double r296736 = 11.1667541262;
        double r296737 = r296735 + r296736;
        double r296738 = r296737 * r296733;
        double r296739 = t;
        double r296740 = r296738 + r296739;
        double r296741 = r296740 * r296733;
        double r296742 = a;
        double r296743 = r296741 + r296742;
        double r296744 = r296743 * r296733;
        double r296745 = b;
        double r296746 = r296744 + r296745;
        double r296747 = r296732 * r296746;
        double r296748 = 15.234687407;
        double r296749 = r296733 + r296748;
        double r296750 = r296749 * r296733;
        double r296751 = 31.4690115749;
        double r296752 = r296750 + r296751;
        double r296753 = r296752 * r296733;
        double r296754 = 11.9400905721;
        double r296755 = r296753 + r296754;
        double r296756 = r296755 * r296733;
        double r296757 = 0.607771387771;
        double r296758 = r296756 + r296757;
        double r296759 = r296747 / r296758;
        double r296760 = r296731 + r296759;
        return r296760;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r296761 = z;
        double r296762 = -1.2458395986064981e+53;
        bool r296763 = r296761 <= r296762;
        double r296764 = 1.6181471283118757e+58;
        bool r296765 = r296761 <= r296764;
        double r296766 = !r296765;
        bool r296767 = r296763 || r296766;
        double r296768 = x;
        double r296769 = y;
        double r296770 = 3.13060547623;
        double r296771 = t;
        double r296772 = r296771 * r296769;
        double r296773 = 2.0;
        double r296774 = pow(r296761, r296773);
        double r296775 = r296772 / r296774;
        double r296776 = 36.527041698806414;
        double r296777 = r296769 / r296761;
        double r296778 = r296776 * r296777;
        double r296779 = r296775 - r296778;
        double r296780 = fma(r296769, r296770, r296779);
        double r296781 = r296768 + r296780;
        double r296782 = 15.234687407;
        double r296783 = r296761 + r296782;
        double r296784 = r296783 * r296761;
        double r296785 = 31.4690115749;
        double r296786 = r296784 + r296785;
        double r296787 = r296786 * r296761;
        double r296788 = 11.9400905721;
        double r296789 = r296787 + r296788;
        double r296790 = r296789 * r296761;
        double r296791 = 0.607771387771;
        double r296792 = r296790 + r296791;
        double r296793 = cbrt(r296792);
        double r296794 = r296793 * r296793;
        double r296795 = r296769 / r296794;
        double r296796 = r296761 * r296770;
        double r296797 = 11.1667541262;
        double r296798 = r296796 + r296797;
        double r296799 = r296798 * r296761;
        double r296800 = r296799 + r296771;
        double r296801 = r296800 * r296761;
        double r296802 = a;
        double r296803 = r296801 + r296802;
        double r296804 = r296803 * r296761;
        double r296805 = b;
        double r296806 = r296804 + r296805;
        double r296807 = r296806 / r296793;
        double r296808 = r296795 * r296807;
        double r296809 = r296768 + r296808;
        double r296810 = r296767 ? r296781 : r296809;
        return r296810;
}

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

Original29.7
Target1.0
Herbie4.6
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.2458395986064981e+53 or 1.6181471283118757e+58 < z

    1. Initial program 61.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]

    2. Taylor expanded around inf 8.3

      \[\leadsto x + \color{blue}{\left(\left(3.13060547622999996 \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    3. Simplified8.3

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)}\]

    if -1.2458395986064981e+53 < z < 1.6181471283118757e+58

    1. Initial program 2.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt3.0

      \[\leadsto x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\color{blue}{\left(\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\right) \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\]

    4. Applied times-frac1.5

      \[\leadsto x + \color{blue}{\frac{y}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.24583959860649812 \cdot 10^{53} \lor \neg \left(z \le 1.6181471283118757 \cdot 10^{58}\right):\\ \;\;\;\;x + \mathsf{fma}\left(y, 3.13060547622999996, \frac{t \cdot y}{{z}^{2}} - 36.527041698806414 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004} \cdot \sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}} \cdot \frac{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}{\sqrt[3]{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))))