Average Error: 29.5 → 1.5
Time: 6.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.1286680735006301 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\frac{{z}^{2}}{t}}, x\right)\\ \mathbf{elif}\;z \le 495619.95934221218:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547622999996, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)\\ \end{array}\]
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 -2.1286680735006301 \cdot 10^{49}:\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\frac{{z}^{2}}{t}}, x\right)\\

\mathbf{elif}\;z \le 495619.95934221218:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(3.13060547622999996, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r314729 = x;
        double r314730 = y;
        double r314731 = z;
        double r314732 = 3.13060547623;
        double r314733 = r314731 * r314732;
        double r314734 = 11.1667541262;
        double r314735 = r314733 + r314734;
        double r314736 = r314735 * r314731;
        double r314737 = t;
        double r314738 = r314736 + r314737;
        double r314739 = r314738 * r314731;
        double r314740 = a;
        double r314741 = r314739 + r314740;
        double r314742 = r314741 * r314731;
        double r314743 = b;
        double r314744 = r314742 + r314743;
        double r314745 = r314730 * r314744;
        double r314746 = 15.234687407;
        double r314747 = r314731 + r314746;
        double r314748 = r314747 * r314731;
        double r314749 = 31.4690115749;
        double r314750 = r314748 + r314749;
        double r314751 = r314750 * r314731;
        double r314752 = 11.9400905721;
        double r314753 = r314751 + r314752;
        double r314754 = r314753 * r314731;
        double r314755 = 0.607771387771;
        double r314756 = r314754 + r314755;
        double r314757 = r314745 / r314756;
        double r314758 = r314729 + r314757;
        return r314758;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r314759 = z;
        double r314760 = -2.12866807350063e+49;
        bool r314761 = r314759 <= r314760;
        double r314762 = y;
        double r314763 = 3.13060547623;
        double r314764 = 1.0;
        double r314765 = 2.0;
        double r314766 = pow(r314759, r314765);
        double r314767 = t;
        double r314768 = r314766 / r314767;
        double r314769 = r314764 / r314768;
        double r314770 = r314763 + r314769;
        double r314771 = x;
        double r314772 = fma(r314762, r314770, r314771);
        double r314773 = 495619.9593422122;
        bool r314774 = r314759 <= r314773;
        double r314775 = 15.234687407;
        double r314776 = r314759 + r314775;
        double r314777 = 31.4690115749;
        double r314778 = fma(r314776, r314759, r314777);
        double r314779 = 11.9400905721;
        double r314780 = fma(r314778, r314759, r314779);
        double r314781 = 0.607771387771;
        double r314782 = fma(r314780, r314759, r314781);
        double r314783 = r314762 / r314782;
        double r314784 = 11.1667541262;
        double r314785 = fma(r314759, r314763, r314784);
        double r314786 = fma(r314785, r314759, r314767);
        double r314787 = a;
        double r314788 = fma(r314786, r314759, r314787);
        double r314789 = b;
        double r314790 = fma(r314788, r314759, r314789);
        double r314791 = fma(r314783, r314790, r314771);
        double r314792 = r314762 / r314759;
        double r314793 = r314767 / r314759;
        double r314794 = fma(r314792, r314793, r314771);
        double r314795 = fma(r314763, r314762, r314794);
        double r314796 = r314774 ? r314791 : r314795;
        double r314797 = r314761 ? r314772 : r314796;
        return r314797;
}

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

Original29.5
Target1.2
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < -2.12866807350063e+49

    1. Initial program 61.4

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

    2. Simplified59.9

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied clear-num1.3

      \[\leadsto \mathsf{fma}\left(y, 3.13060547622999996 + \color{blue}{\frac{1}{\frac{{z}^{2}}{t}}}, x\right)\]

    if -2.12866807350063e+49 < z < 495619.9593422122

    1. Initial program 1.4

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

    2. Simplified0.7

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

    if 495619.9593422122 < z

    1. Initial program 56.3

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 10.9

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

    4. Simplified3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547622999996 + \frac{t}{{z}^{2}}, x\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt4.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\sqrt[3]{3.13060547622999996} \cdot \sqrt[3]{3.13060547622999996}\right) \cdot \sqrt[3]{3.13060547622999996}} + \frac{t}{{z}^{2}}, x\right)\]

    7. Applied fma-def4.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\sqrt[3]{3.13060547622999996} \cdot \sqrt[3]{3.13060547622999996}, \sqrt[3]{3.13060547622999996}, \frac{t}{{z}^{2}}\right)}, x\right)\]

    8. Taylor expanded around 0 10.9

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

    9. Simplified3.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(3.13060547622999996, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.1286680735006301 \cdot 10^{49}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{1}{\frac{{z}^{2}}{t}}, x\right)\\ \mathbf{elif}\;z \le 495619.95934221218:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547622999996, y, \mathsf{fma}\left(\frac{y}{z}, \frac{t}{z}, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))