Average Error: 28.2 → 1.0
Time: 29.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -5.657439373470419 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\ \mathbf{elif}\;z \le 4.025561367592441 \cdot 10^{+50}:\\ \;\;\;\;x + \left(\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\ \end{array}\]
x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}
↓
\begin{array}{l}
\mathbf{if}\;z \le -5.657439373470419 \cdot 10^{+35}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\

\mathbf{elif}\;z \le 4.025561367592441 \cdot 10^{+50}:\\
\;\;\;\;x + \left(\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)\right) \cdot y\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r18888747 = x;
        double r18888748 = y;
        double r18888749 = z;
        double r18888750 = 3.13060547623;
        double r18888751 = r18888749 * r18888750;
        double r18888752 = 11.1667541262;
        double r18888753 = r18888751 + r18888752;
        double r18888754 = r18888753 * r18888749;
        double r18888755 = t;
        double r18888756 = r18888754 + r18888755;
        double r18888757 = r18888756 * r18888749;
        double r18888758 = a;
        double r18888759 = r18888757 + r18888758;
        double r18888760 = r18888759 * r18888749;
        double r18888761 = b;
        double r18888762 = r18888760 + r18888761;
        double r18888763 = r18888748 * r18888762;
        double r18888764 = 15.234687407;
        double r18888765 = r18888749 + r18888764;
        double r18888766 = r18888765 * r18888749;
        double r18888767 = 31.4690115749;
        double r18888768 = r18888766 + r18888767;
        double r18888769 = r18888768 * r18888749;
        double r18888770 = 11.9400905721;
        double r18888771 = r18888769 + r18888770;
        double r18888772 = r18888771 * r18888749;
        double r18888773 = 0.607771387771;
        double r18888774 = r18888772 + r18888773;
        double r18888775 = r18888763 / r18888774;
        double r18888776 = r18888747 + r18888775;
        return r18888776;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r18888777 = z;
        double r18888778 = -5.657439373470419e+35;
        bool r18888779 = r18888777 <= r18888778;
        double r18888780 = y;
        double r18888781 = r18888780 / r18888777;
        double r18888782 = t;
        double r18888783 = 1.0;
        double r18888784 = r18888783 / r18888777;
        double r18888785 = 36.527041698806414;
        double r18888786 = -r18888785;
        double r18888787 = fma(r18888782, r18888784, r18888786);
        double r18888788 = 3.13060547623;
        double r18888789 = r18888780 * r18888788;
        double r18888790 = fma(r18888781, r18888787, r18888789);
        double r18888791 = x;
        double r18888792 = r18888790 + r18888791;
        double r18888793 = 4.025561367592441e+50;
        bool r18888794 = r18888777 <= r18888793;
        double r18888795 = 15.234687407;
        double r18888796 = r18888795 + r18888777;
        double r18888797 = 31.4690115749;
        double r18888798 = fma(r18888777, r18888796, r18888797);
        double r18888799 = 11.9400905721;
        double r18888800 = fma(r18888777, r18888798, r18888799);
        double r18888801 = 0.607771387771;
        double r18888802 = fma(r18888777, r18888800, r18888801);
        double r18888803 = r18888783 / r18888802;
        double r18888804 = 11.1667541262;
        double r18888805 = fma(r18888788, r18888777, r18888804);
        double r18888806 = fma(r18888805, r18888777, r18888782);
        double r18888807 = a;
        double r18888808 = fma(r18888777, r18888806, r18888807);
        double r18888809 = b;
        double r18888810 = fma(r18888808, r18888777, r18888809);
        double r18888811 = r18888803 * r18888810;
        double r18888812 = r18888811 * r18888780;
        double r18888813 = r18888791 + r18888812;
        double r18888814 = r18888794 ? r18888813 : r18888792;
        double r18888815 = r18888779 ? r18888792 : r18888814;
        return r18888815;
}

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

Original28.2
Target1.0
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -5.657439373470419e+35 or 4.025561367592441e+50 < z

    1. Initial program 58.9

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Simplified56.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef56.9

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} + x}\]

    5. Taylor expanded around inf 7.4

      \[\leadsto \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.13060547623 \cdot y\right) - 36.527041698806414 \cdot \frac{y}{z}\right)} + x\]

    6. Simplified1.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, \frac{t}{z} - 36.527041698806414, y \cdot 3.13060547623\right)} + x\]
    7. Using strategy rm

    8. Applied div-inv1.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{z}, \color{blue}{t \cdot \frac{1}{z}} - 36.527041698806414, y \cdot 3.13060547623\right) + x\]

    9. Applied fma-neg1.1

      \[\leadsto \mathsf{fma}\left(\frac{y}{z}, \color{blue}{\mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right)}, y \cdot 3.13060547623\right) + x\]

    if -5.657439373470419e+35 < z < 4.025561367592441e+50

    1. Initial program 2.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]

    2. Simplified0.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}\]
    3. Using strategy rm

    4. Applied fma-udef0.9

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} + x}\]
    5. Using strategy rm

    6. Applied div-inv1.0

      \[\leadsto y \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right) \cdot \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} + x\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -5.657439373470419 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\ \mathbf{elif}\;z \le 4.025561367592441 \cdot 10^{+50}:\\ \;\;\;\;x + \left(\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, \mathsf{fma}\left(t, \frac{1}{z}, -36.527041698806414\right), y \cdot 3.13060547623\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))) (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))))