Average Error: 29.3 → 1.5
Time: 18.4s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -4.56197502220436 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \mathbf{elif}\;z \le 3.6239765426493 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right) \cdot y}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \end{array}\]
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 -4.56197502220436 \cdot 10^{+30}:\\
\;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\

\mathbf{elif}\;z \le 3.6239765426493 \cdot 10^{+38}:\\
\;\;\;\;\frac{\left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right) \cdot y}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r6286064 = x;
        double r6286065 = y;
        double r6286066 = z;
        double r6286067 = 3.13060547623;
        double r6286068 = r6286066 * r6286067;
        double r6286069 = 11.1667541262;
        double r6286070 = r6286068 + r6286069;
        double r6286071 = r6286070 * r6286066;
        double r6286072 = t;
        double r6286073 = r6286071 + r6286072;
        double r6286074 = r6286073 * r6286066;
        double r6286075 = a;
        double r6286076 = r6286074 + r6286075;
        double r6286077 = r6286076 * r6286066;
        double r6286078 = b;
        double r6286079 = r6286077 + r6286078;
        double r6286080 = r6286065 * r6286079;
        double r6286081 = 15.234687407;
        double r6286082 = r6286066 + r6286081;
        double r6286083 = r6286082 * r6286066;
        double r6286084 = 31.4690115749;
        double r6286085 = r6286083 + r6286084;
        double r6286086 = r6286085 * r6286066;
        double r6286087 = 11.9400905721;
        double r6286088 = r6286086 + r6286087;
        double r6286089 = r6286088 * r6286066;
        double r6286090 = 0.607771387771;
        double r6286091 = r6286089 + r6286090;
        double r6286092 = r6286080 / r6286091;
        double r6286093 = r6286064 + r6286092;
        return r6286093;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r6286094 = z;
        double r6286095 = -4.56197502220436e+30;
        bool r6286096 = r6286094 <= r6286095;
        double r6286097 = t;
        double r6286098 = r6286097 / r6286094;
        double r6286099 = y;
        double r6286100 = r6286099 / r6286094;
        double r6286101 = 3.13060547623;
        double r6286102 = x;
        double r6286103 = fma(r6286101, r6286099, r6286102);
        double r6286104 = fma(r6286098, r6286100, r6286103);
        double r6286105 = 3.6239765426493e+38;
        bool r6286106 = r6286094 <= r6286105;
        double r6286107 = b;
        double r6286108 = a;
        double r6286109 = r6286101 * r6286094;
        double r6286110 = 11.1667541262;
        double r6286111 = r6286109 + r6286110;
        double r6286112 = r6286111 * r6286094;
        double r6286113 = r6286112 + r6286097;
        double r6286114 = r6286094 * r6286113;
        double r6286115 = r6286108 + r6286114;
        double r6286116 = r6286094 * r6286115;
        double r6286117 = r6286107 + r6286116;
        double r6286118 = r6286117 * r6286099;
        double r6286119 = 15.234687407;
        double r6286120 = r6286119 + r6286094;
        double r6286121 = r6286094 * r6286120;
        double r6286122 = 31.4690115749;
        double r6286123 = r6286121 + r6286122;
        double r6286124 = r6286123 * r6286094;
        double r6286125 = 11.9400905721;
        double r6286126 = r6286124 + r6286125;
        double r6286127 = r6286126 * r6286094;
        double r6286128 = 0.607771387771;
        double r6286129 = r6286127 + r6286128;
        double r6286130 = r6286118 / r6286129;
        double r6286131 = r6286130 + r6286102;
        double r6286132 = r6286106 ? r6286131 : r6286104;
        double r6286133 = r6286096 ? r6286104 : r6286132;
        return r6286133;
}

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

Original29.3
Target1.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -4.56197502220436e+30 or 3.6239765426493e+38 < z

    1. Initial program 58.3

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

    2. Simplified56.6

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

    3. Taylor expanded around inf 8.2

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

    4. Simplified1.6

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

    if -4.56197502220436e+30 < z < 3.6239765426493e+38

    1. Initial program 1.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.56197502220436 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \mathbf{elif}\;z \le 3.6239765426493 \cdot 10^{+38}:\\ \;\;\;\;\frac{\left(b + z \cdot \left(a + z \cdot \left(\left(3.13060547623 \cdot z + 11.1667541262\right) \cdot z + t\right)\right)\right) \cdot y}{\left(\left(z \cdot \left(15.234687407 + z\right) + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, \frac{y}{z}, \mathsf{fma}\left(3.13060547623, y, x\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))))