Average Error: 29.7 → 1.4
Time: 5.5s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -37920752987725400 \lor \neg \left(z \le 7.35027383630455016 \cdot 10^{56}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{\frac{t}{z}}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -37920752987725400 \lor \neg \left(z \le 7.35027383630455016 \cdot 10^{56}\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{\frac{t}{z}}{z}, x\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r331006 = x;
        double r331007 = y;
        double r331008 = z;
        double r331009 = 3.13060547623;
        double r331010 = r331008 * r331009;
        double r331011 = 11.1667541262;
        double r331012 = r331010 + r331011;
        double r331013 = r331012 * r331008;
        double r331014 = t;
        double r331015 = r331013 + r331014;
        double r331016 = r331015 * r331008;
        double r331017 = a;
        double r331018 = r331016 + r331017;
        double r331019 = r331018 * r331008;
        double r331020 = b;
        double r331021 = r331019 + r331020;
        double r331022 = r331007 * r331021;
        double r331023 = 15.234687407;
        double r331024 = r331008 + r331023;
        double r331025 = r331024 * r331008;
        double r331026 = 31.4690115749;
        double r331027 = r331025 + r331026;
        double r331028 = r331027 * r331008;
        double r331029 = 11.9400905721;
        double r331030 = r331028 + r331029;
        double r331031 = r331030 * r331008;
        double r331032 = 0.607771387771;
        double r331033 = r331031 + r331032;
        double r331034 = r331022 / r331033;
        double r331035 = r331006 + r331034;
        return r331035;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r331036 = z;
        double r331037 = -3.79207529877254e+16;
        bool r331038 = r331036 <= r331037;
        double r331039 = 7.35027383630455e+56;
        bool r331040 = r331036 <= r331039;
        double r331041 = !r331040;
        bool r331042 = r331038 || r331041;
        double r331043 = y;
        double r331044 = 3.13060547623;
        double r331045 = t;
        double r331046 = r331045 / r331036;
        double r331047 = r331046 / r331036;
        double r331048 = r331044 + r331047;
        double r331049 = x;
        double r331050 = fma(r331043, r331048, r331049);
        double r331051 = 15.234687407;
        double r331052 = r331036 + r331051;
        double r331053 = 31.4690115749;
        double r331054 = fma(r331052, r331036, r331053);
        double r331055 = 11.9400905721;
        double r331056 = fma(r331054, r331036, r331055);
        double r331057 = 0.607771387771;
        double r331058 = fma(r331056, r331036, r331057);
        double r331059 = r331043 / r331058;
        double r331060 = 11.1667541262;
        double r331061 = fma(r331036, r331044, r331060);
        double r331062 = fma(r331061, r331036, r331045);
        double r331063 = a;
        double r331064 = fma(r331062, r331036, r331063);
        double r331065 = b;
        double r331066 = fma(r331064, r331036, r331065);
        double r331067 = fma(r331059, r331066, r331049);
        double r331068 = r331042 ? r331050 : r331067;
        return r331068;
}

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
Target1.3
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt -6.4993449962526318 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.0669654369142868 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}{\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547622999996 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -3.79207529877254e+16 or 7.35027383630455e+56 < z

    1. Initial program 59.5

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

    2. Simplified57.3

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

    3. Taylor expanded around inf 8.5

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

    4. Simplified1.7

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

    6. Applied unpow21.7

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

    7. Applied associate-/r*1.7

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

    if -3.79207529877254e+16 < z < 7.35027383630455e+56

    1. Initial program 1.9

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

    2. Simplified1.1

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

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -37920752987725400 \lor \neg \left(z \le 7.35027383630455016 \cdot 10^{56}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996 + \frac{\frac{t}{z}}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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))))