Average Error: 29.8 → 1.4
Time: 15.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547622999996 + 11.166754126200001\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687406999999\right) \cdot z + 31.469011574900001\right) \cdot z + 11.940090572100001\right) \cdot z + 0.60777138777100004}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.7042356486663339 \cdot 10^{47} \lor \neg \left(z \le 5.7420122539198766 \cdot 10^{73}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996, \mathsf{fma}\left(\frac{t}{{z}^{2}}, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]
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 -1.7042356486663339 \cdot 10^{47} \lor \neg \left(z \le 5.7420122539198766 \cdot 10^{73}\right):\\
\;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996, \mathsf{fma}\left(\frac{t}{{z}^{2}}, y, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}, \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 r484970 = x;
        double r484971 = y;
        double r484972 = z;
        double r484973 = 3.13060547623;
        double r484974 = r484972 * r484973;
        double r484975 = 11.1667541262;
        double r484976 = r484974 + r484975;
        double r484977 = r484976 * r484972;
        double r484978 = t;
        double r484979 = r484977 + r484978;
        double r484980 = r484979 * r484972;
        double r484981 = a;
        double r484982 = r484980 + r484981;
        double r484983 = r484982 * r484972;
        double r484984 = b;
        double r484985 = r484983 + r484984;
        double r484986 = r484971 * r484985;
        double r484987 = 15.234687407;
        double r484988 = r484972 + r484987;
        double r484989 = r484988 * r484972;
        double r484990 = 31.4690115749;
        double r484991 = r484989 + r484990;
        double r484992 = r484991 * r484972;
        double r484993 = 11.9400905721;
        double r484994 = r484992 + r484993;
        double r484995 = r484994 * r484972;
        double r484996 = 0.607771387771;
        double r484997 = r484995 + r484996;
        double r484998 = r484986 / r484997;
        double r484999 = r484970 + r484998;
        return r484999;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r485000 = z;
        double r485001 = -1.704235648666334e+47;
        bool r485002 = r485000 <= r485001;
        double r485003 = 5.742012253919877e+73;
        bool r485004 = r485000 <= r485003;
        double r485005 = !r485004;
        bool r485006 = r485002 || r485005;
        double r485007 = y;
        double r485008 = 3.13060547623;
        double r485009 = t;
        double r485010 = 2.0;
        double r485011 = pow(r485000, r485010);
        double r485012 = r485009 / r485011;
        double r485013 = x;
        double r485014 = fma(r485012, r485007, r485013);
        double r485015 = fma(r485007, r485008, r485014);
        double r485016 = 1.0;
        double r485017 = 15.234687407;
        double r485018 = r485000 + r485017;
        double r485019 = 31.4690115749;
        double r485020 = fma(r485018, r485000, r485019);
        double r485021 = 11.9400905721;
        double r485022 = fma(r485020, r485000, r485021);
        double r485023 = 0.607771387771;
        double r485024 = fma(r485022, r485000, r485023);
        double r485025 = r485024 / r485007;
        double r485026 = r485016 / r485025;
        double r485027 = 11.1667541262;
        double r485028 = fma(r485000, r485008, r485027);
        double r485029 = fma(r485028, r485000, r485009);
        double r485030 = a;
        double r485031 = fma(r485029, r485000, r485030);
        double r485032 = b;
        double r485033 = fma(r485031, r485000, r485032);
        double r485034 = fma(r485026, r485033, r485013);
        double r485035 = r485006 ? r485015 : r485034;
        return r485035;
}

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

Original29.8
Target1.0
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 < -1.704235648666334e+47 or 5.742012253919877e+73 < z

    1. Initial program 62.2

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

    2. Simplified61.3

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

    3. Taylor expanded around inf 8.4

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

    4. Simplified0.8

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

    if -1.704235648666334e+47 < z < 5.742012253919877e+73

    1. Initial program 3.6

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

    2. Simplified1.8

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

    4. Applied clear-num2.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.7042356486663339 \cdot 10^{47} \lor \neg \left(z \le 5.7420122539198766 \cdot 10^{73}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547622999996, \mathsf{fma}\left(\frac{t}{{z}^{2}}, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z + 15.234687406999999, z, 31.469011574900001\right), z, 11.940090572100001\right), z, 0.60777138777100004\right)}{y}}, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547622999996, 11.166754126200001\right), z, t\right), z, a\right), z, b\right), x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +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))))