Average Error: 28.5 → 1.3
Time: 29.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.0275167136366416 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 2533004793734.77:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]
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 -1.0275167136366416 \cdot 10^{+34}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\

\mathbf{elif}\;z \le 2533004793734.77:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\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 \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)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r14766890 = x;
        double r14766891 = y;
        double r14766892 = z;
        double r14766893 = 3.13060547623;
        double r14766894 = r14766892 * r14766893;
        double r14766895 = 11.1667541262;
        double r14766896 = r14766894 + r14766895;
        double r14766897 = r14766896 * r14766892;
        double r14766898 = t;
        double r14766899 = r14766897 + r14766898;
        double r14766900 = r14766899 * r14766892;
        double r14766901 = a;
        double r14766902 = r14766900 + r14766901;
        double r14766903 = r14766902 * r14766892;
        double r14766904 = b;
        double r14766905 = r14766903 + r14766904;
        double r14766906 = r14766891 * r14766905;
        double r14766907 = 15.234687407;
        double r14766908 = r14766892 + r14766907;
        double r14766909 = r14766908 * r14766892;
        double r14766910 = 31.4690115749;
        double r14766911 = r14766909 + r14766910;
        double r14766912 = r14766911 * r14766892;
        double r14766913 = 11.9400905721;
        double r14766914 = r14766912 + r14766913;
        double r14766915 = r14766914 * r14766892;
        double r14766916 = 0.607771387771;
        double r14766917 = r14766915 + r14766916;
        double r14766918 = r14766906 / r14766917;
        double r14766919 = r14766890 + r14766918;
        return r14766919;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r14766920 = z;
        double r14766921 = -1.0275167136366416e+34;
        bool r14766922 = r14766920 <= r14766921;
        double r14766923 = y;
        double r14766924 = t;
        double r14766925 = r14766924 / r14766920;
        double r14766926 = r14766925 / r14766920;
        double r14766927 = 3.13060547623;
        double r14766928 = r14766926 + r14766927;
        double r14766929 = x;
        double r14766930 = fma(r14766923, r14766928, r14766929);
        double r14766931 = 2533004793734.77;
        bool r14766932 = r14766920 <= r14766931;
        double r14766933 = 1.0;
        double r14766934 = 15.234687407;
        double r14766935 = r14766934 + r14766920;
        double r14766936 = 31.4690115749;
        double r14766937 = fma(r14766920, r14766935, r14766936);
        double r14766938 = 11.9400905721;
        double r14766939 = fma(r14766920, r14766937, r14766938);
        double r14766940 = 0.607771387771;
        double r14766941 = fma(r14766920, r14766939, r14766940);
        double r14766942 = sqrt(r14766941);
        double r14766943 = r14766933 / r14766942;
        double r14766944 = 11.1667541262;
        double r14766945 = fma(r14766927, r14766920, r14766944);
        double r14766946 = fma(r14766945, r14766920, r14766924);
        double r14766947 = a;
        double r14766948 = fma(r14766920, r14766946, r14766947);
        double r14766949 = b;
        double r14766950 = fma(r14766948, r14766920, r14766949);
        double r14766951 = r14766950 / r14766942;
        double r14766952 = r14766943 * r14766951;
        double r14766953 = fma(r14766923, r14766952, r14766929);
        double r14766954 = r14766932 ? r14766953 : r14766930;
        double r14766955 = r14766922 ? r14766930 : r14766954;
        return r14766955;
}

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

Original28.5
Target1.0
Herbie1.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0275167136366416e+34 or 2533004793734.77 < z

    1. Initial program 56.5

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

    2. Simplified53.8

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

    3. Taylor expanded around inf 9.1

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

    4. Simplified2.0

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

    if -1.0275167136366416e+34 < z < 2533004793734.77

    1. Initial program 0.8

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

    2. Simplified0.4

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

    4. Applied add-sqr-sqrt0.9

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

    5. Applied *-un-lft-identity0.9

      \[\leadsto \mathsf{fma}\left(y, \frac{\color{blue}{1 \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\]

    6. Applied times-frac0.7

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

  4. Final simplification1.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.0275167136366416 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 2533004793734.77:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{1}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +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))))