Average Error: 28.8 → 1.1
Time: 38.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]
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 -1.0597997332978914 \cdot 10^{+45}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\

\mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\
\;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\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{\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)}}}}{\sqrt{\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)}}}}{\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 r19758967 = x;
        double r19758968 = y;
        double r19758969 = z;
        double r19758970 = 3.13060547623;
        double r19758971 = r19758969 * r19758970;
        double r19758972 = 11.1667541262;
        double r19758973 = r19758971 + r19758972;
        double r19758974 = r19758973 * r19758969;
        double r19758975 = t;
        double r19758976 = r19758974 + r19758975;
        double r19758977 = r19758976 * r19758969;
        double r19758978 = a;
        double r19758979 = r19758977 + r19758978;
        double r19758980 = r19758979 * r19758969;
        double r19758981 = b;
        double r19758982 = r19758980 + r19758981;
        double r19758983 = r19758968 * r19758982;
        double r19758984 = 15.234687407;
        double r19758985 = r19758969 + r19758984;
        double r19758986 = r19758985 * r19758969;
        double r19758987 = 31.4690115749;
        double r19758988 = r19758986 + r19758987;
        double r19758989 = r19758988 * r19758969;
        double r19758990 = 11.9400905721;
        double r19758991 = r19758989 + r19758990;
        double r19758992 = r19758991 * r19758969;
        double r19758993 = 0.607771387771;
        double r19758994 = r19758992 + r19758993;
        double r19758995 = r19758983 / r19758994;
        double r19758996 = r19758967 + r19758995;
        return r19758996;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r19758997 = z;
        double r19758998 = -1.0597997332978914e+45;
        bool r19758999 = r19758997 <= r19758998;
        double r19759000 = y;
        double r19759001 = t;
        double r19759002 = r19759001 / r19758997;
        double r19759003 = r19759002 / r19758997;
        double r19759004 = 3.13060547623;
        double r19759005 = r19759003 + r19759004;
        double r19759006 = x;
        double r19759007 = fma(r19759000, r19759005, r19759006);
        double r19759008 = 8.317246802743899e+42;
        bool r19759009 = r19758997 <= r19759008;
        double r19759010 = 11.1667541262;
        double r19759011 = fma(r19759004, r19758997, r19759010);
        double r19759012 = fma(r19759011, r19758997, r19759001);
        double r19759013 = a;
        double r19759014 = fma(r19758997, r19759012, r19759013);
        double r19759015 = b;
        double r19759016 = fma(r19759014, r19758997, r19759015);
        double r19759017 = 15.234687407;
        double r19759018 = r19759017 + r19758997;
        double r19759019 = 31.4690115749;
        double r19759020 = fma(r19758997, r19759018, r19759019);
        double r19759021 = 11.9400905721;
        double r19759022 = fma(r19758997, r19759020, r19759021);
        double r19759023 = 0.607771387771;
        double r19759024 = fma(r19758997, r19759022, r19759023);
        double r19759025 = sqrt(r19759024);
        double r19759026 = sqrt(r19759025);
        double r19759027 = r19759016 / r19759026;
        double r19759028 = r19759027 / r19759026;
        double r19759029 = r19759028 / r19759025;
        double r19759030 = fma(r19759000, r19759029, r19759006);
        double r19759031 = r19759009 ? r19759030 : r19759007;
        double r19759032 = r19758999 ? r19759007 : r19759031;
        return r19759032;
}

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

Original28.8
Target1.0
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -1.0597997332978914e+45 or 8.317246802743899e+42 < z

    1. Initial program 58.8

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

    2. Simplified56.7

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

    3. Taylor expanded around inf 7.9

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

    4. Simplified1.1

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

    if -1.0597997332978914e+45 < z < 8.317246802743899e+42

    1. Initial program 1.6

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

    2. Simplified0.8

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

    4. Applied add-sqr-sqrt1.2

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

    5. Applied associate-/r*1.1

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

    7. Applied add-sqr-sqrt1.1

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

    8. Applied sqrt-prod1.1

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

    9. Applied associate-/r*1.0

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

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))))