Average Error: 28.8 → 1.1
Time: 42.1s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.0597997332978914 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \mathbf{elif}\;z \le 8.317246802743899 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), z, t\right), a\right), z, b\right)}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}}}{\sqrt{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407 + z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\frac{t}{z}}{z} + 3.13060547623, x\right)\\ \end{array}\]
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 r11960030 = x;
        double r11960031 = y;
        double r11960032 = z;
        double r11960033 = 3.13060547623;
        double r11960034 = r11960032 * r11960033;
        double r11960035 = 11.1667541262;
        double r11960036 = r11960034 + r11960035;
        double r11960037 = r11960036 * r11960032;
        double r11960038 = t;
        double r11960039 = r11960037 + r11960038;
        double r11960040 = r11960039 * r11960032;
        double r11960041 = a;
        double r11960042 = r11960040 + r11960041;
        double r11960043 = r11960042 * r11960032;
        double r11960044 = b;
        double r11960045 = r11960043 + r11960044;
        double r11960046 = r11960031 * r11960045;
        double r11960047 = 15.234687407;
        double r11960048 = r11960032 + r11960047;
        double r11960049 = r11960048 * r11960032;
        double r11960050 = 31.4690115749;
        double r11960051 = r11960049 + r11960050;
        double r11960052 = r11960051 * r11960032;
        double r11960053 = 11.9400905721;
        double r11960054 = r11960052 + r11960053;
        double r11960055 = r11960054 * r11960032;
        double r11960056 = 0.607771387771;
        double r11960057 = r11960055 + r11960056;
        double r11960058 = r11960046 / r11960057;
        double r11960059 = r11960030 + r11960058;
        return r11960059;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r11960060 = z;
        double r11960061 = -1.0597997332978914e+45;
        bool r11960062 = r11960060 <= r11960061;
        double r11960063 = y;
        double r11960064 = t;
        double r11960065 = r11960064 / r11960060;
        double r11960066 = r11960065 / r11960060;
        double r11960067 = 3.13060547623;
        double r11960068 = r11960066 + r11960067;
        double r11960069 = x;
        double r11960070 = fma(r11960063, r11960068, r11960069);
        double r11960071 = 8.317246802743899e+42;
        bool r11960072 = r11960060 <= r11960071;
        double r11960073 = 11.1667541262;
        double r11960074 = fma(r11960067, r11960060, r11960073);
        double r11960075 = fma(r11960074, r11960060, r11960064);
        double r11960076 = a;
        double r11960077 = fma(r11960060, r11960075, r11960076);
        double r11960078 = b;
        double r11960079 = fma(r11960077, r11960060, r11960078);
        double r11960080 = 15.234687407;
        double r11960081 = r11960080 + r11960060;
        double r11960082 = 31.4690115749;
        double r11960083 = fma(r11960060, r11960081, r11960082);
        double r11960084 = 11.9400905721;
        double r11960085 = fma(r11960060, r11960083, r11960084);
        double r11960086 = 0.607771387771;
        double r11960087 = fma(r11960060, r11960085, r11960086);
        double r11960088 = sqrt(r11960087);
        double r11960089 = sqrt(r11960088);
        double r11960090 = r11960079 / r11960089;
        double r11960091 = r11960090 / r11960089;
        double r11960092 = r11960091 / r11960088;
        double r11960093 = fma(r11960063, r11960092, r11960069);
        double r11960094 = r11960072 ? r11960093 : r11960070;
        double r11960095 = r11960062 ? r11960070 : r11960094;
        return r11960095;
}

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))))