Average Error: 28.8 → 1.0
Time: 26.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -3.9856954798303585 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\ \mathbf{elif}\;z \le 4.8590408174333496 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), z, a\right), b\right)}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\ \end{array}\]
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 -3.9856954798303585 \cdot 10^{+39}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\

\mathbf{elif}\;z \le 4.8590408174333496 \cdot 10^{+55}:\\
\;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), z, a\right), b\right)}} + x\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r11639092 = x;
        double r11639093 = y;
        double r11639094 = z;
        double r11639095 = 3.13060547623;
        double r11639096 = r11639094 * r11639095;
        double r11639097 = 11.1667541262;
        double r11639098 = r11639096 + r11639097;
        double r11639099 = r11639098 * r11639094;
        double r11639100 = t;
        double r11639101 = r11639099 + r11639100;
        double r11639102 = r11639101 * r11639094;
        double r11639103 = a;
        double r11639104 = r11639102 + r11639103;
        double r11639105 = r11639104 * r11639094;
        double r11639106 = b;
        double r11639107 = r11639105 + r11639106;
        double r11639108 = r11639093 * r11639107;
        double r11639109 = 15.234687407;
        double r11639110 = r11639094 + r11639109;
        double r11639111 = r11639110 * r11639094;
        double r11639112 = 31.4690115749;
        double r11639113 = r11639111 + r11639112;
        double r11639114 = r11639113 * r11639094;
        double r11639115 = 11.9400905721;
        double r11639116 = r11639114 + r11639115;
        double r11639117 = r11639116 * r11639094;
        double r11639118 = 0.607771387771;
        double r11639119 = r11639117 + r11639118;
        double r11639120 = r11639108 / r11639119;
        double r11639121 = r11639092 + r11639120;
        return r11639121;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r11639122 = z;
        double r11639123 = -3.9856954798303585e+39;
        bool r11639124 = r11639122 <= r11639123;
        double r11639125 = y;
        double r11639126 = t;
        double r11639127 = r11639122 * r11639122;
        double r11639128 = r11639126 / r11639127;
        double r11639129 = 3.13060547623;
        double r11639130 = r11639128 + r11639129;
        double r11639131 = 36.527041698806414;
        double r11639132 = r11639131 / r11639122;
        double r11639133 = r11639130 - r11639132;
        double r11639134 = x;
        double r11639135 = fma(r11639125, r11639133, r11639134);
        double r11639136 = 4.8590408174333496e+55;
        bool r11639137 = r11639122 <= r11639136;
        double r11639138 = 15.234687407;
        double r11639139 = r11639138 + r11639122;
        double r11639140 = 31.4690115749;
        double r11639141 = fma(r11639139, r11639122, r11639140);
        double r11639142 = 11.9400905721;
        double r11639143 = fma(r11639122, r11639141, r11639142);
        double r11639144 = 0.607771387771;
        double r11639145 = fma(r11639122, r11639143, r11639144);
        double r11639146 = 11.1667541262;
        double r11639147 = fma(r11639122, r11639129, r11639146);
        double r11639148 = fma(r11639147, r11639122, r11639126);
        double r11639149 = a;
        double r11639150 = fma(r11639148, r11639122, r11639149);
        double r11639151 = b;
        double r11639152 = fma(r11639122, r11639150, r11639151);
        double r11639153 = r11639145 / r11639152;
        double r11639154 = r11639125 / r11639153;
        double r11639155 = r11639154 + r11639134;
        double r11639156 = r11639137 ? r11639155 : r11639135;
        double r11639157 = r11639124 ? r11639135 : r11639156;
        return r11639157;
}

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

Derivation

  1. Split input into 2 regimes

  2. if z < -3.9856954798303585e+39 or 4.8590408174333496e+55 < z

    1. Initial program 58.9

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

    2. Simplified56.8

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

    4. Applied add-cube-cbrt56.8

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

    5. Taylor expanded around inf 1.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(\frac{t}{{z}^{2}} + 3.13060547623\right) - 36.527041698806414 \cdot \frac{1}{z}}, x\right)\]

    6. Simplified1.3

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\left(3.13060547623 + \frac{t}{z \cdot z}\right) - \frac{36.527041698806414}{z}}, x\right)\]

    if -3.9856954798303585e+39 < z < 4.8590408174333496e+55

    1. Initial program 2.2

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

    2. Simplified0.9

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

    4. Applied clear-num0.9

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

    6. Applied fma-udef0.9

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

    7. Simplified0.9

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.9856954798303585 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\ \mathbf{elif}\;z \le 4.8590408174333496 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(15.234687407 + z, z, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), z, t\right), z, a\right), b\right)}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \left(\frac{t}{z \cdot z} + 3.13060547623\right) - \frac{36.527041698806414}{z}, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +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))))