Average Error: 0.3 → 0.3
Time: 10.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left({\left(x + y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left({\left(x + y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r83148 = x;
        double r83149 = y;
        double r83150 = r83148 + r83149;
        double r83151 = log(r83150);
        double r83152 = z;
        double r83153 = log(r83152);
        double r83154 = r83151 + r83153;
        double r83155 = t;
        double r83156 = r83154 - r83155;
        double r83157 = a;
        double r83158 = 0.5;
        double r83159 = r83157 - r83158;
        double r83160 = log(r83155);
        double r83161 = r83159 * r83160;
        double r83162 = r83156 + r83161;
        return r83162;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r83163 = t;
        double r83164 = log(r83163);
        double r83165 = a;
        double r83166 = 0.5;
        double r83167 = r83165 - r83166;
        double r83168 = x;
        double r83169 = y;
        double r83170 = r83168 + r83169;
        double r83171 = 0.3333333333333333;
        double r83172 = pow(r83170, r83171);
        double r83173 = cbrt(r83170);
        double r83174 = r83172 * r83173;
        double r83175 = log(r83174);
        double r83176 = log(r83173);
        double r83177 = z;
        double r83178 = log(r83177);
        double r83179 = r83176 + r83178;
        double r83180 = r83175 + r83179;
        double r83181 = r83180 - r83163;
        double r83182 = fma(r83164, r83167, r83181);
        return r83182;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Derivation

  1. Initial program 0.3

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \color{blue}{\left(\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}\right)} + \log z\right) - t\right)\]

  5. Applied log-prod0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \log \left(\sqrt[3]{x + y}\right)\right)} + \log z\right) - t\right)\]

  6. Applied associate-+l+0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right)} - t\right)\]
  7. Using strategy rm

  8. Applied pow1/30.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\color{blue}{{\left(x + y\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]

  9. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left({\left(x + y\right)}^{\frac{1}{3}} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))