Average Error: 0.3 → 0.3
Time: 36.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log t \cdot \left(a - 0.5\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log t \cdot \left(a - 0.5\right)
double f(double x, double y, double z, double t, double a) {
        double r72167 = x;
        double r72168 = y;
        double r72169 = r72167 + r72168;
        double r72170 = log(r72169);
        double r72171 = z;
        double r72172 = log(r72171);
        double r72173 = r72170 + r72172;
        double r72174 = t;
        double r72175 = r72173 - r72174;
        double r72176 = a;
        double r72177 = 0.5;
        double r72178 = r72176 - r72177;
        double r72179 = log(r72174);
        double r72180 = r72178 * r72179;
        double r72181 = r72175 + r72180;
        return r72181;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r72182 = 2.0;
        double r72183 = z;
        double r72184 = cbrt(r72183);
        double r72185 = log(r72184);
        double r72186 = y;
        double r72187 = x;
        double r72188 = r72186 + r72187;
        double r72189 = log(r72188);
        double r72190 = fma(r72182, r72185, r72189);
        double r72191 = t;
        double r72192 = r72185 - r72191;
        double r72193 = r72190 + r72192;
        double r72194 = log(r72191);
        double r72195 = a;
        double r72196 = 0.5;
        double r72197 = r72195 - r72196;
        double r72198 = r72194 * r72197;
        double r72199 = r72193 + r72198;
        return r72199;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Applied associate-+r+0.3

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

  8. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]
  9. Using strategy rm

  10. Applied fma-udef0.3

    \[\leadsto \color{blue}{\log t \cdot \left(a - 0.5\right) + \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)}\]

  11. Final simplification0.3

    \[\leadsto \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log t \cdot \left(a - 0.5\right)\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))