Average Error: 0.3 → 0.3
Time: 42.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{{z}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{{z}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r72176 = x;
        double r72177 = y;
        double r72178 = r72176 + r72177;
        double r72179 = log(r72178);
        double r72180 = z;
        double r72181 = log(r72180);
        double r72182 = r72179 + r72181;
        double r72183 = t;
        double r72184 = r72182 - r72183;
        double r72185 = a;
        double r72186 = 0.5;
        double r72187 = r72185 - r72186;
        double r72188 = log(r72183);
        double r72189 = r72187 * r72188;
        double r72190 = r72184 + r72189;
        return r72190;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r72191 = a;
        double r72192 = 0.5;
        double r72193 = r72191 - r72192;
        double r72194 = t;
        double r72195 = log(r72194);
        double r72196 = 2.0;
        double r72197 = z;
        double r72198 = 0.6666666666666666;
        double r72199 = pow(r72197, r72198);
        double r72200 = cbrt(r72199);
        double r72201 = cbrt(r72197);
        double r72202 = cbrt(r72201);
        double r72203 = r72200 * r72202;
        double r72204 = log(r72203);
        double r72205 = x;
        double r72206 = y;
        double r72207 = r72205 + r72206;
        double r72208 = log(r72207);
        double r72209 = fma(r72196, r72204, r72208);
        double r72210 = log(r72201);
        double r72211 = r72209 + r72210;
        double r72212 = r72211 - r72194;
        double r72213 = fma(r72193, r72195, r72212);
        return r72213;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied add-cube-cbrt0.3

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

  10. Applied cbrt-prod0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +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))))