Average Error: 0.3 → 0.3
Time: 12.2s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{1} \cdot {z}^{\frac{1}{3}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{1} \cdot {z}^{\frac{1}{3}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t
double f(double x, double y, double z, double t, double a) {
        double r66010 = x;
        double r66011 = y;
        double r66012 = r66010 + r66011;
        double r66013 = log(r66012);
        double r66014 = z;
        double r66015 = log(r66014);
        double r66016 = r66013 + r66015;
        double r66017 = t;
        double r66018 = r66016 - r66017;
        double r66019 = a;
        double r66020 = 0.5;
        double r66021 = r66019 - r66020;
        double r66022 = log(r66017);
        double r66023 = r66021 * r66022;
        double r66024 = r66018 + r66023;
        return r66024;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r66025 = 2.0;
        double r66026 = 1.0;
        double r66027 = cbrt(r66026);
        double r66028 = z;
        double r66029 = 0.3333333333333333;
        double r66030 = pow(r66028, r66029);
        double r66031 = r66027 * r66030;
        double r66032 = log(r66031);
        double r66033 = x;
        double r66034 = y;
        double r66035 = r66033 + r66034;
        double r66036 = log(r66035);
        double r66037 = fma(r66025, r66032, r66036);
        double r66038 = cbrt(r66028);
        double r66039 = log(r66038);
        double r66040 = r66037 + r66039;
        double r66041 = t;
        double r66042 = r66040 - r66041;
        double r66043 = a;
        double r66044 = 0.5;
        double r66045 = r66043 - r66044;
        double r66046 = log(r66041);
        double r66047 = r66045 * r66046;
        double r66048 = r66042 + r66047;
        return r66048;
}

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. Using strategy rm

  3. Applied add-cube-cbrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied associate-+r+0.3

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

  6. Simplified0.3

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

  8. Applied *-un-lft-identity0.3

    \[\leadsto \left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{\color{blue}{1 \cdot z}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  9. Applied cbrt-prod0.3

    \[\leadsto \left(\left(\mathsf{fma}\left(2, \log \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{z}\right)}, \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  10. Simplified0.3

    \[\leadsto \left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{1} \cdot \color{blue}{{z}^{\frac{1}{3}}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  11. Final simplification0.3

    \[\leadsto \left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{1} \cdot {z}^{\frac{1}{3}}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

Reproduce

herbie shell --seed 2020065 +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))))