Average Error: 0.3 → 0.3
Time: 37.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log z - t\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)\]
\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(\left(\log z - t\right) + \log \left(\sqrt[3]{y + x}\right)\right) + \log \left(\sqrt[3]{y + x} \cdot \sqrt[3]{y + x}\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r92161 = x;
        double r92162 = y;
        double r92163 = r92161 + r92162;
        double r92164 = log(r92163);
        double r92165 = z;
        double r92166 = log(r92165);
        double r92167 = r92164 + r92166;
        double r92168 = t;
        double r92169 = r92167 - r92168;
        double r92170 = a;
        double r92171 = 0.5;
        double r92172 = r92170 - r92171;
        double r92173 = log(r92168);
        double r92174 = r92172 * r92173;
        double r92175 = r92169 + r92174;
        return r92175;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r92176 = t;
        double r92177 = log(r92176);
        double r92178 = a;
        double r92179 = 0.5;
        double r92180 = r92178 - r92179;
        double r92181 = z;
        double r92182 = log(r92181);
        double r92183 = r92182 - r92176;
        double r92184 = y;
        double r92185 = x;
        double r92186 = r92184 + r92185;
        double r92187 = cbrt(r92186);
        double r92188 = log(r92187);
        double r92189 = r92183 + r92188;
        double r92190 = r92187 * r92187;
        double r92191 = log(r92190);
        double r92192 = r92189 + r92191;
        double r92193 = fma(r92177, r92180, r92192);
        return r92193;
}

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. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 +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))))