Average Error: 0.2 → 0.3
Time: 40.4s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left({t}^{\frac{1}{3}}\right) \cdot \left(a - 0.5\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left({t}^{\frac{1}{3}}\right) \cdot \left(a - 0.5\right) + \mathsf{fma}\left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r2169178 = x;
        double r2169179 = y;
        double r2169180 = r2169178 + r2169179;
        double r2169181 = log(r2169180);
        double r2169182 = z;
        double r2169183 = log(r2169182);
        double r2169184 = r2169181 + r2169183;
        double r2169185 = t;
        double r2169186 = r2169184 - r2169185;
        double r2169187 = a;
        double r2169188 = 0.5;
        double r2169189 = r2169187 - r2169188;
        double r2169190 = log(r2169185);
        double r2169191 = r2169189 * r2169190;
        double r2169192 = r2169186 + r2169191;
        return r2169192;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2169193 = t;
        double r2169194 = 0.3333333333333333;
        double r2169195 = pow(r2169193, r2169194);
        double r2169196 = log(r2169195);
        double r2169197 = a;
        double r2169198 = 0.5;
        double r2169199 = r2169197 - r2169198;
        double r2169200 = r2169196 * r2169199;
        double r2169201 = cbrt(r2169193);
        double r2169202 = r2169201 * r2169201;
        double r2169203 = log(r2169202);
        double r2169204 = x;
        double r2169205 = y;
        double r2169206 = r2169204 + r2169205;
        double r2169207 = log(r2169206);
        double r2169208 = z;
        double r2169209 = log(r2169208);
        double r2169210 = r2169207 + r2169209;
        double r2169211 = r2169210 - r2169193;
        double r2169212 = fma(r2169203, r2169199, r2169211);
        double r2169213 = r2169200 + r2169212;
        return r2169213;
}

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 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.2

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

  4. Applied log-prod0.3

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

  5. Applied distribute-rgt-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied pow1/30.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019142 +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))))