Average Error: 0.3 → 0.3
Time: 34.4s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + \log \left(y + x\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\left(\log t\right), \left(a - 0.5\right), \left(\left(\log \left(\sqrt[3]{z}\right) - t\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)\right) + \log \left(y + x\right)
double f(double x, double y, double z, double t, double a) {
        double r1559261 = x;
        double r1559262 = y;
        double r1559263 = r1559261 + r1559262;
        double r1559264 = log(r1559263);
        double r1559265 = z;
        double r1559266 = log(r1559265);
        double r1559267 = r1559264 + r1559266;
        double r1559268 = t;
        double r1559269 = r1559267 - r1559268;
        double r1559270 = a;
        double r1559271 = 0.5;
        double r1559272 = r1559270 - r1559271;
        double r1559273 = log(r1559268);
        double r1559274 = r1559272 * r1559273;
        double r1559275 = r1559269 + r1559274;
        return r1559275;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1559276 = t;
        double r1559277 = log(r1559276);
        double r1559278 = a;
        double r1559279 = 0.5;
        double r1559280 = r1559278 - r1559279;
        double r1559281 = z;
        double r1559282 = cbrt(r1559281);
        double r1559283 = log(r1559282);
        double r1559284 = r1559283 - r1559276;
        double r1559285 = r1559282 * r1559282;
        double r1559286 = log(r1559285);
        double r1559287 = r1559284 + r1559286;
        double r1559288 = fma(r1559277, r1559280, r1559287);
        double r1559289 = y;
        double r1559290 = x;
        double r1559291 = r1559289 + r1559290;
        double r1559292 = log(r1559291);
        double r1559293 = r1559288 + r1559292;
        return r1559293;
}

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 associate--l+0.3

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

  4. Applied associate-+l+0.3

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

  5. Simplified0.3

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied log-prod0.3

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

  9. Applied associate--l+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019132 +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))))