Average Error: 0.3 → 0.3
Time: 10.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}, \log z \cdot \log z\right)} - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}, \log z \cdot \log z\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r57367 = x;
        double r57368 = y;
        double r57369 = r57367 + r57368;
        double r57370 = log(r57369);
        double r57371 = z;
        double r57372 = log(r57371);
        double r57373 = r57370 + r57372;
        double r57374 = t;
        double r57375 = r57373 - r57374;
        double r57376 = a;
        double r57377 = 0.5;
        double r57378 = r57376 - r57377;
        double r57379 = log(r57374);
        double r57380 = r57378 * r57379;
        double r57381 = r57375 + r57380;
        return r57381;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r57382 = t;
        double r57383 = log(r57382);
        double r57384 = a;
        double r57385 = 0.5;
        double r57386 = r57384 - r57385;
        double r57387 = x;
        double r57388 = y;
        double r57389 = r57387 + r57388;
        double r57390 = log(r57389);
        double r57391 = 3.0;
        double r57392 = pow(r57390, r57391);
        double r57393 = z;
        double r57394 = log(r57393);
        double r57395 = pow(r57394, r57391);
        double r57396 = r57392 + r57395;
        double r57397 = r57390 - r57394;
        double r57398 = pow(r57397, r57391);
        double r57399 = cbrt(r57398);
        double r57400 = r57394 * r57394;
        double r57401 = fma(r57390, r57399, r57400);
        double r57402 = r57396 / r57401;
        double r57403 = r57402 - r57382;
        double r57404 = fma(r57383, r57386, r57403);
        return r57404;
}

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

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

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

    \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right)}\]
  5. Using strategy rm

  6. Applied flip3-+0.3

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

  7. Simplified0.3

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

  9. Applied add-cbrt-cube0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +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))))