Average Error: 0.3 → 0.3
Time: 15.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}\right)} - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r57368 = x;
        double r57369 = y;
        double r57370 = r57368 + r57369;
        double r57371 = log(r57370);
        double r57372 = z;
        double r57373 = log(r57372);
        double r57374 = r57371 + r57373;
        double r57375 = t;
        double r57376 = r57374 - r57375;
        double r57377 = a;
        double r57378 = 0.5;
        double r57379 = r57377 - r57378;
        double r57380 = log(r57375);
        double r57381 = r57379 * r57380;
        double r57382 = r57376 + r57381;
        return r57382;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r57383 = a;
        double r57384 = 0.5;
        double r57385 = r57383 - r57384;
        double r57386 = t;
        double r57387 = log(r57386);
        double r57388 = x;
        double r57389 = y;
        double r57390 = r57388 + r57389;
        double r57391 = log(r57390);
        double r57392 = 3.0;
        double r57393 = pow(r57391, r57392);
        double r57394 = z;
        double r57395 = log(r57394);
        double r57396 = pow(r57395, r57392);
        double r57397 = r57393 + r57396;
        double r57398 = r57391 - r57395;
        double r57399 = pow(r57398, r57392);
        double r57400 = cbrt(r57399);
        double r57401 = r57391 * r57400;
        double r57402 = fma(r57395, r57395, r57401);
        double r57403 = r57397 / r57402;
        double r57404 = r57403 - r57386;
        double r57405 = fma(r57385, r57387, r57404);
        return r57405;
}

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

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

  4. Applied pow10.3

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

  6. Applied flip3-+0.3

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

  7. Simplified0.3

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

  9. Applied add-cbrt-cube0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

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))))