Average Error: 0.3 → 0.3
Time: 42.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]
\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, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r52370 = x;
        double r52371 = y;
        double r52372 = r52370 + r52371;
        double r52373 = log(r52372);
        double r52374 = z;
        double r52375 = log(r52374);
        double r52376 = r52373 + r52375;
        double r52377 = t;
        double r52378 = r52376 - r52377;
        double r52379 = a;
        double r52380 = 0.5;
        double r52381 = r52379 - r52380;
        double r52382 = log(r52377);
        double r52383 = r52381 * r52382;
        double r52384 = r52378 + r52383;
        return r52384;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r52385 = a;
        double r52386 = 0.5;
        double r52387 = r52385 - r52386;
        double r52388 = t;
        double r52389 = log(r52388);
        double r52390 = 2.0;
        double r52391 = z;
        double r52392 = cbrt(r52391);
        double r52393 = log(r52392);
        double r52394 = r52390 * r52393;
        double r52395 = x;
        double r52396 = y;
        double r52397 = r52395 + r52396;
        double r52398 = log(r52397);
        double r52399 = r52393 + r52398;
        double r52400 = r52394 + r52399;
        double r52401 = r52400 - r52388;
        double r52402 = fma(r52387, r52389, r52401);
        return r52402;
}

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 add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
  8. Using strategy rm

  9. Applied fma-udef0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)} + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]

  10. Applied associate-+l+0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(x + y\right) + \log \left(\sqrt[3]{z}\right)\right)\right)} - t\right)\]

  11. Simplified0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \color{blue}{\left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)}\right) - t\right)\]

  12. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]

Reproduce

herbie shell --seed 2019323 +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))))