Average Error: 0.3 → 0.3
Time: 19.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right)} - t\right)\]
\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, \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 \left(\log \left(x + y\right) - \log z\right)\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r59429 = x;
        double r59430 = y;
        double r59431 = r59429 + r59430;
        double r59432 = log(r59431);
        double r59433 = z;
        double r59434 = log(r59433);
        double r59435 = r59432 + r59434;
        double r59436 = t;
        double r59437 = r59435 - r59436;
        double r59438 = a;
        double r59439 = 0.5;
        double r59440 = r59438 - r59439;
        double r59441 = log(r59436);
        double r59442 = r59440 * r59441;
        double r59443 = r59437 + r59442;
        return r59443;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r59444 = a;
        double r59445 = 0.5;
        double r59446 = r59444 - r59445;
        double r59447 = t;
        double r59448 = log(r59447);
        double r59449 = x;
        double r59450 = y;
        double r59451 = r59449 + r59450;
        double r59452 = log(r59451);
        double r59453 = 3.0;
        double r59454 = pow(r59452, r59453);
        double r59455 = z;
        double r59456 = log(r59455);
        double r59457 = pow(r59456, r59453);
        double r59458 = r59454 + r59457;
        double r59459 = r59452 - r59456;
        double r59460 = r59452 * r59459;
        double r59461 = fma(r59456, r59456, r59460);
        double r59462 = r59458 / r59461;
        double r59463 = r59462 - r59447;
        double r59464 = fma(r59446, r59448, r59463);
        return r59464;
}

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. Final simplification0.3

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

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))))