Average Error: 0.3 → 0.3
Time: 20.0s
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 r338402 = x;
        double r338403 = y;
        double r338404 = r338402 + r338403;
        double r338405 = log(r338404);
        double r338406 = z;
        double r338407 = log(r338406);
        double r338408 = r338405 + r338407;
        double r338409 = t;
        double r338410 = r338408 - r338409;
        double r338411 = a;
        double r338412 = 0.5;
        double r338413 = r338411 - r338412;
        double r338414 = log(r338409);
        double r338415 = r338413 * r338414;
        double r338416 = r338410 + r338415;
        return r338416;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r338417 = a;
        double r338418 = 0.5;
        double r338419 = r338417 - r338418;
        double r338420 = t;
        double r338421 = log(r338420);
        double r338422 = x;
        double r338423 = y;
        double r338424 = r338422 + r338423;
        double r338425 = log(r338424);
        double r338426 = 3.0;
        double r338427 = pow(r338425, r338426);
        double r338428 = z;
        double r338429 = log(r338428);
        double r338430 = pow(r338429, r338426);
        double r338431 = r338427 + r338430;
        double r338432 = r338425 - r338429;
        double r338433 = r338425 * r338432;
        double r338434 = fma(r338429, r338429, r338433);
        double r338435 = r338431 / r338434;
        double r338436 = r338435 - r338420;
        double r338437 = fma(r338419, r338421, r338436);
        return r338437;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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 *-un-lft-identity0.3

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

  6. Applied flip3-+0.3

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

  7. Simplified0.3

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

  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

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))