Average Error: 0.3 → 0.3
Time: 1.0m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z - \log \left(x + y\right), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)} - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \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), \log \left(x + y\right) \cdot \log \left(x + y\right)\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r210 = x;
        double r211 = y;
        double r212 = r210 + r211;
        double r213 = log(r212);
        double r214 = z;
        double r215 = log(r214);
        double r216 = r213 + r215;
        double r217 = t;
        double r218 = r216 - r217;
        double r219 = a;
        double r220 = 0.5;
        double r221 = r219 - r220;
        double r222 = log(r217);
        double r223 = r221 * r222;
        double r224 = r218 + r223;
        return r224;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r225 = t;
        double r226 = log(r225);
        double r227 = a;
        double r228 = 0.5;
        double r229 = r227 - r228;
        double r230 = x;
        double r231 = y;
        double r232 = r230 + r231;
        double r233 = log(r232);
        double r234 = 3.0;
        double r235 = pow(r233, r234);
        double r236 = z;
        double r237 = log(r236);
        double r238 = pow(r237, r234);
        double r239 = r235 + r238;
        double r240 = r237 - r233;
        double r241 = r233 * r233;
        double r242 = fma(r237, r240, r241);
        double r243 = r239 / r242;
        double r244 = r243 - r225;
        double r245 = fma(r226, r229, r244);
        return r245;
}

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 flip3-+0.3

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

  5. Simplified0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020025 +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))))