Average Error: 0.3 → 0.3
Time: 11.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log \left(\sqrt{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt{t}\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log \left(\sqrt{t}\right), a - 0.5, \left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt{t}\right)
double f(double x, double y, double z, double t, double a) {
        double r134262 = x;
        double r134263 = y;
        double r134264 = r134262 + r134263;
        double r134265 = log(r134264);
        double r134266 = z;
        double r134267 = log(r134266);
        double r134268 = r134265 + r134267;
        double r134269 = t;
        double r134270 = r134268 - r134269;
        double r134271 = a;
        double r134272 = 0.5;
        double r134273 = r134271 - r134272;
        double r134274 = log(r134269);
        double r134275 = r134273 * r134274;
        double r134276 = r134270 + r134275;
        return r134276;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r134277 = t;
        double r134278 = sqrt(r134277);
        double r134279 = log(r134278);
        double r134280 = a;
        double r134281 = 0.5;
        double r134282 = r134280 - r134281;
        double r134283 = x;
        double r134284 = y;
        double r134285 = r134283 + r134284;
        double r134286 = log(r134285);
        double r134287 = z;
        double r134288 = log(r134287);
        double r134289 = r134286 + r134288;
        double r134290 = r134289 - r134277;
        double r134291 = fma(r134279, r134282, r134290);
        double r134292 = r134282 * r134279;
        double r134293 = r134291 + r134292;
        return r134293;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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))))