Average Error: 0.3 → 0.3
Time: 15.3s
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 r58242 = x;
        double r58243 = y;
        double r58244 = r58242 + r58243;
        double r58245 = log(r58244);
        double r58246 = z;
        double r58247 = log(r58246);
        double r58248 = r58245 + r58247;
        double r58249 = t;
        double r58250 = r58248 - r58249;
        double r58251 = a;
        double r58252 = 0.5;
        double r58253 = r58251 - r58252;
        double r58254 = log(r58249);
        double r58255 = r58253 * r58254;
        double r58256 = r58250 + r58255;
        return r58256;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r58257 = t;
        double r58258 = sqrt(r58257);
        double r58259 = log(r58258);
        double r58260 = a;
        double r58261 = 0.5;
        double r58262 = r58260 - r58261;
        double r58263 = x;
        double r58264 = y;
        double r58265 = r58263 + r58264;
        double r58266 = log(r58265);
        double r58267 = z;
        double r58268 = log(r58267);
        double r58269 = r58266 + r58268;
        double r58270 = r58269 - r58257;
        double r58271 = fma(r58259, r58262, r58270);
        double r58272 = r58262 * r58259;
        double r58273 = r58271 + r58272;
        return r58273;
}

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))))