Average Error: 0.3 → 0.3
Time: 34.5s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\left(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\right)\]
\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, \left(\left(\log \left(\sqrt{z}\right) - t\right) + \log \left(\sqrt{z}\right)\right) + \log \left(y + x\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r2154168 = x;
        double r2154169 = y;
        double r2154170 = r2154168 + r2154169;
        double r2154171 = log(r2154170);
        double r2154172 = z;
        double r2154173 = log(r2154172);
        double r2154174 = r2154171 + r2154173;
        double r2154175 = t;
        double r2154176 = r2154174 - r2154175;
        double r2154177 = a;
        double r2154178 = 0.5;
        double r2154179 = r2154177 - r2154178;
        double r2154180 = log(r2154175);
        double r2154181 = r2154179 * r2154180;
        double r2154182 = r2154176 + r2154181;
        return r2154182;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2154183 = t;
        double r2154184 = log(r2154183);
        double r2154185 = a;
        double r2154186 = 0.5;
        double r2154187 = r2154185 - r2154186;
        double r2154188 = z;
        double r2154189 = sqrt(r2154188);
        double r2154190 = log(r2154189);
        double r2154191 = r2154190 - r2154183;
        double r2154192 = r2154191 + r2154190;
        double r2154193 = y;
        double r2154194 = x;
        double r2154195 = r2154193 + r2154194;
        double r2154196 = log(r2154195);
        double r2154197 = r2154192 + r2154196;
        double r2154198 = fma(r2154184, r2154187, r2154197);
        return r2154198;
}

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

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

  4. Applied add-sqr-sqrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))