Average Error: 0.3 → 0.3
Time: 13.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(1, \log \left(x + y\right) + \log z, -t\right)\right)\]
\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, \mathsf{fma}\left(1, \log \left(x + y\right) + \log z, -t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r338144 = x;
        double r338145 = y;
        double r338146 = r338144 + r338145;
        double r338147 = log(r338146);
        double r338148 = z;
        double r338149 = log(r338148);
        double r338150 = r338147 + r338149;
        double r338151 = t;
        double r338152 = r338150 - r338151;
        double r338153 = a;
        double r338154 = 0.5;
        double r338155 = r338153 - r338154;
        double r338156 = log(r338151);
        double r338157 = r338155 * r338156;
        double r338158 = r338152 + r338157;
        return r338158;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r338159 = t;
        double r338160 = log(r338159);
        double r338161 = a;
        double r338162 = 0.5;
        double r338163 = r338161 - r338162;
        double r338164 = 1.0;
        double r338165 = x;
        double r338166 = y;
        double r338167 = r338165 + r338166;
        double r338168 = log(r338167);
        double r338169 = z;
        double r338170 = log(r338169);
        double r338171 = r338168 + r338170;
        double r338172 = -r338159;
        double r338173 = fma(r338164, r338171, r338172);
        double r338174 = fma(r338160, r338163, r338173);
        return r338174;
}

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

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

  4. Applied *-un-lft-identity0.3

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

  5. Applied fma-neg0.3

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

  6. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(1, \log \left(x + y\right) + \log z, -t\right)\right)\]

Reproduce

herbie shell --seed 2019356 +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))))