Average Error: 0.3 → 0.3
Time: 2.2m
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 r1026233 = x;
        double r1026234 = y;
        double r1026235 = r1026233 + r1026234;
        double r1026236 = log(r1026235);
        double r1026237 = z;
        double r1026238 = log(r1026237);
        double r1026239 = r1026236 + r1026238;
        double r1026240 = t;
        double r1026241 = r1026239 - r1026240;
        double r1026242 = a;
        double r1026243 = 0.5;
        double r1026244 = r1026242 - r1026243;
        double r1026245 = log(r1026240);
        double r1026246 = r1026244 * r1026245;
        double r1026247 = r1026241 + r1026246;
        return r1026247;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r1026248 = t;
        double r1026249 = log(r1026248);
        double r1026250 = a;
        double r1026251 = 0.5;
        double r1026252 = r1026250 - r1026251;
        double r1026253 = x;
        double r1026254 = y;
        double r1026255 = r1026253 + r1026254;
        double r1026256 = log(r1026255);
        double r1026257 = 3.0;
        double r1026258 = pow(r1026256, r1026257);
        double r1026259 = z;
        double r1026260 = log(r1026259);
        double r1026261 = pow(r1026260, r1026257);
        double r1026262 = r1026258 + r1026261;
        double r1026263 = r1026260 - r1026256;
        double r1026264 = r1026256 * r1026256;
        double r1026265 = fma(r1026260, r1026263, r1026264);
        double r1026266 = r1026262 / r1026265;
        double r1026267 = r1026266 - r1026248;
        double r1026268 = fma(r1026249, r1026252, r1026267);
        return r1026268;
}

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 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

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))