Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 10.5s

Precision: 64

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

↓

\[\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}, \log z \cdot \log z\right)} - t\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r306169 = x;
        double r306170 = y;
        double r306171 = r306169 + r306170;
        double r306172 = log(r306171);
        double r306173 = z;
        double r306174 = log(r306173);
        double r306175 = r306172 + r306174;
        double r306176 = t;
        double r306177 = r306175 - r306176;
        double r306178 = a;
        double r306179 = 0.5;
        double r306180 = r306178 - r306179;
        double r306181 = log(r306176);
        double r306182 = r306180 * r306181;
        double r306183 = r306177 + r306182;
        return r306183;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r306184 = t;
        double r306185 = log(r306184);
        double r306186 = a;
        double r306187 = 0.5;
        double r306188 = r306186 - r306187;
        double r306189 = x;
        double r306190 = y;
        double r306191 = r306189 + r306190;
        double r306192 = log(r306191);
        double r306193 = 3.0;
        double r306194 = pow(r306192, r306193);
        double r306195 = z;
        double r306196 = log(r306195);
        double r306197 = pow(r306196, r306193);
        double r306198 = r306194 + r306197;
        double r306199 = r306192 - r306196;
        double r306200 = pow(r306199, r306193);
        double r306201 = cbrt(r306200);
        double r306202 = r306196 * r306196;
        double r306203 = fma(r306192, r306201, r306202);
        double r306204 = r306198 / r306203;
        double r306205 = r306204 - r306184;
        double r306206 = fma(r306185, r306188, r306205);
        return r306206;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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. Simplified 0.3

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

4. Applied pow1 0.3

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

6. Applied flip3-+ 0.3

   \[\leadsto {\left(\mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\log \left(x + y\right) \cdot \log \left(x + y\right) + \left(\log z \cdot \log z - \log \left(x + y\right) \cdot \log z\right)}} - t\right)\right)}^{1}\]

7. Simplified 0.3

   \[\leadsto {\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\log \left(x + y\right), \log \left(x + y\right) - \log z, \log z \cdot \log z\right)}} - t\right)\right)}^{1}\]
8. Using strategy rm

9. Applied add-cbrt-cube 0.3

   \[\leadsto {\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \color{blue}{\sqrt[3]{\left(\left(\log \left(x + y\right) - \log z\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right) \cdot \left(\log \left(x + y\right) - \log z\right)}}, \log z \cdot \log z\right)} - t\right)\right)}^{1}\]

10. Simplified 0.3

    \[\leadsto {\left(\mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \sqrt[3]{\color{blue}{{\left(\log \left(x + y\right) - \log z\right)}^{3}}}, \log z \cdot \log z\right)} - t\right)\right)}^{1}\]

11. Final simplification 0.3

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \sqrt[3]{{\left(\log \left(x + y\right) - \log z\right)}^{3}}, \log z \cdot \log z\right)} - t\right)\]

Reproduce

herbie shell --seed 2020047 +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))))