Average Error: 0.3 → 0.3
Time: 3.0m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + \left(\log \left(\sqrt{\sqrt[3]{x + y}}\right) + \left(\log \left(\sqrt{x + y}\right) + \log z\right)\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) + \left(\log \left(\sqrt{\sqrt[3]{x + y}}\right) + \left(\log \left(\sqrt{x + y}\right) + \log z\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r466172 = x;
        double r466173 = y;
        double r466174 = r466172 + r466173;
        double r466175 = log(r466174);
        double r466176 = z;
        double r466177 = log(r466176);
        double r466178 = r466175 + r466177;
        double r466179 = t;
        double r466180 = r466178 - r466179;
        double r466181 = a;
        double r466182 = 0.5;
        double r466183 = r466181 - r466182;
        double r466184 = log(r466179);
        double r466185 = r466183 * r466184;
        double r466186 = r466180 + r466185;
        return r466186;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r466187 = a;
        double r466188 = 0.5;
        double r466189 = r466187 - r466188;
        double r466190 = t;
        double r466191 = log(r466190);
        double r466192 = x;
        double r466193 = y;
        double r466194 = r466192 + r466193;
        double r466195 = cbrt(r466194);
        double r466196 = r466195 * r466195;
        double r466197 = sqrt(r466196);
        double r466198 = log(r466197);
        double r466199 = sqrt(r466195);
        double r466200 = log(r466199);
        double r466201 = sqrt(r466194);
        double r466202 = log(r466201);
        double r466203 = z;
        double r466204 = log(r466203);
        double r466205 = r466202 + r466204;
        double r466206 = r466200 + r466205;
        double r466207 = r466198 + r466206;
        double r466208 = r466207 - r466190;
        double r466209 = fma(r466189, r466191, r466208);
        return r466209;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied log-prod0.2

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

  6. Applied associate-+l+0.3

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

  8. Applied add-cube-cbrt0.3

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

  9. Applied sqrt-prod0.3

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

  10. Applied log-prod0.3

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

  11. Applied associate-+l+0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019305 +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))))