Average Error: 0.3 → 0.3
Time: 32.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \mathsf{fma}\left(a - 0.5, \log t, \log z - t\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \mathsf{fma}\left(a - 0.5, \log t, \log z - t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r53191 = x;
        double r53192 = y;
        double r53193 = r53191 + r53192;
        double r53194 = log(r53193);
        double r53195 = z;
        double r53196 = log(r53195);
        double r53197 = r53194 + r53196;
        double r53198 = t;
        double r53199 = r53197 - r53198;
        double r53200 = a;
        double r53201 = 0.5;
        double r53202 = r53200 - r53201;
        double r53203 = log(r53198);
        double r53204 = r53202 * r53203;
        double r53205 = r53199 + r53204;
        return r53205;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r53206 = x;
        double r53207 = y;
        double r53208 = r53206 + r53207;
        double r53209 = cbrt(r53208);
        double r53210 = r53209 * r53209;
        double r53211 = log(r53210);
        double r53212 = log(r53209);
        double r53213 = a;
        double r53214 = 0.5;
        double r53215 = r53213 - r53214;
        double r53216 = t;
        double r53217 = log(r53216);
        double r53218 = z;
        double r53219 = log(r53218);
        double r53220 = r53219 - r53216;
        double r53221 = fma(r53215, r53217, r53220);
        double r53222 = r53212 + r53221;
        double r53223 = r53211 + r53222;
        return r53223;
}

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. Using strategy rm

  3. Applied associate--l+0.3

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

  4. Applied associate-+l+0.3

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

  5. Simplified0.3

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied log-prod0.3

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

  9. Applied associate-+l+0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019195 +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))))