Average Error: 0.3 → 0.3
Time: 2.2m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(x + y\right) - \left(\left(\mathsf{fma}\left(\log t, 0.5 - a, t\right) - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) - \log \left(\sqrt[3]{z}\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(x + y\right) - \left(\left(\mathsf{fma}\left(\log t, 0.5 - a, t\right) - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) - \log \left(\sqrt[3]{z}\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r18821305 = x;
        double r18821306 = y;
        double r18821307 = r18821305 + r18821306;
        double r18821308 = log(r18821307);
        double r18821309 = z;
        double r18821310 = log(r18821309);
        double r18821311 = r18821308 + r18821310;
        double r18821312 = t;
        double r18821313 = r18821311 - r18821312;
        double r18821314 = a;
        double r18821315 = 0.5;
        double r18821316 = r18821314 - r18821315;
        double r18821317 = log(r18821312);
        double r18821318 = r18821316 * r18821317;
        double r18821319 = r18821313 + r18821318;
        return r18821319;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r18821320 = x;
        double r18821321 = y;
        double r18821322 = r18821320 + r18821321;
        double r18821323 = log(r18821322);
        double r18821324 = t;
        double r18821325 = log(r18821324);
        double r18821326 = 0.5;
        double r18821327 = a;
        double r18821328 = r18821326 - r18821327;
        double r18821329 = fma(r18821325, r18821328, r18821324);
        double r18821330 = z;
        double r18821331 = cbrt(r18821330);
        double r18821332 = r18821331 * r18821331;
        double r18821333 = log(r18821332);
        double r18821334 = r18821329 - r18821333;
        double r18821335 = log(r18821331);
        double r18821336 = r18821334 - r18821335;
        double r18821337 = r18821323 - r18821336;
        return r18821337;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate--r+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"

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

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