Average Error: 0.3 → 0.3
Time: 38.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\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[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r238379 = x;
        double r238380 = y;
        double r238381 = r238379 + r238380;
        double r238382 = log(r238381);
        double r238383 = z;
        double r238384 = log(r238383);
        double r238385 = r238382 + r238384;
        double r238386 = t;
        double r238387 = r238385 - r238386;
        double r238388 = a;
        double r238389 = 0.5;
        double r238390 = r238388 - r238389;
        double r238391 = log(r238386);
        double r238392 = r238390 * r238391;
        double r238393 = r238387 + r238392;
        return r238393;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r238394 = a;
        double r238395 = 0.5;
        double r238396 = r238394 - r238395;
        double r238397 = t;
        double r238398 = log(r238397);
        double r238399 = x;
        double r238400 = y;
        double r238401 = r238399 + r238400;
        double r238402 = cbrt(r238401);
        double r238403 = r238402 * r238402;
        double r238404 = log(r238403);
        double r238405 = log(r238402);
        double r238406 = z;
        double r238407 = log(r238406);
        double r238408 = r238405 + r238407;
        double r238409 = r238404 + r238408;
        double r238410 = r238409 - r238397;
        double r238411 = fma(r238396, r238398, r238410);
        return r238411;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019208 +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))))