Average Error: 0.2 → 0.2
Time: 30.7s
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 r217273 = x;
        double r217274 = y;
        double r217275 = r217273 + r217274;
        double r217276 = log(r217275);
        double r217277 = z;
        double r217278 = log(r217277);
        double r217279 = r217276 + r217278;
        double r217280 = t;
        double r217281 = r217279 - r217280;
        double r217282 = a;
        double r217283 = 0.5;
        double r217284 = r217282 - r217283;
        double r217285 = log(r217280);
        double r217286 = r217284 * r217285;
        double r217287 = r217281 + r217286;
        return r217287;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r217288 = a;
        double r217289 = 0.5;
        double r217290 = r217288 - r217289;
        double r217291 = t;
        double r217292 = log(r217291);
        double r217293 = x;
        double r217294 = y;
        double r217295 = r217293 + r217294;
        double r217296 = cbrt(r217295);
        double r217297 = r217296 * r217296;
        double r217298 = log(r217297);
        double r217299 = log(r217296);
        double r217300 = z;
        double r217301 = log(r217300);
        double r217302 = r217299 + r217301;
        double r217303 = r217298 + r217302;
        double r217304 = r217303 - r217291;
        double r217305 = fma(r217290, r217292, r217304);
        return r217305;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
Herbie0.2
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

  1. Initial program 0.2

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

  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-cube-cbrt0.2

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

  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+0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019235 +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))))