Average Error: 0.3 → 0.3
Time: 15.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log t \cdot \left(a - 0.5\right) + \left(\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \log \left(x + y\right), \log z \cdot \left(\log z - \log \left(x + y\right)\right)\right)} - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log t \cdot \left(a - 0.5\right) + \left(\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log \left(x + y\right), \log \left(x + y\right), \log z \cdot \left(\log z - \log \left(x + y\right)\right)\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r354276 = x;
        double r354277 = y;
        double r354278 = r354276 + r354277;
        double r354279 = log(r354278);
        double r354280 = z;
        double r354281 = log(r354280);
        double r354282 = r354279 + r354281;
        double r354283 = t;
        double r354284 = r354282 - r354283;
        double r354285 = a;
        double r354286 = 0.5;
        double r354287 = r354285 - r354286;
        double r354288 = log(r354283);
        double r354289 = r354287 * r354288;
        double r354290 = r354284 + r354289;
        return r354290;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r354291 = t;
        double r354292 = log(r354291);
        double r354293 = a;
        double r354294 = 0.5;
        double r354295 = r354293 - r354294;
        double r354296 = r354292 * r354295;
        double r354297 = x;
        double r354298 = y;
        double r354299 = r354297 + r354298;
        double r354300 = log(r354299);
        double r354301 = 3.0;
        double r354302 = pow(r354300, r354301);
        double r354303 = z;
        double r354304 = log(r354303);
        double r354305 = pow(r354304, r354301);
        double r354306 = r354302 + r354305;
        double r354307 = r354304 - r354300;
        double r354308 = r354304 * r354307;
        double r354309 = fma(r354300, r354300, r354308);
        double r354310 = r354306 / r354309;
        double r354311 = r354310 - r354291;
        double r354312 = r354296 + r354311;
        return r354312;
}

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 fma-udef0.3

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

  5. Simplified0.3

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

  7. Applied flip3-+0.3

    \[\leadsto \log t \cdot \left(a - 0.5\right) + \left(\color{blue}{\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\log \left(x + y\right) \cdot \log \left(x + y\right) + \left(\log z \cdot \log z - \log \left(x + y\right) \cdot \log z\right)}} - t\right)\]

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020045 +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))))