Average Error: 0.3 → 0.3
Time: 1.1m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(\sqrt[3]{\sqrt{z}}\right) + \left(\log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) + \left(\left(\log t \cdot \left(a - 0.5\right) + \left(\log \left(x + y\right) - t\right)\right) + \log \left(\sqrt{z}\right)\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(\sqrt[3]{\sqrt{z}}\right) + \left(\log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right) + \left(\left(\log t \cdot \left(a - 0.5\right) + \left(\log \left(x + y\right) - t\right)\right) + \log \left(\sqrt{z}\right)\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r4308361 = x;
        double r4308362 = y;
        double r4308363 = r4308361 + r4308362;
        double r4308364 = log(r4308363);
        double r4308365 = z;
        double r4308366 = log(r4308365);
        double r4308367 = r4308364 + r4308366;
        double r4308368 = t;
        double r4308369 = r4308367 - r4308368;
        double r4308370 = a;
        double r4308371 = 0.5;
        double r4308372 = r4308370 - r4308371;
        double r4308373 = log(r4308368);
        double r4308374 = r4308372 * r4308373;
        double r4308375 = r4308369 + r4308374;
        return r4308375;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r4308376 = z;
        double r4308377 = sqrt(r4308376);
        double r4308378 = cbrt(r4308377);
        double r4308379 = log(r4308378);
        double r4308380 = r4308378 * r4308378;
        double r4308381 = log(r4308380);
        double r4308382 = t;
        double r4308383 = log(r4308382);
        double r4308384 = a;
        double r4308385 = 0.5;
        double r4308386 = r4308384 - r4308385;
        double r4308387 = r4308383 * r4308386;
        double r4308388 = x;
        double r4308389 = y;
        double r4308390 = r4308388 + r4308389;
        double r4308391 = log(r4308390);
        double r4308392 = r4308391 - r4308382;
        double r4308393 = r4308387 + r4308392;
        double r4308394 = log(r4308377);
        double r4308395 = r4308393 + r4308394;
        double r4308396 = r4308381 + r4308395;
        double r4308397 = r4308379 + r4308396;
        return r4308397;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied fma-udef0.3

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

  5. Applied associate-+r+0.3

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

  7. Applied add-sqr-sqrt0.3

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

  8. Applied log-prod0.3

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

  9. Applied associate-+r+0.3

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

  11. Applied add-cube-cbrt0.3

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

  12. Applied log-prod0.3

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

  13. Applied associate-+r+0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019120 +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))))