Average Error: 0.3 → 0.3
Time: 12.8s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right) + \left(\left(a - 0.5\right) \cdot \log \left(\sqrt{t}\right) + \left(\left(a - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{t}}\right)\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right) + \left(\left(a - 0.5\right) \cdot \log \left(\sqrt{t}\right) + \left(\left(a - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) + \left(a - 0.5\right) \cdot \log \left(\sqrt{\sqrt[3]{t}}\right)\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r96405 = x;
        double r96406 = y;
        double r96407 = r96405 + r96406;
        double r96408 = log(r96407);
        double r96409 = z;
        double r96410 = log(r96409);
        double r96411 = r96408 + r96410;
        double r96412 = t;
        double r96413 = r96411 - r96412;
        double r96414 = a;
        double r96415 = 0.5;
        double r96416 = r96414 - r96415;
        double r96417 = log(r96412);
        double r96418 = r96416 * r96417;
        double r96419 = r96413 + r96418;
        return r96419;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r96420 = x;
        double r96421 = y;
        double r96422 = r96420 + r96421;
        double r96423 = log(r96422);
        double r96424 = z;
        double r96425 = sqrt(r96424);
        double r96426 = log(r96425);
        double r96427 = r96423 + r96426;
        double r96428 = r96427 + r96426;
        double r96429 = t;
        double r96430 = r96428 - r96429;
        double r96431 = a;
        double r96432 = 0.5;
        double r96433 = r96431 - r96432;
        double r96434 = sqrt(r96429);
        double r96435 = log(r96434);
        double r96436 = r96433 * r96435;
        double r96437 = cbrt(r96429);
        double r96438 = r96437 * r96437;
        double r96439 = sqrt(r96438);
        double r96440 = log(r96439);
        double r96441 = r96433 * r96440;
        double r96442 = sqrt(r96437);
        double r96443 = log(r96442);
        double r96444 = r96433 * r96443;
        double r96445 = r96441 + r96444;
        double r96446 = r96436 + r96445;
        double r96447 = r96430 + r96446;
        return r96447;
}

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. Using strategy rm

  3. Applied add-sqr-sqrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  7. Applied add-cube-cbrt0.3

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

  8. Applied sqrt-prod0.3

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

  9. Applied log-prod0.3

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

  10. Applied distribute-lft-in0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied log-prod0.3

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

  14. Applied associate-+r+0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))