Average Error: 0.3 → 0.3
Time: 46.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(a - 0.5\right) \cdot \log t + \mathsf{fma}\left(\left(-\sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)\right)\right) + \left(\log \left(y + x\right) + \left(\log z - t\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(a - 0.5\right) \cdot \log t + \mathsf{fma}\left(\left(-\sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), \left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)\right)\right) + \left(\log \left(y + x\right) + \left(\log z - t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r2249622 = x;
        double r2249623 = y;
        double r2249624 = r2249622 + r2249623;
        double r2249625 = log(r2249624);
        double r2249626 = z;
        double r2249627 = log(r2249626);
        double r2249628 = r2249625 + r2249627;
        double r2249629 = t;
        double r2249630 = r2249628 - r2249629;
        double r2249631 = a;
        double r2249632 = 0.5;
        double r2249633 = r2249631 - r2249632;
        double r2249634 = log(r2249629);
        double r2249635 = r2249633 * r2249634;
        double r2249636 = r2249630 + r2249635;
        return r2249636;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r2249637 = a;
        double r2249638 = 0.5;
        double r2249639 = r2249637 - r2249638;
        double r2249640 = t;
        double r2249641 = log(r2249640);
        double r2249642 = r2249639 * r2249641;
        double r2249643 = cbrt(r2249640);
        double r2249644 = -r2249643;
        double r2249645 = r2249643 * r2249643;
        double r2249646 = r2249645 * r2249643;
        double r2249647 = fma(r2249644, r2249645, r2249646);
        double r2249648 = r2249642 + r2249647;
        double r2249649 = y;
        double r2249650 = x;
        double r2249651 = r2249649 + r2249650;
        double r2249652 = log(r2249651);
        double r2249653 = z;
        double r2249654 = log(r2249653);
        double r2249655 = r2249654 - r2249640;
        double r2249656 = r2249652 + r2249655;
        double r2249657 = r2249648 + r2249656;
        return r2249657;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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-cube-cbrt0.7

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

  4. Applied pow10.7

    \[\leadsto \left(\left(\log \left(x + y\right) + \log \color{blue}{\left({z}^{1}\right)}\right) - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) + \left(a - 0.5\right) \cdot \log t\]

  5. Applied log-pow0.7

    \[\leadsto \left(\left(\log \left(x + y\right) + \color{blue}{1 \cdot \log z}\right) - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) + \left(a - 0.5\right) \cdot \log t\]

  6. Applied *-un-lft-identity0.7

    \[\leadsto \left(\left(\color{blue}{1 \cdot \log \left(x + y\right)} + 1 \cdot \log z\right) - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) + \left(a - 0.5\right) \cdot \log t\]

  7. Applied distribute-lft-out0.7

    \[\leadsto \left(\color{blue}{1 \cdot \left(\log \left(x + y\right) + \log z\right)} - \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right) + \left(a - 0.5\right) \cdot \log t\]

  8. Applied prod-diff0.7

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(1, \left(\log \left(x + y\right) + \log z\right), \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) + \mathsf{fma}\left(\left(-\sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right)\right)} + \left(a - 0.5\right) \cdot \log t\]

  9. Applied associate-+l+0.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \left(\log \left(x + y\right) + \log z\right), \left(-\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) + \left(\mathsf{fma}\left(\left(-\sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right), \left(\sqrt[3]{t} \cdot \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right)\right) + \left(a - 0.5\right) \cdot \log t\right)}\]

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019121 +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))))