Average Error: 0.3 → 0.3
Time: 30.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\log \left(\sqrt[3]{t}\right) \cdot \left(a - 0.5\right) + \mathsf{fma}\left(\log \left({t}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{t}\right), a - 0.5, \left(\log z + \log \left(x + y\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\log \left(\sqrt[3]{t}\right) \cdot \left(a - 0.5\right) + \mathsf{fma}\left(\log \left({t}^{\frac{1}{3}}\right) + \log \left(\sqrt[3]{t}\right), a - 0.5, \left(\log z + \log \left(x + y\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r979701 = x;
        double r979702 = y;
        double r979703 = r979701 + r979702;
        double r979704 = log(r979703);
        double r979705 = z;
        double r979706 = log(r979705);
        double r979707 = r979704 + r979706;
        double r979708 = t;
        double r979709 = r979707 - r979708;
        double r979710 = a;
        double r979711 = 0.5;
        double r979712 = r979710 - r979711;
        double r979713 = log(r979708);
        double r979714 = r979712 * r979713;
        double r979715 = r979709 + r979714;
        return r979715;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r979716 = t;
        double r979717 = cbrt(r979716);
        double r979718 = log(r979717);
        double r979719 = a;
        double r979720 = 0.5;
        double r979721 = r979719 - r979720;
        double r979722 = r979718 * r979721;
        double r979723 = 0.3333333333333333;
        double r979724 = pow(r979716, r979723);
        double r979725 = log(r979724);
        double r979726 = r979725 + r979718;
        double r979727 = z;
        double r979728 = log(r979727);
        double r979729 = x;
        double r979730 = y;
        double r979731 = r979729 + r979730;
        double r979732 = log(r979731);
        double r979733 = r979728 + r979732;
        double r979734 = r979733 - r979716;
        double r979735 = fma(r979726, r979721, r979734);
        double r979736 = r979722 + r979735;
        return r979736;
}

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-cbrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied pow1/30.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019153 +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))))