Average Error: 0.2 → 0.3
Time: 11.6s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(\log t, a - 0.5, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r65754 = x;
        double r65755 = y;
        double r65756 = r65754 + r65755;
        double r65757 = log(r65756);
        double r65758 = z;
        double r65759 = log(r65758);
        double r65760 = r65757 + r65759;
        double r65761 = t;
        double r65762 = r65760 - r65761;
        double r65763 = a;
        double r65764 = 0.5;
        double r65765 = r65763 - r65764;
        double r65766 = log(r65761);
        double r65767 = r65765 * r65766;
        double r65768 = r65762 + r65767;
        return r65768;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r65769 = t;
        double r65770 = log(r65769);
        double r65771 = a;
        double r65772 = 0.5;
        double r65773 = r65771 - r65772;
        double r65774 = x;
        double r65775 = y;
        double r65776 = r65774 + r65775;
        double r65777 = cbrt(r65776);
        double r65778 = r65777 * r65777;
        double r65779 = log(r65778);
        double r65780 = log(r65777);
        double r65781 = z;
        double r65782 = log(r65781);
        double r65783 = r65780 + r65782;
        double r65784 = r65779 + r65783;
        double r65785 = r65784 - r65769;
        double r65786 = fma(r65770, r65773, r65785);
        return r65786;
}

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 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  2. Simplified0.2

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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))))