Average Error: 0.2 → 0.2
Time: 45.1s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]
\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, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r3690648 = x;
        double r3690649 = y;
        double r3690650 = r3690648 + r3690649;
        double r3690651 = log(r3690650);
        double r3690652 = z;
        double r3690653 = log(r3690652);
        double r3690654 = r3690651 + r3690653;
        double r3690655 = t;
        double r3690656 = r3690654 - r3690655;
        double r3690657 = a;
        double r3690658 = 0.5;
        double r3690659 = r3690657 - r3690658;
        double r3690660 = log(r3690655);
        double r3690661 = r3690659 * r3690660;
        double r3690662 = r3690656 + r3690661;
        return r3690662;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r3690663 = t;
        double r3690664 = log(r3690663);
        double r3690665 = a;
        double r3690666 = 0.5;
        double r3690667 = r3690665 - r3690666;
        double r3690668 = 2.0;
        double r3690669 = z;
        double r3690670 = cbrt(r3690669);
        double r3690671 = log(r3690670);
        double r3690672 = y;
        double r3690673 = x;
        double r3690674 = r3690672 + r3690673;
        double r3690675 = log(r3690674);
        double r3690676 = fma(r3690668, r3690671, r3690675);
        double r3690677 = r3690671 - r3690663;
        double r3690678 = r3690676 + r3690677;
        double r3690679 = fma(r3690664, r3690667, r3690678);
        return r3690679;
}

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

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

  4. Applied add-cube-cbrt0.2

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

  5. Applied log-prod0.3

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

  6. Applied associate--l+0.3

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

  7. Applied associate-+r+0.2

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

  8. Simplified0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right)} + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]

  9. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]

Reproduce

herbie shell --seed 2019174 +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))))