Average Error: 0.2 → 0.2
Time: 33.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\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(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r22629549 = x;
        double r22629550 = y;
        double r22629551 = r22629549 + r22629550;
        double r22629552 = log(r22629551);
        double r22629553 = z;
        double r22629554 = log(r22629553);
        double r22629555 = r22629552 + r22629554;
        double r22629556 = t;
        double r22629557 = r22629555 - r22629556;
        double r22629558 = a;
        double r22629559 = 0.5;
        double r22629560 = r22629558 - r22629559;
        double r22629561 = log(r22629556);
        double r22629562 = r22629560 * r22629561;
        double r22629563 = r22629557 + r22629562;
        return r22629563;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r22629564 = t;
        double r22629565 = log(r22629564);
        double r22629566 = a;
        double r22629567 = 0.5;
        double r22629568 = r22629566 - r22629567;
        double r22629569 = 2.0;
        double r22629570 = z;
        double r22629571 = cbrt(r22629570);
        double r22629572 = log(r22629571);
        double r22629573 = y;
        double r22629574 = x;
        double r22629575 = r22629573 + r22629574;
        double r22629576 = log(r22629575);
        double r22629577 = fma(r22629569, r22629572, r22629576);
        double r22629578 = r22629577 + r22629572;
        double r22629579 = r22629578 - r22629564;
        double r22629580 = fma(r22629565, r22629568, r22629579);
        return r22629580;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.2
Target0.2
Herbie0.2
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

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

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

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"

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))