Average Error: 0.3 → 0.3
Time: 41.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(2 \cdot \log \left(\sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) + \log \left(x + y\right)\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r216564 = x;
        double r216565 = y;
        double r216566 = r216564 + r216565;
        double r216567 = log(r216566);
        double r216568 = z;
        double r216569 = log(r216568);
        double r216570 = r216567 + r216569;
        double r216571 = t;
        double r216572 = r216570 - r216571;
        double r216573 = a;
        double r216574 = 0.5;
        double r216575 = r216573 - r216574;
        double r216576 = log(r216571);
        double r216577 = r216575 * r216576;
        double r216578 = r216572 + r216577;
        return r216578;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r216579 = a;
        double r216580 = 0.5;
        double r216581 = r216579 - r216580;
        double r216582 = t;
        double r216583 = log(r216582);
        double r216584 = 2.0;
        double r216585 = z;
        double r216586 = cbrt(r216585);
        double r216587 = log(r216586);
        double r216588 = r216584 * r216587;
        double r216589 = x;
        double r216590 = y;
        double r216591 = r216589 + r216590;
        double r216592 = log(r216591);
        double r216593 = r216587 + r216592;
        double r216594 = r216588 + r216593;
        double r216595 = r216594 - r216582;
        double r216596 = fma(r216581, r216583, r216595);
        return r216596;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original0.3
Target0.3
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

  1. Initial program 0.3

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

  2. Simplified0.3

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  9. Applied fma-udef0.3

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

  10. Applied associate-+l+0.3

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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