Average Error: 0.2 → 0.2
Time: 41.0s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]
\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 r14532651 = x;
        double r14532652 = y;
        double r14532653 = r14532651 + r14532652;
        double r14532654 = log(r14532653);
        double r14532655 = z;
        double r14532656 = log(r14532655);
        double r14532657 = r14532654 + r14532656;
        double r14532658 = t;
        double r14532659 = r14532657 - r14532658;
        double r14532660 = a;
        double r14532661 = 0.5;
        double r14532662 = r14532660 - r14532661;
        double r14532663 = log(r14532658);
        double r14532664 = r14532662 * r14532663;
        double r14532665 = r14532659 + r14532664;
        return r14532665;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r14532666 = t;
        double r14532667 = log(r14532666);
        double r14532668 = a;
        double r14532669 = 0.5;
        double r14532670 = r14532668 - r14532669;
        double r14532671 = 2.0;
        double r14532672 = z;
        double r14532673 = cbrt(r14532672);
        double r14532674 = log(r14532673);
        double r14532675 = y;
        double r14532676 = x;
        double r14532677 = r14532675 + r14532676;
        double r14532678 = log(r14532677);
        double r14532679 = fma(r14532671, r14532674, r14532678);
        double r14532680 = r14532674 - r14532666;
        double r14532681 = r14532679 + r14532680;
        double r14532682 = fma(r14532667, r14532670, r14532681);
        return r14532682;
}

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

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

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

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