Average Error: 0.3 → 0.3
Time: 20.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right)} - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right)} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r292768 = x;
        double r292769 = y;
        double r292770 = r292768 + r292769;
        double r292771 = log(r292770);
        double r292772 = z;
        double r292773 = log(r292772);
        double r292774 = r292771 + r292773;
        double r292775 = t;
        double r292776 = r292774 - r292775;
        double r292777 = a;
        double r292778 = 0.5;
        double r292779 = r292777 - r292778;
        double r292780 = log(r292775);
        double r292781 = r292779 * r292780;
        double r292782 = r292776 + r292781;
        return r292782;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r292783 = a;
        double r292784 = 0.5;
        double r292785 = r292783 - r292784;
        double r292786 = t;
        double r292787 = log(r292786);
        double r292788 = x;
        double r292789 = y;
        double r292790 = r292788 + r292789;
        double r292791 = log(r292790);
        double r292792 = 3.0;
        double r292793 = pow(r292791, r292792);
        double r292794 = z;
        double r292795 = log(r292794);
        double r292796 = pow(r292795, r292792);
        double r292797 = r292793 + r292796;
        double r292798 = r292791 - r292795;
        double r292799 = r292791 * r292798;
        double r292800 = fma(r292795, r292795, r292799);
        double r292801 = r292797 / r292800;
        double r292802 = r292801 - r292786;
        double r292803 = fma(r292785, r292787, r292802);
        return r292803;
}

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 pow10.3

    \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log z\right) - t\right)\right)}^{1}}\]
  5. Using strategy rm

  6. Applied flip3-+0.3

    \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\log \left(x + y\right) \cdot \log \left(x + y\right) + \left(\log z \cdot \log z - \log \left(x + y\right) \cdot \log z\right)}} - t\right)\right)}^{1}\]

  7. Simplified0.3

    \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right)}} - t\right)\right)}^{1}\]

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2020047 +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))))