Average Error: 0.3 → 0.3
Time: 11.4s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{\log \left(\sqrt{x + y}\right) \cdot \log \left(x + y\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)\]
\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, \frac{\log \left(\sqrt{x + y}\right) \cdot \log \left(x + y\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r295004 = x;
        double r295005 = y;
        double r295006 = r295004 + r295005;
        double r295007 = log(r295006);
        double r295008 = z;
        double r295009 = log(r295008);
        double r295010 = r295007 + r295009;
        double r295011 = t;
        double r295012 = r295010 - r295011;
        double r295013 = a;
        double r295014 = 0.5;
        double r295015 = r295013 - r295014;
        double r295016 = log(r295011);
        double r295017 = r295015 * r295016;
        double r295018 = r295012 + r295017;
        return r295018;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r295019 = t;
        double r295020 = log(r295019);
        double r295021 = a;
        double r295022 = 0.5;
        double r295023 = r295021 - r295022;
        double r295024 = x;
        double r295025 = y;
        double r295026 = r295024 + r295025;
        double r295027 = sqrt(r295026);
        double r295028 = log(r295027);
        double r295029 = log(r295026);
        double r295030 = r295028 * r295029;
        double r295031 = z;
        double r295032 = log(r295031);
        double r295033 = 2.0;
        double r295034 = pow(r295032, r295033);
        double r295035 = -r295034;
        double r295036 = fma(r295028, r295029, r295035);
        double r295037 = r295030 + r295036;
        double r295038 = r295029 - r295032;
        double r295039 = r295037 / r295038;
        double r295040 = r295039 - r295019;
        double r295041 = fma(r295020, r295023, r295040);
        return r295041;
}

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

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

  4. Applied flip-+0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied log-prod0.3

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

  8. Applied distribute-rgt-in0.3

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

  9. Applied associate--l+0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +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))))