Average Error: 0.3 → 0.3
Time: 12.2s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) + \log \left(\sqrt{\sqrt[3]{z}}\right)\right) + \log \left(\sqrt{z}\right)\right) + \mathsf{fma}\left(\log t, a - 0.5, -t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(\left(\left(\log \left(x + y\right) + \log \left(\sqrt{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right)\right) + \log \left(\sqrt{\sqrt[3]{z}}\right)\right) + \log \left(\sqrt{z}\right)\right) + \mathsf{fma}\left(\log t, a - 0.5, -t\right)
double f(double x, double y, double z, double t, double a) {
        double r339034 = x;
        double r339035 = y;
        double r339036 = r339034 + r339035;
        double r339037 = log(r339036);
        double r339038 = z;
        double r339039 = log(r339038);
        double r339040 = r339037 + r339039;
        double r339041 = t;
        double r339042 = r339040 - r339041;
        double r339043 = a;
        double r339044 = 0.5;
        double r339045 = r339043 - r339044;
        double r339046 = log(r339041);
        double r339047 = r339045 * r339046;
        double r339048 = r339042 + r339047;
        return r339048;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r339049 = x;
        double r339050 = y;
        double r339051 = r339049 + r339050;
        double r339052 = log(r339051);
        double r339053 = z;
        double r339054 = cbrt(r339053);
        double r339055 = r339054 * r339054;
        double r339056 = sqrt(r339055);
        double r339057 = log(r339056);
        double r339058 = r339052 + r339057;
        double r339059 = sqrt(r339054);
        double r339060 = log(r339059);
        double r339061 = r339058 + r339060;
        double r339062 = sqrt(r339053);
        double r339063 = log(r339062);
        double r339064 = r339061 + r339063;
        double r339065 = t;
        double r339066 = log(r339065);
        double r339067 = a;
        double r339068 = 0.5;
        double r339069 = r339067 - r339068;
        double r339070 = -r339065;
        double r339071 = fma(r339066, r339069, r339070);
        double r339072 = r339064 + r339071;
        return r339072;
}

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. Using strategy rm

  3. Applied sub-neg0.3

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

  4. Applied associate-+l+0.3

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

  5. Simplified0.3

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

  7. Applied add-sqr-sqrt0.3

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

  8. Applied log-prod0.3

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

  9. Applied associate-+r+0.3

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

  11. Applied add-cube-cbrt0.3

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

  12. Applied sqrt-prod0.3

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

  13. Applied log-prod0.3

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

  14. Applied associate-+r+0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019353 +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))))