Average Error: 0.3 → 0.3
Time: 11.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(x + y\right)\right) + \log \left(\sqrt[3]{z}\right)\right) - t\right)\]
\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, \left(\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)
double f(double x, double y, double z, double t, double a) {
        double r67061 = x;
        double r67062 = y;
        double r67063 = r67061 + r67062;
        double r67064 = log(r67063);
        double r67065 = z;
        double r67066 = log(r67065);
        double r67067 = r67064 + r67066;
        double r67068 = t;
        double r67069 = r67067 - r67068;
        double r67070 = a;
        double r67071 = 0.5;
        double r67072 = r67070 - r67071;
        double r67073 = log(r67068);
        double r67074 = r67072 * r67073;
        double r67075 = r67069 + r67074;
        return r67075;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r67076 = t;
        double r67077 = log(r67076);
        double r67078 = a;
        double r67079 = 0.5;
        double r67080 = r67078 - r67079;
        double r67081 = 2.0;
        double r67082 = z;
        double r67083 = cbrt(r67082);
        double r67084 = log(r67083);
        double r67085 = x;
        double r67086 = y;
        double r67087 = r67085 + r67086;
        double r67088 = log(r67087);
        double r67089 = fma(r67081, r67084, r67088);
        double r67090 = r67089 + r67084;
        double r67091 = r67090 - r67076;
        double r67092 = fma(r67077, r67080, r67091);
        return r67092;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

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 add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019346 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))