Average Error: 0.2 → 0.3
Time: 36.7s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), \log \left(y + x\right)\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)\]
\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 r241071 = x;
        double r241072 = y;
        double r241073 = r241071 + r241072;
        double r241074 = log(r241073);
        double r241075 = z;
        double r241076 = log(r241075);
        double r241077 = r241074 + r241076;
        double r241078 = t;
        double r241079 = r241077 - r241078;
        double r241080 = a;
        double r241081 = 0.5;
        double r241082 = r241080 - r241081;
        double r241083 = log(r241078);
        double r241084 = r241082 * r241083;
        double r241085 = r241079 + r241084;
        return r241085;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r241086 = t;
        double r241087 = log(r241086);
        double r241088 = a;
        double r241089 = 0.5;
        double r241090 = r241088 - r241089;
        double r241091 = 2.0;
        double r241092 = z;
        double r241093 = cbrt(r241092);
        double r241094 = log(r241093);
        double r241095 = y;
        double r241096 = x;
        double r241097 = r241095 + r241096;
        double r241098 = log(r241097);
        double r241099 = fma(r241091, r241094, r241098);
        double r241100 = r241094 - r241086;
        double r241101 = r241099 + r241100;
        double r241102 = fma(r241087, r241090, r241101);
        return r241102;
}

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
Herbie0.3
\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)\]

Derivation

  1. Initial program 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]

  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+0.3

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

  8. Simplified0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019196 +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))))