Average Error: 0.2 → 0.3
Time: 1.6m
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right) + \left(\mathsf{fma}\left(\left(a - 0.5\right), \left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right), \left(\log \left(x + y\right)\right)\right) + \left(\log z - t\right)\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right) + \left(\mathsf{fma}\left(\left(a - 0.5\right), \left(\log \left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)\right), \left(\log \left(x + y\right)\right)\right) + \left(\log z - t\right)\right)
double f(double x, double y, double z, double t, double a) {
        double r8779088 = x;
        double r8779089 = y;
        double r8779090 = r8779088 + r8779089;
        double r8779091 = log(r8779090);
        double r8779092 = z;
        double r8779093 = log(r8779092);
        double r8779094 = r8779091 + r8779093;
        double r8779095 = t;
        double r8779096 = r8779094 - r8779095;
        double r8779097 = a;
        double r8779098 = 0.5;
        double r8779099 = r8779097 - r8779098;
        double r8779100 = log(r8779095);
        double r8779101 = r8779099 * r8779100;
        double r8779102 = r8779096 + r8779101;
        return r8779102;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r8779103 = a;
        double r8779104 = 0.5;
        double r8779105 = r8779103 - r8779104;
        double r8779106 = t;
        double r8779107 = cbrt(r8779106);
        double r8779108 = log(r8779107);
        double r8779109 = r8779105 * r8779108;
        double r8779110 = r8779107 * r8779107;
        double r8779111 = log(r8779110);
        double r8779112 = x;
        double r8779113 = y;
        double r8779114 = r8779112 + r8779113;
        double r8779115 = log(r8779114);
        double r8779116 = fma(r8779105, r8779111, r8779115);
        double r8779117 = z;
        double r8779118 = log(r8779117);
        double r8779119 = r8779118 - r8779106;
        double r8779120 = r8779116 + r8779119;
        double r8779121 = r8779109 + r8779120;
        return r8779121;
}

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 0.2

    \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.3

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

  4. Applied log-prod0.3

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

  5. Applied distribute-lft-in0.3

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

  6. Applied associate-+r+0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))