Average Error: 0.3 → 0.3
Time: 21.9s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)\]
\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t
↓
\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) + \left(\log \left(\sqrt[3]{x + y}\right) + \log z\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r57110 = x;
        double r57111 = y;
        double r57112 = r57110 + r57111;
        double r57113 = log(r57112);
        double r57114 = z;
        double r57115 = log(r57114);
        double r57116 = r57113 + r57115;
        double r57117 = t;
        double r57118 = r57116 - r57117;
        double r57119 = a;
        double r57120 = 0.5;
        double r57121 = r57119 - r57120;
        double r57122 = log(r57117);
        double r57123 = r57121 * r57122;
        double r57124 = r57118 + r57123;
        return r57124;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r57125 = a;
        double r57126 = 0.5;
        double r57127 = r57125 - r57126;
        double r57128 = t;
        double r57129 = log(r57128);
        double r57130 = x;
        double r57131 = y;
        double r57132 = r57130 + r57131;
        double r57133 = cbrt(r57132);
        double r57134 = r57133 * r57133;
        double r57135 = log(r57134);
        double r57136 = log(r57133);
        double r57137 = z;
        double r57138 = log(r57137);
        double r57139 = r57136 + r57138;
        double r57140 = r57135 + r57139;
        double r57141 = r57140 - r57128;
        double r57142 = fma(r57127, r57129, r57141);
        return r57142;
}

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

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.3

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

  6. Applied associate-+l+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2020046 +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))))