Average Error: 0.2 → 0.3
Time: 42.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(a - 0.5, \log t, \left(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)\]
\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(\left(\log \left(x + y\right) + \log \left(\sqrt{z}\right)\right) + \log \left(\sqrt{z}\right)\right) - t\right)
double f(double x, double y, double z, double t, double a) {
        double r77095 = x;
        double r77096 = y;
        double r77097 = r77095 + r77096;
        double r77098 = log(r77097);
        double r77099 = z;
        double r77100 = log(r77099);
        double r77101 = r77098 + r77100;
        double r77102 = t;
        double r77103 = r77101 - r77102;
        double r77104 = a;
        double r77105 = 0.5;
        double r77106 = r77104 - r77105;
        double r77107 = log(r77102);
        double r77108 = r77106 * r77107;
        double r77109 = r77103 + r77108;
        return r77109;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r77110 = a;
        double r77111 = 0.5;
        double r77112 = r77110 - r77111;
        double r77113 = t;
        double r77114 = log(r77113);
        double r77115 = x;
        double r77116 = y;
        double r77117 = r77115 + r77116;
        double r77118 = log(r77117);
        double r77119 = z;
        double r77120 = sqrt(r77119);
        double r77121 = log(r77120);
        double r77122 = r77118 + r77121;
        double r77123 = r77122 + r77121;
        double r77124 = r77123 - r77113;
        double r77125 = fma(r77112, r77114, r77124);
        return r77125;
}

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. Simplified0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied log-prod0.2

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

  6. Applied associate-+r+0.3

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

  7. Final simplification0.3

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

Reproduce

herbie shell --seed 2019347 +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))))