Average Error: 0.3 → 0.3
Time: 11.3s
Precision: 64
\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t\]
↓
\[\mathsf{fma}\left(\log t, a - 0.5, \frac{\log \left(x + y\right) \cdot \log \left(\sqrt{x + y}\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)\]
\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, \frac{\log \left(x + y\right) \cdot \log \left(\sqrt{x + y}\right) + \mathsf{fma}\left(\log \left(\sqrt{x + y}\right), \log \left(x + y\right), -{\left(\log z\right)}^{2}\right)}{\log \left(x + y\right) - \log z} - t\right)
double f(double x, double y, double z, double t, double a) {
        double r57106 = x;
        double r57107 = y;
        double r57108 = r57106 + r57107;
        double r57109 = log(r57108);
        double r57110 = z;
        double r57111 = log(r57110);
        double r57112 = r57109 + r57111;
        double r57113 = t;
        double r57114 = r57112 - r57113;
        double r57115 = a;
        double r57116 = 0.5;
        double r57117 = r57115 - r57116;
        double r57118 = log(r57113);
        double r57119 = r57117 * r57118;
        double r57120 = r57114 + r57119;
        return r57120;
}
↓
double f(double x, double y, double z, double t, double a) {
        double r57121 = t;
        double r57122 = log(r57121);
        double r57123 = a;
        double r57124 = 0.5;
        double r57125 = r57123 - r57124;
        double r57126 = x;
        double r57127 = y;
        double r57128 = r57126 + r57127;
        double r57129 = log(r57128);
        double r57130 = sqrt(r57128);
        double r57131 = log(r57130);
        double r57132 = r57129 * r57131;
        double r57133 = z;
        double r57134 = log(r57133);
        double r57135 = 2.0;
        double r57136 = pow(r57134, r57135);
        double r57137 = -r57136;
        double r57138 = fma(r57131, r57129, r57137);
        double r57139 = r57132 + r57138;
        double r57140 = r57129 - r57134;
        double r57141 = r57139 / r57140;
        double r57142 = r57141 - r57121;
        double r57143 = fma(r57122, r57125, r57142);
        return r57143;
}

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 flip-+0.3

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

  6. Applied add-sqr-sqrt0.3

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

  7. Applied log-prod0.3

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

  8. Applied distribute-lft-in0.3

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

  9. Applied associate--l+0.3

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

  10. Simplified0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2020062 +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))))