Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 15.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r399139 = x;
        double r399140 = y;
        double r399141 = r399139 + r399140;
        double r399142 = log(r399141);
        double r399143 = z;
        double r399144 = log(r399143);
        double r399145 = r399142 + r399144;
        double r399146 = t;
        double r399147 = r399145 - r399146;
        double r399148 = a;
        double r399149 = 0.5;
        double r399150 = r399148 - r399149;
        double r399151 = log(r399146);
        double r399152 = r399150 * r399151;
        double r399153 = r399147 + r399152;
        return r399153;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r399154 = a;
        double r399155 = 0.5;
        double r399156 = r399154 - r399155;
        double r399157 = t;
        double r399158 = log(r399157);
        double r399159 = x;
        double r399160 = y;
        double r399161 = r399159 + r399160;
        double r399162 = log(r399161);
        double r399163 = 3.0;
        double r399164 = pow(r399162, r399163);
        double r399165 = z;
        double r399166 = log(r399165);
        double r399167 = pow(r399166, r399163);
        double r399168 = r399164 + r399167;
        double r399169 = r399162 - r399166;
        double r399170 = pow(r399169, r399163);
        double r399171 = cbrt(r399170);
        double r399172 = r399162 * r399171;
        double r399173 = fma(r399166, r399166, r399172);
        double r399174 = r399168 / r399173;
        double r399175 = r399174 - r399157;
        double r399176 = fma(r399156, r399158, r399175);
        return r399176;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	0.3
Target	0.3
Herbie	0.3

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

Derivation

1. Initial program 0.3

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

2. Simplified 0.3

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

4. Applied pow1 0.3

   \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(a - 0.5, \log t, \left(\log \left(x + y\right) + \log z\right) - t\right)\right)}^{1}}\]
5. Using strategy rm

6. Applied flip3-+ 0.3

   \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \color{blue}{\frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\log \left(x + y\right) \cdot \log \left(x + y\right) + \left(\log z \cdot \log z - \log \left(x + y\right) \cdot \log z\right)}} - t\right)\right)}^{1}\]

7. Simplified 0.3

   \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right)}} - t\right)\right)}^{1}\]
8. Using strategy rm

9. Applied add-cbrt-cube 0.3

   \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\log \left(x + y\right) - \log z\right) \cdot \left(\log \left(x + y\right) - \log z\right)\right) \cdot \left(\log \left(x + y\right) - \log z\right)}}\right)} - t\right)\right)}^{1}\]

10. Simplified 0.3

    \[\leadsto {\left(\mathsf{fma}\left(a - 0.5, \log t, \frac{{\left(\log \left(x + y\right)\right)}^{3} + {\left(\log z\right)}^{3}}{\mathsf{fma}\left(\log z, \log z, \log \left(x + y\right) \cdot \sqrt[3]{\color{blue}{{\left(\log \left(x + y\right) - \log z\right)}^{3}}}\right)} - t\right)\right)}^{1}\]

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))