Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 36.9s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r16404244 = x;
        double r16404245 = y;
        double r16404246 = r16404244 + r16404245;
        double r16404247 = log(r16404246);
        double r16404248 = z;
        double r16404249 = log(r16404248);
        double r16404250 = r16404247 + r16404249;
        double r16404251 = t;
        double r16404252 = r16404250 - r16404251;
        double r16404253 = a;
        double r16404254 = 0.5;
        double r16404255 = r16404253 - r16404254;
        double r16404256 = log(r16404251);
        double r16404257 = r16404255 * r16404256;
        double r16404258 = r16404252 + r16404257;
        return r16404258;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r16404259 = a;
        double r16404260 = 0.5;
        double r16404261 = r16404259 - r16404260;
        double r16404262 = t;
        double r16404263 = cbrt(r16404262);
        double r16404264 = log(r16404263);
        double r16404265 = r16404261 * r16404264;
        double r16404266 = r16404263 * r16404263;
        double r16404267 = log(r16404266);
        double r16404268 = z;
        double r16404269 = log(r16404268);
        double r16404270 = x;
        double r16404271 = y;
        double r16404272 = r16404270 + r16404271;
        double r16404273 = cbrt(r16404272);
        double r16404274 = r16404273 * r16404273;
        double r16404275 = log(r16404274);
        double r16404276 = r16404269 + r16404275;
        double r16404277 = log(r16404273);
        double r16404278 = r16404276 + r16404277;
        double r16404279 = fma(r16404261, r16404267, r16404278);
        double r16404280 = r16404279 - r16404262;
        double r16404281 = r16404265 + r16404280;
        return r16404281;
}

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. Using strategy rm

3. Applied add-cube-cbrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-rgt-in 0.3

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

6. Applied associate-+r+ 0.3

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

7. Simplified 0.3

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

9. Applied add-cube-cbrt 0.3

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

10. Applied log-prod 0.3

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

11. Applied associate-+r+ 0.3

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

12. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"

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

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