Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 32.8s

Precision: 64

Math

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

↓

\[\log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]

C

double f(double x, double y, double z, double t, double a) {
        double r276398 = x;
        double r276399 = y;
        double r276400 = r276398 + r276399;
        double r276401 = log(r276400);
        double r276402 = z;
        double r276403 = log(r276402);
        double r276404 = r276401 + r276403;
        double r276405 = t;
        double r276406 = r276404 - r276405;
        double r276407 = a;
        double r276408 = 0.5;
        double r276409 = r276407 - r276408;
        double r276410 = log(r276405);
        double r276411 = r276409 * r276410;
        double r276412 = r276406 + r276411;
        return r276412;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r276413 = x;
        double r276414 = y;
        double r276415 = r276413 + r276414;
        double r276416 = log(r276415);
        double r276417 = t;
        double r276418 = z;
        double r276419 = log(r276418);
        double r276420 = r276417 - r276419;
        double r276421 = a;
        double r276422 = 0.5;
        double r276423 = r276421 - r276422;
        double r276424 = 2.0;
        double r276425 = cbrt(r276417);
        double r276426 = log(r276425);
        double r276427 = r276424 * r276426;
        double r276428 = r276423 * r276427;
        double r276429 = 0.3333333333333333;
        double r276430 = log(r276417);
        double r276431 = r276429 * r276430;
        double r276432 = r276423 * r276431;
        double r276433 = r276428 + r276432;
        double r276434 = r276420 - r276433;
        double r276435 = r276416 - r276434;
        return r276435;
}

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

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

4. Applied add-cube-cbrt 0.3

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

5. Applied log-prod 0.3

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

6. Applied distribute-lft-in 0.3

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

7. Simplified 0.3

   \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right)} + \left(a - 0.5\right) \cdot \log \left(\sqrt[3]{t}\right)\right)\right)\]

8. Taylor expanded around inf 0.3

   \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right) - 0.5 \cdot \log \left({\left(\frac{1}{t}\right)}^{\frac{-1}{3}}\right)\right)}\right)\right)\]

9. Simplified 0.3

   \[\leadsto \log \left(y + x\right) - \left(\left(t - \log z\right) - \left(\left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) \cdot \left(a - 0.5\right) + \color{blue}{\left(a - 0.5\right) \cdot \left(-\frac{-1}{3} \cdot \log t\right)}\right)\right)\]

10. Final simplification 0.3

    \[\leadsto \log \left(x + y\right) - \left(\left(t - \log z\right) - \left(\left(a - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{t}\right)\right) + \left(a - 0.5\right) \cdot \left(\frac{1}{3} \cdot \log t\right)\right)\right)\]

Reproduce

herbie shell --seed 2019194 
(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))))