Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 35.1s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r15743580 = x;
        double r15743581 = y;
        double r15743582 = r15743580 + r15743581;
        double r15743583 = log(r15743582);
        double r15743584 = z;
        double r15743585 = log(r15743584);
        double r15743586 = r15743583 + r15743585;
        double r15743587 = t;
        double r15743588 = r15743586 - r15743587;
        double r15743589 = a;
        double r15743590 = 0.5;
        double r15743591 = r15743589 - r15743590;
        double r15743592 = log(r15743587);
        double r15743593 = r15743591 * r15743592;
        double r15743594 = r15743588 + r15743593;
        return r15743594;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r15743595 = a;
        double r15743596 = 0.5;
        double r15743597 = r15743595 - r15743596;
        double r15743598 = t;
        double r15743599 = 0.3333333333333333;
        double r15743600 = pow(r15743598, r15743599);
        double r15743601 = log(r15743600);
        double r15743602 = r15743597 * r15743601;
        double r15743603 = cbrt(r15743598);
        double r15743604 = log(r15743603);
        double r15743605 = r15743597 * r15743604;
        double r15743606 = r15743605 + r15743605;
        double r15743607 = r15743602 + r15743606;
        double r15743608 = y;
        double r15743609 = x;
        double r15743610 = r15743608 + r15743609;
        double r15743611 = log(r15743610);
        double r15743612 = z;
        double r15743613 = log(r15743612);
        double r15743614 = r15743611 + r15743613;
        double r15743615 = r15743614 - r15743598;
        double r15743616 = r15743607 + r15743615;
        return r15743616;
}

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-lft-in 0.3

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

6. Simplified 0.3

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

8. Applied pow1/3 0.3

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

9. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019163 +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))))