Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 37.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r262517 = x;
        double r262518 = y;
        double r262519 = r262517 + r262518;
        double r262520 = log(r262519);
        double r262521 = z;
        double r262522 = log(r262521);
        double r262523 = r262520 + r262522;
        double r262524 = t;
        double r262525 = r262523 - r262524;
        double r262526 = a;
        double r262527 = 0.5;
        double r262528 = r262526 - r262527;
        double r262529 = log(r262524);
        double r262530 = r262528 * r262529;
        double r262531 = r262525 + r262530;
        return r262531;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r262532 = x;
        double r262533 = y;
        double r262534 = r262532 + r262533;
        double r262535 = log(r262534);
        double r262536 = z;
        double r262537 = log(r262536);
        double r262538 = r262535 + r262537;
        double r262539 = t;
        double r262540 = r262538 - r262539;
        double r262541 = a;
        double r262542 = 0.5;
        double r262543 = r262541 - r262542;
        double r262544 = sqrt(r262539);
        double r262545 = log(r262544);
        double r262546 = r262543 * r262545;
        double r262547 = 2.0;
        double r262548 = cbrt(r262544);
        double r262549 = log(r262548);
        double r262550 = r262547 * r262549;
        double r262551 = r262543 * r262550;
        double r262552 = r262543 * r262549;
        double r262553 = r262551 + r262552;
        double r262554 = r262546 + r262553;
        double r262555 = r262540 + r262554;
        return r262555;
}

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-sqr-sqrt 0.3

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

4. Applied log-prod 0.3

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

5. Applied distribute-lft-in 0.3

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

7. Applied add-cube-cbrt 0.3

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

8. Applied log-prod 0.3

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

9. Applied distribute-lft-in 0.3

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

10. Simplified 0.3

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

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019306 
(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))))