Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.1

Time: 31.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r182658 = x;
        double r182659 = y;
        double r182660 = r182658 + r182659;
        double r182661 = z;
        double r182662 = r182660 + r182661;
        double r182663 = t;
        double r182664 = log(r182663);
        double r182665 = r182661 * r182664;
        double r182666 = r182662 - r182665;
        double r182667 = a;
        double r182668 = 0.5;
        double r182669 = r182667 - r182668;
        double r182670 = b;
        double r182671 = r182669 * r182670;
        double r182672 = r182666 + r182671;
        return r182672;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r182673 = z;
        double r182674 = y;
        double r182675 = r182673 + r182674;
        double r182676 = x;
        double r182677 = r182675 + r182676;
        double r182678 = 2.0;
        double r182679 = t;
        double r182680 = cbrt(r182679);
        double r182681 = log(r182680);
        double r182682 = r182678 * r182681;
        double r182683 = r182673 * r182682;
        double r182684 = r182677 - r182683;
        double r182685 = 0.6666666666666666;
        double r182686 = pow(r182679, r182685);
        double r182687 = cbrt(r182686);
        double r182688 = cbrt(r182680);
        double r182689 = r182687 * r182688;
        double r182690 = log(r182689);
        double r182691 = r182690 * r182673;
        double r182692 = r182684 - r182691;
        double r182693 = a;
        double r182694 = 0.5;
        double r182695 = r182693 - r182694;
        double r182696 = b;
        double r182697 = r182695 * r182696;
        double r182698 = r182692 + r182697;
        return r182698;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	0.1
Target	0.3
Herbie	0.1

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

Derivation

1. Initial program 0.1

   \[\left(\left(\left(x + y\right) + z\right) - z \cdot \log t\right) + \left(a - 0.5\right) \cdot b\]
2. Using strategy rm

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-rgt-in 0.1

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

6. Applied associate--r+ 0.1

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

7. Simplified 0.1

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

9. Applied add-cube-cbrt 0.1

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

10. Applied cbrt-prod 0.1

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

11. Simplified 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ (+ (+ x y) (/ (* (- 1 (pow (log t) 2)) z) (+ 1 (log t)))) (* (- a 0.5) b))

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