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

Average Error: 0.1 → 0.1

Time: 16.0s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r447481 = x;
        double r447482 = y;
        double r447483 = r447481 + r447482;
        double r447484 = z;
        double r447485 = r447483 + r447484;
        double r447486 = t;
        double r447487 = log(r447486);
        double r447488 = r447484 * r447487;
        double r447489 = r447485 - r447488;
        double r447490 = a;
        double r447491 = 0.5;
        double r447492 = r447490 - r447491;
        double r447493 = b;
        double r447494 = r447492 * r447493;
        double r447495 = r447489 + r447494;
        return r447495;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r447496 = z;
        double r447497 = y;
        double r447498 = r447496 + r447497;
        double r447499 = x;
        double r447500 = t;
        double r447501 = sqrt(r447500);
        double r447502 = log(r447501);
        double r447503 = cbrt(r447500);
        double r447504 = fabs(r447503);
        double r447505 = log(r447504);
        double r447506 = r447502 + r447505;
        double r447507 = r447496 * r447506;
        double r447508 = r447499 - r447507;
        double r447509 = r447498 + r447508;
        double r447510 = 0.6666666666666666;
        double r447511 = pow(r447500, r447510);
        double r447512 = cbrt(r447511);
        double r447513 = cbrt(r447503);
        double r447514 = r447512 * r447513;
        double r447515 = sqrt(r447514);
        double r447516 = log(r447515);
        double r447517 = r447516 * r447496;
        double r447518 = r447509 - r447517;
        double r447519 = a;
        double r447520 = 0.5;
        double r447521 = r447519 - r447520;
        double r447522 = b;
        double r447523 = r447521 * r447522;
        double r447524 = r447518 + r447523;
        return r447524;
}

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	0.4
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-sqr-sqrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Applied associate--r+ 0.1

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

8. Applied add-cube-cbrt 0.1

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

9. Applied sqrt-prod 0.1

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

10. Applied log-prod 0.1

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

11. Applied distribute-rgt-in 0.1

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

12. Applied associate--r+ 0.1

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

13. Simplified 0.1

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

15. Applied add-cube-cbrt 0.1

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

16. Applied cbrt-prod 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020047 
(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)))