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

Average Error: 0.1 → 0.1

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b) {
        double r244464 = x;
        double r244465 = y;
        double r244466 = r244464 + r244465;
        double r244467 = z;
        double r244468 = r244466 + r244467;
        double r244469 = t;
        double r244470 = log(r244469);
        double r244471 = r244467 * r244470;
        double r244472 = r244468 - r244471;
        double r244473 = a;
        double r244474 = 0.5;
        double r244475 = r244473 - r244474;
        double r244476 = b;
        double r244477 = r244475 * r244476;
        double r244478 = r244472 + r244477;
        return r244478;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r244479 = x;
        double r244480 = z;
        double r244481 = y;
        double r244482 = r244480 + r244481;
        double r244483 = r244479 + r244482;
        double r244484 = 2.0;
        double r244485 = t;
        double r244486 = cbrt(r244485);
        double r244487 = log(r244486);
        double r244488 = r244484 * r244487;
        double r244489 = r244488 * r244480;
        double r244490 = r244483 - r244489;
        double r244491 = 0.3333333333333333;
        double r244492 = sqrt(r244491);
        double r244493 = log(r244485);
        double r244494 = r244493 * r244480;
        double r244495 = r244492 * r244494;
        double r244496 = r244492 * r244495;
        double r244497 = r244490 - r244496;
        double r244498 = a;
        double r244499 = 0.5;
        double r244500 = r244498 - r244499;
        double r244501 = b;
        double r244502 = r244500 * r244501;
        double r244503 = r244497 + r244502;
        return r244503;
}

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-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

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

9. Applied pow1/3 0.1

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

10. Applied log-pow 0.1

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

11. Applied associate-*l* 0.1

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

13. Applied add-sqr-sqrt 0.1

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

14. Applied associate-*l* 0.1

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

15. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019212 
(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)))