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

Average Error: 0.1 → 0.1

Time: 13.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77385 = x;
        double r77386 = y;
        double r77387 = log(r77386);
        double r77388 = r77385 * r77387;
        double r77389 = z;
        double r77390 = r77388 + r77389;
        double r77391 = t;
        double r77392 = r77390 + r77391;
        double r77393 = a;
        double r77394 = r77392 + r77393;
        double r77395 = b;
        double r77396 = 0.5;
        double r77397 = r77395 - r77396;
        double r77398 = c;
        double r77399 = log(r77398);
        double r77400 = r77397 * r77399;
        double r77401 = r77394 + r77400;
        double r77402 = i;
        double r77403 = r77386 * r77402;
        double r77404 = r77401 + r77403;
        return r77404;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r77405 = x;
        double r77406 = 2.0;
        double r77407 = y;
        double r77408 = cbrt(r77407);
        double r77409 = log(r77408);
        double r77410 = r77406 * r77409;
        double r77411 = 0.3333333333333333;
        double r77412 = pow(r77407, r77411);
        double r77413 = log(r77412);
        double r77414 = r77405 * r77413;
        double r77415 = 1.0;
        double r77416 = pow(r77414, r77415);
        double r77417 = fma(r77405, r77410, r77416);
        double r77418 = z;
        double r77419 = r77417 + r77418;
        double r77420 = t;
        double r77421 = r77419 + r77420;
        double r77422 = a;
        double r77423 = r77421 + r77422;
        double r77424 = b;
        double r77425 = 0.5;
        double r77426 = r77424 - r77425;
        double r77427 = c;
        double r77428 = log(r77427);
        double r77429 = r77426 * r77428;
        double r77430 = r77423 + r77429;
        double r77431 = i;
        double r77432 = r77407 * r77431;
        double r77433 = r77430 + r77432;
        return r77433;
}

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

Bits error versus c

Bits error versus i

Derivation

1. Initial program 0.1

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

3. Applied add-cube-cbrt 0.1

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

4. Applied log-prod 0.1

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

5. Applied distribute-lft-in 0.1

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

6. Simplified 0.1

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

8. Applied fma-def 0.1

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

10. Applied pow1 0.1

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

11. Applied pow1 0.1

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

12. Applied pow-prod-down 0.1

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

13. Simplified 0.1

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

14. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  :precision binary64
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))