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

Average Error: 0.1 → 0.1

Time: 29.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r89201 = x;
        double r89202 = y;
        double r89203 = log(r89202);
        double r89204 = r89201 * r89203;
        double r89205 = z;
        double r89206 = r89204 + r89205;
        double r89207 = t;
        double r89208 = r89206 + r89207;
        double r89209 = a;
        double r89210 = r89208 + r89209;
        double r89211 = b;
        double r89212 = 0.5;
        double r89213 = r89211 - r89212;
        double r89214 = c;
        double r89215 = log(r89214);
        double r89216 = r89213 * r89215;
        double r89217 = r89210 + r89216;
        double r89218 = i;
        double r89219 = r89202 * r89218;
        double r89220 = r89217 + r89219;
        return r89220;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r89221 = c;
        double r89222 = log(r89221);
        double r89223 = b;
        double r89224 = 0.5;
        double r89225 = r89223 - r89224;
        double r89226 = r89222 * r89225;
        double r89227 = t;
        double r89228 = x;
        double r89229 = y;
        double r89230 = cbrt(r89229);
        double r89231 = r89230 * r89230;
        double r89232 = cbrt(r89231);
        double r89233 = log(r89232);
        double r89234 = 2.0;
        double r89235 = r89233 * r89234;
        double r89236 = r89228 * r89235;
        double r89237 = log(r89230);
        double r89238 = 1.0;
        double r89239 = r89238 / r89229;
        double r89240 = -0.3333333333333333;
        double r89241 = pow(r89239, r89240);
        double r89242 = cbrt(r89241);
        double r89243 = log(r89242);
        double r89244 = r89243 * r89234;
        double r89245 = r89237 + r89244;
        double r89246 = r89245 * r89228;
        double r89247 = r89236 + r89246;
        double r89248 = z;
        double r89249 = r89247 + r89248;
        double r89250 = r89227 + r89249;
        double r89251 = a;
        double r89252 = i;
        double r89253 = r89252 * r89229;
        double r89254 = r89251 + r89253;
        double r89255 = r89250 + r89254;
        double r89256 = r89226 + r89255;
        return r89256;
}

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. Simplified 0.1

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

4. Applied add-cube-cbrt 0.1

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

5. Applied log-prod 0.1

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

6. Applied distribute-lft-in 0.1

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

7. Simplified 0.1

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

9. Applied add-cube-cbrt 0.1

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

10. Applied cbrt-prod 0.1

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

11. Applied log-prod 0.1

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

12. Applied distribute-rgt-in 0.1

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

13. Applied distribute-lft-in 0.1

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

14. Applied associate-+l+ 0.1

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

15. Simplified 0.1

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

16. Taylor expanded around inf 0.1

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

17. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logBeta from math-functions-0.1.5.2, B"
  (+ (+ (+ (+ (+ (* x (log y)) z) t) a) (* (- b 0.5) (log c))) (* y i)))