Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 1.7 → 1.1

Time: 18.8s

Precision: 64

Math

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

↓

\[\left(\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right) \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r114218 = x;
        double r114219 = y;
        double r114220 = z;
        double r114221 = log(r114220);
        double r114222 = r114219 * r114221;
        double r114223 = t;
        double r114224 = 1.0;
        double r114225 = r114223 - r114224;
        double r114226 = a;
        double r114227 = log(r114226);
        double r114228 = r114225 * r114227;
        double r114229 = r114222 + r114228;
        double r114230 = b;
        double r114231 = r114229 - r114230;
        double r114232 = exp(r114231);
        double r114233 = r114218 * r114232;
        double r114234 = r114233 / r114219;
        return r114234;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r114235 = 1.0;
        double r114236 = a;
        double r114237 = r114235 / r114236;
        double r114238 = sqrt(r114237);
        double r114239 = 1.0;
        double r114240 = pow(r114238, r114239);
        double r114241 = y;
        double r114242 = z;
        double r114243 = r114235 / r114242;
        double r114244 = log(r114243);
        double r114245 = log(r114237);
        double r114246 = t;
        double r114247 = b;
        double r114248 = fma(r114245, r114246, r114247);
        double r114249 = fma(r114241, r114244, r114248);
        double r114250 = exp(r114249);
        double r114251 = sqrt(r114250);
        double r114252 = r114240 / r114251;
        double r114253 = x;
        double r114254 = r114252 * r114253;
        double r114255 = r114252 / r114241;
        double r114256 = r114254 * r114255;
        return r114256;
}

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

Derivation

1. Initial program 1.7

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

2. Taylor expanded around inf 1.7

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

3. Simplified 1.1

   \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]
4. Using strategy rm

5. Applied *-un-lft-identity 1.1

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

6. Applied times-frac 1.6

   \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}}\]

7. Simplified 1.6

   \[\leadsto \color{blue}{x} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\]
8. Using strategy rm

9. Applied *-un-lft-identity 1.6

   \[\leadsto x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{\color{blue}{1 \cdot y}}\]

10. Applied add-sqr-sqrt 1.6

    \[\leadsto x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{1 \cdot y}\]

11. Applied add-sqr-sqrt 1.6

    \[\leadsto x \cdot \frac{\frac{{\color{blue}{\left(\sqrt{\frac{1}{a}} \cdot \sqrt{\frac{1}{a}}\right)}}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{1 \cdot y}\]

12. Applied unpow-prod-down 1.6

    \[\leadsto x \cdot \frac{\frac{\color{blue}{{\left(\sqrt{\frac{1}{a}}\right)}^{1} \cdot {\left(\sqrt{\frac{1}{a}}\right)}^{1}}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{1 \cdot y}\]

13. Applied times-frac 1.6

    \[\leadsto x \cdot \frac{\color{blue}{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot \frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}}{1 \cdot y}\]

14. Applied times-frac 1.6

    \[\leadsto x \cdot \color{blue}{\left(\frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{1} \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\right)}\]

15. Applied associate-*r* 1.1

    \[\leadsto \color{blue}{\left(x \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{1}\right) \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}}\]

16. Simplified 1.1

    \[\leadsto \color{blue}{\left(\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right)} \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

17. Final simplification 1.1

    \[\leadsto \left(\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}} \cdot x\right) \cdot \frac{\frac{{\left(\sqrt{\frac{1}{a}}\right)}^{1}}{\sqrt{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}}{y}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))