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

Average Error: 9.6 → 0.4

Time: 22.8s

Precision: 64

Math

\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

↓

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

C

double f(double x, double y, double z, double t) {
        double r317175 = x;
        double r317176 = y;
        double r317177 = log(r317176);
        double r317178 = r317175 * r317177;
        double r317179 = z;
        double r317180 = 1.0;
        double r317181 = r317180 - r317176;
        double r317182 = log(r317181);
        double r317183 = r317179 * r317182;
        double r317184 = r317178 + r317183;
        double r317185 = t;
        double r317186 = r317184 - r317185;
        return r317186;
}

↓

double f(double x, double y, double z, double t) {
        double r317187 = x;
        double r317188 = 1.0;
        double r317189 = y;
        double r317190 = 0.6666666666666666;
        double r317191 = pow(r317189, r317190);
        double r317192 = r317188 / r317191;
        double r317193 = -0.3333333333333333;
        double r317194 = pow(r317192, r317193);
        double r317195 = cbrt(r317189);
        double r317196 = r317188 / r317195;
        double r317197 = pow(r317196, r317193);
        double r317198 = r317194 * r317197;
        double r317199 = r317198 * r317195;
        double r317200 = log(r317199);
        double r317201 = r317187 * r317200;
        double r317202 = log(r317195);
        double r317203 = r317202 * r317187;
        double r317204 = z;
        double r317205 = 1.0;
        double r317206 = log(r317205);
        double r317207 = r317205 * r317189;
        double r317208 = r317206 - r317207;
        double r317209 = r317204 * r317208;
        double r317210 = 0.5;
        double r317211 = 2.0;
        double r317212 = pow(r317189, r317211);
        double r317213 = r317204 * r317212;
        double r317214 = pow(r317205, r317211);
        double r317215 = r317213 / r317214;
        double r317216 = r317210 * r317215;
        double r317217 = r317209 - r317216;
        double r317218 = r317203 + r317217;
        double r317219 = r317201 + r317218;
        double r317220 = t;
        double r317221 = r317219 - r317220;
        return r317221;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	9.6
Target	0.3
Herbie	0.4

\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

1. Initial program 9.6

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

   \[\leadsto \left(x \cdot \log y + \color{blue}{\left(z \cdot \left(\log 1 - 1 \cdot y\right) - \frac{1}{2} \cdot \frac{z \cdot {y}^{2}}{{1}^{2}}\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.4

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

6. Applied log-prod 0.4

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

7. Applied distribute-lft-in 0.4

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

8. Applied associate-+l+ 0.4

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

9. Simplified 0.4

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

10. Taylor expanded around inf 0.4

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

12. Applied add-cube-cbrt 0.4

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

13. Applied add-cube-cbrt 0.4

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

14. Applied times-frac 0.4

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

15. Applied unpow-prod-down 0.4

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

16. Simplified 0.4

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

17. Simplified 0.4

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

18. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))