Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.3 → 0.3

Time: 10.7s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

↓

\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\frac{{1}^{3}}{\frac{3}{{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)}^{3}}}}\]

C

double f(double x, double y, double z, double t) {
        double r894153 = 1.0;
        double r894154 = 3.0;
        double r894155 = r894153 / r894154;
        double r894156 = x;
        double r894157 = y;
        double r894158 = 27.0;
        double r894159 = r894157 * r894158;
        double r894160 = r894156 / r894159;
        double r894161 = r894154 * r894160;
        double r894162 = z;
        double r894163 = 2.0;
        double r894164 = r894162 * r894163;
        double r894165 = r894161 / r894164;
        double r894166 = t;
        double r894167 = sqrt(r894166);
        double r894168 = r894165 * r894167;
        double r894169 = acos(r894168);
        double r894170 = r894155 * r894169;
        return r894170;
}

↓

double f(double x, double y, double z, double t) {
        double r894171 = 1.0;
        double r894172 = 3.0;
        double r894173 = cbrt(r894172);
        double r894174 = r894173 * r894173;
        double r894175 = r894171 / r894174;
        double r894176 = 1.0;
        double r894177 = 3.0;
        double r894178 = pow(r894176, r894177);
        double r894179 = 0.05555555555555555;
        double r894180 = x;
        double r894181 = z;
        double r894182 = y;
        double r894183 = r894181 * r894182;
        double r894184 = r894180 / r894183;
        double r894185 = r894179 * r894184;
        double r894186 = t;
        double r894187 = sqrt(r894186);
        double r894188 = r894185 * r894187;
        double r894189 = acos(r894188);
        double r894190 = pow(r894189, r894177);
        double r894191 = r894172 / r894190;
        double r894192 = r894178 / r894191;
        double r894193 = cbrt(r894192);
        double r894194 = r894175 * r894193;
        return r894194;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.3
Target	1.2
Herbie	0.3

\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

1. Initial program 1.3

   \[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.3

   \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

4. Applied *-un-lft-identity 1.3

   \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

5. Applied times-frac 0.4

   \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]

6. Applied associate-*l* 0.3

   \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)}\]

7. Taylor expanded around 0 0.3

   \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]
8. Using strategy rm

9. Applied add-cbrt-cube 1.3

   \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \color{blue}{\sqrt[3]{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}}\right)\]

10. Applied add-cbrt-cube 1.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\color{blue}{\sqrt[3]{\left(1 \cdot 1\right) \cdot 1}}}{\sqrt[3]{3}} \cdot \sqrt[3]{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}\right)\]

11. Applied cbrt-undiv 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{3}}} \cdot \sqrt[3]{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}\right)\]

12. Applied cbrt-unprod 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\sqrt[3]{\frac{\left(1 \cdot 1\right) \cdot 1}{3} \cdot \left(\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right) \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)}}\]

13. Simplified 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\color{blue}{\frac{{1}^{3}}{\frac{3}{{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)}^{3}}}}}\]

14. Final simplification 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\frac{{1}^{3}}{\frac{3}{{\left(\cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)}^{3}}}}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))