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

Average Error: 1.3 → 0.3

Time: 23.5s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r578172 = 1.0;
        double r578173 = 3.0;
        double r578174 = r578172 / r578173;
        double r578175 = x;
        double r578176 = y;
        double r578177 = 27.0;
        double r578178 = r578176 * r578177;
        double r578179 = r578175 / r578178;
        double r578180 = r578173 * r578179;
        double r578181 = z;
        double r578182 = 2.0;
        double r578183 = r578181 * r578182;
        double r578184 = r578180 / r578183;
        double r578185 = t;
        double r578186 = sqrt(r578185);
        double r578187 = r578184 * r578186;
        double r578188 = acos(r578187);
        double r578189 = r578174 * r578188;
        return r578189;
}

↓

double f(double x, double y, double z, double t) {
        double r578190 = 1.0;
        double r578191 = 3.0;
        double r578192 = cbrt(r578191);
        double r578193 = r578192 * r578192;
        double r578194 = r578190 / r578193;
        double r578195 = 1.0;
        double r578196 = r578195 / r578192;
        double r578197 = t;
        double r578198 = sqrt(r578197);
        double r578199 = z;
        double r578200 = 2.0;
        double r578201 = r578199 * r578200;
        double r578202 = y;
        double r578203 = x;
        double r578204 = r578191 * r578203;
        double r578205 = r578190 / r578204;
        double r578206 = r578202 * r578205;
        double r578207 = 27.0;
        double r578208 = r578206 * r578207;
        double r578209 = r578201 * r578208;
        double r578210 = r578198 / r578209;
        double r578211 = acos(r578210);
        double r578212 = r578196 * r578211;
        double r578213 = r578194 * r578212;
        return r578213;
}

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* 0.4

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

7. Simplified 0.4

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

9. Applied div-inv 0.4

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

10. Simplified 0.3

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

11. Final simplification 0.3

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"

  :herbie-target
  (/ (acos (* (/ (/ x 27.0) (* y z)) (/ (sqrt t) (/ 2.0 3.0)))) 3.0)

  (* (/ 1.0 3.0) (acos (* (/ (* 3.0 (/ x (* y 27.0))) (* z 2.0)) (sqrt t)))))