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

Average Error: 1.3 → 0.3

Time: 13.0s

Precision: 64

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

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r800147 = 1.0;
        double r800148 = 3.0;
        double r800149 = r800147 / r800148;
        double r800150 = x;
        double r800151 = y;
        double r800152 = 27.0;
        double r800153 = r800151 * r800152;
        double r800154 = r800150 / r800153;
        double r800155 = r800148 * r800154;
        double r800156 = z;
        double r800157 = 2.0;
        double r800158 = r800156 * r800157;
        double r800159 = r800155 / r800158;
        double r800160 = t;
        double r800161 = sqrt(r800160);
        double r800162 = r800159 * r800161;
        double r800163 = acos(r800162);
        double r800164 = r800149 * r800163;
        return r800164;
}

↓

double f(double x, double y, double z, double t) {
        double r800165 = 1.0;
        double r800166 = 3.0;
        double r800167 = cbrt(r800166);
        double r800168 = r800167 * r800167;
        double r800169 = r800165 / r800168;
        double r800170 = 1.0;
        double r800171 = r800170 / r800167;
        double r800172 = x;
        double r800173 = y;
        double r800174 = 27.0;
        double r800175 = r800173 * r800174;
        double r800176 = r800172 / r800175;
        double r800177 = r800166 * r800176;
        double r800178 = z;
        double r800179 = 2.0;
        double r800180 = r800178 * r800179;
        double r800181 = r800177 / r800180;
        double r800182 = t;
        double r800183 = sqrt(r800182);
        double r800184 = r800181 * r800183;
        double r800185 = acos(r800184);
        double r800186 = r800171 * r800185;
        double r800187 = r800169 * r800186;
        return r800187;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.3
Target	1.3
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 0.3

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

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. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(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)))))