Average Error: 1.2 → 0.2
Time: 25.7s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)\]
\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r27099206 = 1.0;
        double r27099207 = 3.0;
        double r27099208 = r27099206 / r27099207;
        double r27099209 = x;
        double r27099210 = y;
        double r27099211 = 27.0;
        double r27099212 = r27099210 * r27099211;
        double r27099213 = r27099209 / r27099212;
        double r27099214 = r27099207 * r27099213;
        double r27099215 = z;
        double r27099216 = 2.0;
        double r27099217 = r27099215 * r27099216;
        double r27099218 = r27099214 / r27099217;
        double r27099219 = t;
        double r27099220 = sqrt(r27099219);
        double r27099221 = r27099218 * r27099220;
        double r27099222 = acos(r27099221);
        double r27099223 = r27099208 * r27099222;
        return r27099223;
}
↓
double f(double x, double y, double z, double t) {
        double r27099224 = 1.0;
        double r27099225 = 3.0;
        double r27099226 = cbrt(r27099225);
        double r27099227 = r27099226 * r27099226;
        double r27099228 = r27099224 / r27099227;
        double r27099229 = 1.0;
        double r27099230 = r27099229 / r27099226;
        double r27099231 = x;
        double r27099232 = z;
        double r27099233 = y;
        double r27099234 = r27099232 * r27099233;
        double r27099235 = r27099231 / r27099234;
        double r27099236 = 0.05555555555555555;
        double r27099237 = r27099235 * r27099236;
        double r27099238 = t;
        double r27099239 = sqrt(r27099238);
        double r27099240 = r27099237 * r27099239;
        double r27099241 = acos(r27099240);
        double r27099242 = r27099230 * r27099241;
        double r27099243 = r27099228 * r27099242;
        return r27099243;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original1.2
Target1.1
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.2

    \[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1.0}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  4. Applied *-un-lft-identity1.2

    \[\leadsto \frac{\color{blue}{1 \cdot 1.0}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{1.0}{\sqrt[3]{3.0}}\right)} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}\]

  7. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\color{blue}{\left(0.05555555555555555 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2019158 
(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)))))