Average Error: 1.3 → 0.3
Time: 18.5s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]
\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(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)
double f(double x, double y, double z, double t) {
        double r14071120 = 1.0;
        double r14071121 = 3.0;
        double r14071122 = r14071120 / r14071121;
        double r14071123 = x;
        double r14071124 = y;
        double r14071125 = 27.0;
        double r14071126 = r14071124 * r14071125;
        double r14071127 = r14071123 / r14071126;
        double r14071128 = r14071121 * r14071127;
        double r14071129 = z;
        double r14071130 = 2.0;
        double r14071131 = r14071129 * r14071130;
        double r14071132 = r14071128 / r14071131;
        double r14071133 = t;
        double r14071134 = sqrt(r14071133);
        double r14071135 = r14071132 * r14071134;
        double r14071136 = acos(r14071135);
        double r14071137 = r14071122 * r14071136;
        return r14071137;
}
↓
double f(double x, double y, double z, double t) {
        double r14071138 = 1.0;
        double r14071139 = 3.0;
        double r14071140 = cbrt(r14071139);
        double r14071141 = r14071140 * r14071140;
        double r14071142 = r14071138 / r14071141;
        double r14071143 = 1.0;
        double r14071144 = x;
        double r14071145 = z;
        double r14071146 = y;
        double r14071147 = r14071145 * r14071146;
        double r14071148 = r14071144 / r14071147;
        double r14071149 = t;
        double r14071150 = sqrt(r14071149);
        double r14071151 = r14071148 * r14071150;
        double r14071152 = 0.05555555555555555;
        double r14071153 = r14071151 * r14071152;
        double r14071154 = acos(r14071153);
        double r14071155 = r14071154 / r14071140;
        double r14071156 = r14071143 * r14071155;
        double r14071157 = r14071142 * r14071156;
        return r14071157;
}

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

Original1.3
Target1.2
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.3

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

  3. Applied add-cube-cbrt1.3

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

  4. Applied *-un-lft-identity1.3

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 +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)))))