Average Error: 1.3 → 0.4
Time: 25.1s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r33076132 = 1.0;
        double r33076133 = 3.0;
        double r33076134 = r33076132 / r33076133;
        double r33076135 = x;
        double r33076136 = y;
        double r33076137 = 27.0;
        double r33076138 = r33076136 * r33076137;
        double r33076139 = r33076135 / r33076138;
        double r33076140 = r33076133 * r33076139;
        double r33076141 = z;
        double r33076142 = 2.0;
        double r33076143 = r33076141 * r33076142;
        double r33076144 = r33076140 / r33076143;
        double r33076145 = t;
        double r33076146 = sqrt(r33076145);
        double r33076147 = r33076144 * r33076146;
        double r33076148 = acos(r33076147);
        double r33076149 = r33076134 * r33076148;
        return r33076149;
}
↓
double f(double x, double y, double z, double t) {
        double r33076150 = 1.0;
        double r33076151 = 3.0;
        double r33076152 = cbrt(r33076151);
        double r33076153 = r33076150 / r33076152;
        double r33076154 = x;
        double r33076155 = 27.0;
        double r33076156 = y;
        double r33076157 = r33076155 * r33076156;
        double r33076158 = r33076154 / r33076157;
        double r33076159 = r33076158 * r33076151;
        double r33076160 = 2.0;
        double r33076161 = z;
        double r33076162 = r33076160 * r33076161;
        double r33076163 = r33076159 / r33076162;
        double r33076164 = t;
        double r33076165 = sqrt(r33076164);
        double r33076166 = r33076163 * r33076165;
        double r33076167 = acos(r33076166);
        double r33076168 = r33076153 * r33076167;
        double r33076169 = 1.0;
        double r33076170 = r33076152 * r33076152;
        double r33076171 = r33076169 / r33076170;
        double r33076172 = r33076168 * r33076171;
        return r33076172;
}

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
Herbie0.4
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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-cbrt1.3

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

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

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

  5. Applied times-frac0.4

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

  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. Final simplification0.4

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

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)))))