Average Error: 1.3 → 0.3
Time: 26.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \left(\frac{\frac{\pi}{2}}{\sqrt[3]{3}} - \frac{\sin^{-1} \left(\frac{\frac{\frac{x \cdot 3}{27 \cdot y}}{z} \cdot \sqrt{t}}{2}\right)}{\sqrt[3]{3}}\right)\right)\]
\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(1 \cdot \left(\frac{\frac{\pi}{2}}{\sqrt[3]{3}} - \frac{\sin^{-1} \left(\frac{\frac{\frac{x \cdot 3}{27 \cdot y}}{z} \cdot \sqrt{t}}{2}\right)}{\sqrt[3]{3}}\right)\right)
double f(double x, double y, double z, double t) {
        double r36941121 = 1.0;
        double r36941122 = 3.0;
        double r36941123 = r36941121 / r36941122;
        double r36941124 = x;
        double r36941125 = y;
        double r36941126 = 27.0;
        double r36941127 = r36941125 * r36941126;
        double r36941128 = r36941124 / r36941127;
        double r36941129 = r36941122 * r36941128;
        double r36941130 = z;
        double r36941131 = 2.0;
        double r36941132 = r36941130 * r36941131;
        double r36941133 = r36941129 / r36941132;
        double r36941134 = t;
        double r36941135 = sqrt(r36941134);
        double r36941136 = r36941133 * r36941135;
        double r36941137 = acos(r36941136);
        double r36941138 = r36941123 * r36941137;
        return r36941138;
}
↓
double f(double x, double y, double z, double t) {
        double r36941139 = 1.0;
        double r36941140 = 3.0;
        double r36941141 = cbrt(r36941140);
        double r36941142 = r36941141 * r36941141;
        double r36941143 = r36941139 / r36941142;
        double r36941144 = 1.0;
        double r36941145 = atan2(1.0, 0.0);
        double r36941146 = 2.0;
        double r36941147 = r36941145 / r36941146;
        double r36941148 = r36941147 / r36941141;
        double r36941149 = x;
        double r36941150 = r36941149 * r36941140;
        double r36941151 = 27.0;
        double r36941152 = y;
        double r36941153 = r36941151 * r36941152;
        double r36941154 = r36941150 / r36941153;
        double r36941155 = z;
        double r36941156 = r36941154 / r36941155;
        double r36941157 = t;
        double r36941158 = sqrt(r36941157);
        double r36941159 = r36941156 * r36941158;
        double r36941160 = 2.0;
        double r36941161 = r36941159 / r36941160;
        double r36941162 = asin(r36941161);
        double r36941163 = r36941162 / r36941141;
        double r36941164 = r36941148 - r36941163;
        double r36941165 = r36941144 * r36941164;
        double r36941166 = r36941143 * r36941165;
        return r36941166;
}

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

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-frac0.3

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

  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. Using strategy rm

  8. Applied div-inv0.3

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

  9. Applied associate-*l*0.3

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

  10. Simplified0.3

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

  12. Applied acos-asin0.3

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

  13. Applied div-sub0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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)))))