Average Error: 1.4 → 0.2
Time: 25.8s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t}}{\frac{z \cdot 2}{3 \cdot x}}}{27 \cdot y}\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t}}{\frac{z \cdot 2}{3 \cdot x}}}{27 \cdot y}\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r602801 = 1.0;
        double r602802 = 3.0;
        double r602803 = r602801 / r602802;
        double r602804 = x;
        double r602805 = y;
        double r602806 = 27.0;
        double r602807 = r602805 * r602806;
        double r602808 = r602804 / r602807;
        double r602809 = r602802 * r602808;
        double r602810 = z;
        double r602811 = 2.0;
        double r602812 = r602810 * r602811;
        double r602813 = r602809 / r602812;
        double r602814 = t;
        double r602815 = sqrt(r602814);
        double r602816 = r602813 * r602815;
        double r602817 = acos(r602816);
        double r602818 = r602803 * r602817;
        return r602818;
}
↓
double f(double x, double y, double z, double t) {
        double r602819 = 1.0;
        double r602820 = 3.0;
        double r602821 = cbrt(r602820);
        double r602822 = r602819 / r602821;
        double r602823 = t;
        double r602824 = sqrt(r602823);
        double r602825 = z;
        double r602826 = 2.0;
        double r602827 = r602825 * r602826;
        double r602828 = x;
        double r602829 = r602820 * r602828;
        double r602830 = r602827 / r602829;
        double r602831 = r602824 / r602830;
        double r602832 = 27.0;
        double r602833 = y;
        double r602834 = r602832 * r602833;
        double r602835 = r602831 / r602834;
        double r602836 = acos(r602835);
        double r602837 = r602822 * r602836;
        double r602838 = log1p(r602837);
        double r602839 = expm1(r602838);
        double r602840 = 1.0;
        double r602841 = r602821 * r602821;
        double r602842 = r602840 / r602841;
        double r602843 = r602839 * r602842;
        return r602843;
}

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

Original1.4
Target1.3
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

Derivation

  1. Initial program 1.4

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

  3. Applied add-cube-cbrt1.4

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

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

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

  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. Simplified0.4

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

  9. Applied expm1-log1p-u0.4

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\sqrt{t}}{\frac{z \cdot 2}{\frac{3 \cdot x}{y \cdot 27}}}\right)\right)\right)}\]

  10. Simplified0.2

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t}}{\frac{z \cdot 2}{3 \cdot x}}}{27 \cdot y}\right)\right)}\right)\]

  11. Final simplification0.2

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{\sqrt{t}}{\frac{z \cdot 2}{3 \cdot x}}}{27 \cdot y}\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]

Reproduce

herbie shell --seed 2019194 +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)))))