Average Error: 1.4 → 0.2
Time: 10.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]
\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(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r1849 = 1.0;
        double r1850 = 3.0;
        double r1851 = r1849 / r1850;
        double r1852 = x;
        double r1853 = y;
        double r1854 = 27.0;
        double r1855 = r1853 * r1854;
        double r1856 = r1852 / r1855;
        double r1857 = r1850 * r1856;
        double r1858 = z;
        double r1859 = 2.0;
        double r1860 = r1858 * r1859;
        double r1861 = r1857 / r1860;
        double r1862 = t;
        double r1863 = sqrt(r1862);
        double r1864 = r1861 * r1863;
        double r1865 = acos(r1864);
        double r1866 = r1851 * r1865;
        return r1866;
}
↓
double f(double x, double y, double z, double t) {
        double r1867 = 1.0;
        double r1868 = 3.0;
        double r1869 = cbrt(r1868);
        double r1870 = r1869 * r1869;
        double r1871 = r1867 / r1870;
        double r1872 = 1.0;
        double r1873 = r1872 / r1869;
        double r1874 = 0.05555555555555555;
        double r1875 = x;
        double r1876 = z;
        double r1877 = y;
        double r1878 = r1876 * r1877;
        double r1879 = r1875 / r1878;
        double r1880 = r1874 * r1879;
        double r1881 = t;
        double r1882 = sqrt(r1881);
        double r1883 = r1880 * r1882;
        double r1884 = acos(r1883);
        double r1885 = r1873 * r1884;
        double r1886 = r1871 * r1885;
        return r1886;
}

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
Target1.2
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. Taylor expanded around 0 1.2

    \[\leadsto \frac{1}{3} \cdot \cos^{-1} \left(\color{blue}{\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\]
  3. Using strategy rm

  4. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\]

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

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\]

  6. Applied times-frac0.2

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\sqrt[3]{3}}\right)} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\]

  7. Applied associate-*l*0.2

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))