Average Error: 1.3 → 0.3
Time: 15.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]
\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(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r976813 = 1.0;
        double r976814 = 3.0;
        double r976815 = r976813 / r976814;
        double r976816 = x;
        double r976817 = y;
        double r976818 = 27.0;
        double r976819 = r976817 * r976818;
        double r976820 = r976816 / r976819;
        double r976821 = r976814 * r976820;
        double r976822 = z;
        double r976823 = 2.0;
        double r976824 = r976822 * r976823;
        double r976825 = r976821 / r976824;
        double r976826 = t;
        double r976827 = sqrt(r976826);
        double r976828 = r976825 * r976827;
        double r976829 = acos(r976828);
        double r976830 = r976815 * r976829;
        return r976830;
}
↓
double f(double x, double y, double z, double t) {
        double r976831 = 1.0;
        double r976832 = 3.0;
        double r976833 = cbrt(r976832);
        double r976834 = r976833 * r976833;
        double r976835 = r976831 / r976834;
        double r976836 = 1.0;
        double r976837 = r976836 / r976833;
        double r976838 = 0.05555555555555555;
        double r976839 = x;
        double r976840 = z;
        double r976841 = y;
        double r976842 = r976840 * r976841;
        double r976843 = r976839 / r976842;
        double r976844 = r976838 * r976843;
        double r976845 = t;
        double r976846 = sqrt(r976845);
        double r976847 = r976844 * r976846;
        double r976848 = acos(r976847);
        double r976849 = r976837 * r976848;
        double r976850 = r976835 * r976849;
        return r976850;
}

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. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\left(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]

  8. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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)))))