Average Error: 1.2 → 0.2
Time: 21.0s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\cos^{-1} \left(\log \left(e^{0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right)}{\sqrt[3]{3}}\right)\]
\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 \frac{\cos^{-1} \left(\log \left(e^{0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)}\right)\right)}{\sqrt[3]{3}}\right)
double f(double x, double y, double z, double t) {
        double r488717 = 1.0;
        double r488718 = 3.0;
        double r488719 = r488717 / r488718;
        double r488720 = x;
        double r488721 = y;
        double r488722 = 27.0;
        double r488723 = r488721 * r488722;
        double r488724 = r488720 / r488723;
        double r488725 = r488718 * r488724;
        double r488726 = z;
        double r488727 = 2.0;
        double r488728 = r488726 * r488727;
        double r488729 = r488725 / r488728;
        double r488730 = t;
        double r488731 = sqrt(r488730);
        double r488732 = r488729 * r488731;
        double r488733 = acos(r488732);
        double r488734 = r488719 * r488733;
        return r488734;
}
↓
double f(double x, double y, double z, double t) {
        double r488735 = 1.0;
        double r488736 = 3.0;
        double r488737 = cbrt(r488736);
        double r488738 = r488737 * r488737;
        double r488739 = r488735 / r488738;
        double r488740 = 1.0;
        double r488741 = 0.05555555555555555;
        double r488742 = t;
        double r488743 = sqrt(r488742);
        double r488744 = x;
        double r488745 = z;
        double r488746 = y;
        double r488747 = r488745 * r488746;
        double r488748 = r488744 / r488747;
        double r488749 = r488743 * r488748;
        double r488750 = r488741 * r488749;
        double r488751 = exp(r488750);
        double r488752 = log(r488751);
        double r488753 = acos(r488752);
        double r488754 = r488753 / r488737;
        double r488755 = r488740 * r488754;
        double r488756 = r488739 * r488755;
        return r488756;
}

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

Original1.2
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 1.2

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

  3. Applied add-cube-cbrt1.2

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

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

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

  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 0.2

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

  9. Applied add-log-exp0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019304 
(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)))))