Average Error: 1.2 → 0.2
Time: 21.9s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]
\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)
double f(double x, double y, double z, double t) {
        double r32172609 = 1.0;
        double r32172610 = 3.0;
        double r32172611 = r32172609 / r32172610;
        double r32172612 = x;
        double r32172613 = y;
        double r32172614 = 27.0;
        double r32172615 = r32172613 * r32172614;
        double r32172616 = r32172612 / r32172615;
        double r32172617 = r32172610 * r32172616;
        double r32172618 = z;
        double r32172619 = 2.0;
        double r32172620 = r32172618 * r32172619;
        double r32172621 = r32172617 / r32172620;
        double r32172622 = t;
        double r32172623 = sqrt(r32172622);
        double r32172624 = r32172621 * r32172623;
        double r32172625 = acos(r32172624);
        double r32172626 = r32172611 * r32172625;
        return r32172626;
}
↓
double f(double x, double y, double z, double t) {
        double r32172627 = 1.0;
        double r32172628 = 3.0;
        double r32172629 = cbrt(r32172628);
        double r32172630 = r32172629 * r32172629;
        double r32172631 = r32172627 / r32172630;
        double r32172632 = 1.0;
        double r32172633 = x;
        double r32172634 = z;
        double r32172635 = y;
        double r32172636 = r32172634 * r32172635;
        double r32172637 = r32172633 / r32172636;
        double r32172638 = t;
        double r32172639 = sqrt(r32172638);
        double r32172640 = r32172637 * r32172639;
        double r32172641 = 0.05555555555555555;
        double r32172642 = r32172640 * r32172641;
        double r32172643 = acos(r32172642);
        double r32172644 = r32172643 / r32172629;
        double r32172645 = r32172632 * r32172644;
        double r32172646 = r32172631 * r32172645;
        return r32172646;
}

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
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.2

    \[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1.0}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

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

    \[\leadsto \frac{\color{blue}{1 \cdot 1.0}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{1.0}{\sqrt[3]{3.0}}\right)} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}\]

  7. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \color{blue}{\left(1.0 \cdot \frac{\cos^{-1} \left(0.05555555555555555 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3.0}}\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]

Reproduce

herbie shell --seed 2019162 
(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)))))