Average Error: 1.2 → 0.2
Time: 31.8s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)\]
\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(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r25136625 = 1.0;
        double r25136626 = 3.0;
        double r25136627 = r25136625 / r25136626;
        double r25136628 = x;
        double r25136629 = y;
        double r25136630 = 27.0;
        double r25136631 = r25136629 * r25136630;
        double r25136632 = r25136628 / r25136631;
        double r25136633 = r25136626 * r25136632;
        double r25136634 = z;
        double r25136635 = 2.0;
        double r25136636 = r25136634 * r25136635;
        double r25136637 = r25136633 / r25136636;
        double r25136638 = t;
        double r25136639 = sqrt(r25136638);
        double r25136640 = r25136637 * r25136639;
        double r25136641 = acos(r25136640);
        double r25136642 = r25136627 * r25136641;
        return r25136642;
}
↓
double f(double x, double y, double z, double t) {
        double r25136643 = 1.0;
        double r25136644 = 3.0;
        double r25136645 = cbrt(r25136644);
        double r25136646 = r25136645 * r25136645;
        double r25136647 = r25136643 / r25136646;
        double r25136648 = 1.0;
        double r25136649 = r25136648 / r25136645;
        double r25136650 = x;
        double r25136651 = z;
        double r25136652 = y;
        double r25136653 = r25136651 * r25136652;
        double r25136654 = r25136650 / r25136653;
        double r25136655 = 0.05555555555555555;
        double r25136656 = r25136654 * r25136655;
        double r25136657 = t;
        double r25136658 = sqrt(r25136657);
        double r25136659 = r25136656 * r25136658;
        double r25136660 = acos(r25136659);
        double r25136661 = r25136649 * r25136660;
        double r25136662 = r25136647 * r25136661;
        return r25136662;
}

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
Target1.1
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

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +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)))))