Average Error: 1.3 → 0.3
Time: 20.3s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r35752649 = 1.0;
        double r35752650 = 3.0;
        double r35752651 = r35752649 / r35752650;
        double r35752652 = x;
        double r35752653 = y;
        double r35752654 = 27.0;
        double r35752655 = r35752653 * r35752654;
        double r35752656 = r35752652 / r35752655;
        double r35752657 = r35752650 * r35752656;
        double r35752658 = z;
        double r35752659 = 2.0;
        double r35752660 = r35752658 * r35752659;
        double r35752661 = r35752657 / r35752660;
        double r35752662 = t;
        double r35752663 = sqrt(r35752662);
        double r35752664 = r35752661 * r35752663;
        double r35752665 = acos(r35752664);
        double r35752666 = r35752651 * r35752665;
        return r35752666;
}
↓
double f(double x, double y, double z, double t) {
        double r35752667 = 1.0;
        double r35752668 = 3.0;
        double r35752669 = cbrt(r35752668);
        double r35752670 = r35752667 / r35752669;
        double r35752671 = x;
        double r35752672 = 27.0;
        double r35752673 = y;
        double r35752674 = r35752672 * r35752673;
        double r35752675 = r35752671 / r35752674;
        double r35752676 = r35752675 * r35752668;
        double r35752677 = 2.0;
        double r35752678 = z;
        double r35752679 = r35752677 * r35752678;
        double r35752680 = r35752676 / r35752679;
        double r35752681 = t;
        double r35752682 = sqrt(r35752681);
        double r35752683 = r35752680 * r35752682;
        double r35752684 = acos(r35752683);
        double r35752685 = r35752670 * r35752684;
        double r35752686 = 1.0;
        double r35752687 = r35752669 * r35752669;
        double r35752688 = r35752686 / r35752687;
        double r35752689 = r35752685 * r35752688;
        return r35752689;
}

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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 
(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)))))