Average Error: 1.4 → 0.3
Time: 22.5s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\frac{\left(0.05555555555555555 \cdot \sqrt{t}\right) \cdot \frac{x}{z}}{y}\right)}{\sqrt[3]{3.0}}\right)\]
\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(\frac{\left(0.05555555555555555 \cdot \sqrt{t}\right) \cdot \frac{x}{z}}{y}\right)}{\sqrt[3]{3.0}}\right)
double f(double x, double y, double z, double t) {
        double r36252987 = 1.0;
        double r36252988 = 3.0;
        double r36252989 = r36252987 / r36252988;
        double r36252990 = x;
        double r36252991 = y;
        double r36252992 = 27.0;
        double r36252993 = r36252991 * r36252992;
        double r36252994 = r36252990 / r36252993;
        double r36252995 = r36252988 * r36252994;
        double r36252996 = z;
        double r36252997 = 2.0;
        double r36252998 = r36252996 * r36252997;
        double r36252999 = r36252995 / r36252998;
        double r36253000 = t;
        double r36253001 = sqrt(r36253000);
        double r36253002 = r36252999 * r36253001;
        double r36253003 = acos(r36253002);
        double r36253004 = r36252989 * r36253003;
        return r36253004;
}
↓
double f(double x, double y, double z, double t) {
        double r36253005 = 1.0;
        double r36253006 = 3.0;
        double r36253007 = cbrt(r36253006);
        double r36253008 = r36253007 * r36253007;
        double r36253009 = r36253005 / r36253008;
        double r36253010 = 1.0;
        double r36253011 = 0.05555555555555555;
        double r36253012 = t;
        double r36253013 = sqrt(r36253012);
        double r36253014 = r36253011 * r36253013;
        double r36253015 = x;
        double r36253016 = z;
        double r36253017 = r36253015 / r36253016;
        double r36253018 = r36253014 * r36253017;
        double r36253019 = y;
        double r36253020 = r36253018 / r36253019;
        double r36253021 = acos(r36253020);
        double r36253022 = r36253021 / r36253007;
        double r36253023 = r36253010 * r36253022;
        double r36253024 = r36253009 * r36253023;
        return r36253024;
}

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

Original1.4
Target1.2
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.4

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

  3. Applied add-cube-cbrt1.4

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

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

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  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. Using strategy rm

  9. Applied div-inv0.2

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

  10. Applied associate-*l*0.2

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

  11. Simplified0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019164 
(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)))))