Average Error: 1.4 → 0.3
Time: 57.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\sqrt{t} \cdot \left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\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(\sqrt{t} \cdot \left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r27689998 = 1.0;
        double r27689999 = 3.0;
        double r27690000 = r27689998 / r27689999;
        double r27690001 = x;
        double r27690002 = y;
        double r27690003 = 27.0;
        double r27690004 = r27690002 * r27690003;
        double r27690005 = r27690001 / r27690004;
        double r27690006 = r27689999 * r27690005;
        double r27690007 = z;
        double r27690008 = 2.0;
        double r27690009 = r27690007 * r27690008;
        double r27690010 = r27690006 / r27690009;
        double r27690011 = t;
        double r27690012 = sqrt(r27690011);
        double r27690013 = r27690010 * r27690012;
        double r27690014 = acos(r27690013);
        double r27690015 = r27690000 * r27690014;
        return r27690015;
}
↓
double f(double x, double y, double z, double t) {
        double r27690016 = 1.0;
        double r27690017 = 3.0;
        double r27690018 = cbrt(r27690017);
        double r27690019 = r27690016 / r27690018;
        double r27690020 = t;
        double r27690021 = sqrt(r27690020);
        double r27690022 = x;
        double r27690023 = z;
        double r27690024 = y;
        double r27690025 = r27690023 * r27690024;
        double r27690026 = r27690022 / r27690025;
        double r27690027 = 0.05555555555555555;
        double r27690028 = r27690026 * r27690027;
        double r27690029 = r27690021 * r27690028;
        double r27690030 = acos(r27690029);
        double r27690031 = r27690019 * r27690030;
        double r27690032 = 1.0;
        double r27690033 = r27690018 * r27690018;
        double r27690034 = r27690032 / r27690033;
        double r27690035 = r27690031 * r27690034;
        return r27690035;
}

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

Derivation

  1. Initial program 1.4

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

  3. Applied add-cube-cbrt1.4

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

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

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 +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)))))