Average Error: 1.3 → 0.3
Time: 17.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\frac{\sqrt{t}}{\frac{2 \cdot z}{\frac{x \cdot 3}{27 \cdot y}}}\right) \cdot \sqrt{1}}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(\frac{\sqrt{t}}{\frac{2 \cdot z}{\frac{x \cdot 3}{27 \cdot y}}}\right) \cdot \sqrt{1}}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r520036 = 1.0;
        double r520037 = 3.0;
        double r520038 = r520036 / r520037;
        double r520039 = x;
        double r520040 = y;
        double r520041 = 27.0;
        double r520042 = r520040 * r520041;
        double r520043 = r520039 / r520042;
        double r520044 = r520037 * r520043;
        double r520045 = z;
        double r520046 = 2.0;
        double r520047 = r520045 * r520046;
        double r520048 = r520044 / r520047;
        double r520049 = t;
        double r520050 = sqrt(r520049);
        double r520051 = r520048 * r520050;
        double r520052 = acos(r520051);
        double r520053 = r520038 * r520052;
        return r520053;
}
↓
double f(double x, double y, double z, double t) {
        double r520054 = 1.0;
        double r520055 = sqrt(r520054);
        double r520056 = 3.0;
        double r520057 = cbrt(r520056);
        double r520058 = r520057 * r520057;
        double r520059 = r520055 / r520058;
        double r520060 = t;
        double r520061 = sqrt(r520060);
        double r520062 = 2.0;
        double r520063 = z;
        double r520064 = r520062 * r520063;
        double r520065 = x;
        double r520066 = r520065 * r520056;
        double r520067 = 27.0;
        double r520068 = y;
        double r520069 = r520067 * r520068;
        double r520070 = r520066 / r520069;
        double r520071 = r520064 / r520070;
        double r520072 = r520061 / r520071;
        double r520073 = acos(r520072);
        double r520074 = r520073 * r520055;
        double r520075 = r520074 / r520057;
        double r520076 = r520059 * r520075;
        return r520076;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

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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 add-sqr-sqrt1.3

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +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)))))