Average Error: 1.3 → 0.3
Time: 5.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1 \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r774057 = 1.0;
        double r774058 = 3.0;
        double r774059 = r774057 / r774058;
        double r774060 = x;
        double r774061 = y;
        double r774062 = 27.0;
        double r774063 = r774061 * r774062;
        double r774064 = r774060 / r774063;
        double r774065 = r774058 * r774064;
        double r774066 = z;
        double r774067 = 2.0;
        double r774068 = r774066 * r774067;
        double r774069 = r774065 / r774068;
        double r774070 = t;
        double r774071 = sqrt(r774070);
        double r774072 = r774069 * r774071;
        double r774073 = acos(r774072);
        double r774074 = r774059 * r774073;
        return r774074;
}
↓
double f(double x, double y, double z, double t) {
        double r774075 = 1.0;
        double r774076 = 3.0;
        double r774077 = cbrt(r774076);
        double r774078 = r774077 * r774077;
        double r774079 = r774075 / r774078;
        double r774080 = 1.0;
        double r774081 = x;
        double r774082 = y;
        double r774083 = 27.0;
        double r774084 = r774082 * r774083;
        double r774085 = r774081 / r774084;
        double r774086 = r774076 * r774085;
        double r774087 = z;
        double r774088 = 2.0;
        double r774089 = r774087 * r774088;
        double r774090 = r774086 / r774089;
        double r774091 = t;
        double r774092 = sqrt(r774091);
        double r774093 = r774090 * r774092;
        double r774094 = acos(r774093);
        double r774095 = r774080 * r774094;
        double r774096 = r774095 / r774077;
        double r774097 = r774079 * r774096;
        return r774097;
}

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

Enter valid numbers for all inputs

Target

Original1.3
Target1.1
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. Using strategy rm

  8. Applied associate-*l/0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, D"
  :precision binary64

  :herbie-target
  (/ (acos (* (/ (/ x 27) (* y z)) (/ (sqrt t) (/ 2 3)))) 3)

  (* (/ 1 3) (acos (* (/ (* 3 (/ x (* y 27))) (* z 2)) (sqrt t)))))