Average Error: 1.3 → 0.2
Time: 24.0s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)\right)\]
\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 \left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r36161099 = 1.0;
        double r36161100 = 3.0;
        double r36161101 = r36161099 / r36161100;
        double r36161102 = x;
        double r36161103 = y;
        double r36161104 = 27.0;
        double r36161105 = r36161103 * r36161104;
        double r36161106 = r36161102 / r36161105;
        double r36161107 = r36161100 * r36161106;
        double r36161108 = z;
        double r36161109 = 2.0;
        double r36161110 = r36161108 * r36161109;
        double r36161111 = r36161107 / r36161110;
        double r36161112 = t;
        double r36161113 = sqrt(r36161112);
        double r36161114 = r36161111 * r36161113;
        double r36161115 = acos(r36161114);
        double r36161116 = r36161101 * r36161115;
        return r36161116;
}
↓
double f(double x, double y, double z, double t) {
        double r36161117 = 1.0;
        double r36161118 = 3.0;
        double r36161119 = cbrt(r36161118);
        double r36161120 = r36161119 * r36161119;
        double r36161121 = r36161117 / r36161120;
        double r36161122 = 1.0;
        double r36161123 = r36161122 / r36161119;
        double r36161124 = x;
        double r36161125 = z;
        double r36161126 = y;
        double r36161127 = r36161125 * r36161126;
        double r36161128 = r36161124 / r36161127;
        double r36161129 = 0.05555555555555555;
        double r36161130 = r36161128 * r36161129;
        double r36161131 = t;
        double r36161132 = sqrt(r36161131);
        double r36161133 = r36161130 * r36161132;
        double r36161134 = acos(r36161133);
        double r36161135 = r36161123 * r36161134;
        double r36161136 = r36161121 * r36161135;
        return r36161136;
}

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
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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. Taylor expanded around 0 0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 
(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)))))