Average Error: 1.3 → 0.3
Time: 12.4s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r735117 = 1.0;
        double r735118 = 3.0;
        double r735119 = r735117 / r735118;
        double r735120 = x;
        double r735121 = y;
        double r735122 = 27.0;
        double r735123 = r735121 * r735122;
        double r735124 = r735120 / r735123;
        double r735125 = r735118 * r735124;
        double r735126 = z;
        double r735127 = 2.0;
        double r735128 = r735126 * r735127;
        double r735129 = r735125 / r735128;
        double r735130 = t;
        double r735131 = sqrt(r735130);
        double r735132 = r735129 * r735131;
        double r735133 = acos(r735132);
        double r735134 = r735119 * r735133;
        return r735134;
}
↓
double f(double x, double y, double z, double t) {
        double r735135 = 1.0;
        double r735136 = cbrt(r735135);
        double r735137 = r735136 * r735136;
        double r735138 = 3.0;
        double r735139 = cbrt(r735138);
        double r735140 = r735139 * r735139;
        double r735141 = r735137 / r735140;
        double r735142 = r735136 / r735139;
        double r735143 = 0.05555555555555555;
        double r735144 = x;
        double r735145 = z;
        double r735146 = y;
        double r735147 = r735145 * r735146;
        double r735148 = r735144 / r735147;
        double r735149 = r735143 * r735148;
        double r735150 = t;
        double r735151 = sqrt(r735150);
        double r735152 = r735149 * r735151;
        double r735153 = acos(r735152);
        double r735154 = r735142 * r735153;
        double r735155 = r735141 * r735154;
        return r735155;
}

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

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-cube-cbrt1.3

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.3

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]

  8. Final simplification0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.055555555555555552 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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)))))