Average Error: 1.3 → 0.4
Time: 25.8s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{x \cdot \frac{3}{z}}{2 \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\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(\frac{x \cdot \frac{3}{z}}{2 \cdot \left(y \cdot 27\right)} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r34009146 = 1.0;
        double r34009147 = 3.0;
        double r34009148 = r34009146 / r34009147;
        double r34009149 = x;
        double r34009150 = y;
        double r34009151 = 27.0;
        double r34009152 = r34009150 * r34009151;
        double r34009153 = r34009149 / r34009152;
        double r34009154 = r34009147 * r34009153;
        double r34009155 = z;
        double r34009156 = 2.0;
        double r34009157 = r34009155 * r34009156;
        double r34009158 = r34009154 / r34009157;
        double r34009159 = t;
        double r34009160 = sqrt(r34009159);
        double r34009161 = r34009158 * r34009160;
        double r34009162 = acos(r34009161);
        double r34009163 = r34009148 * r34009162;
        return r34009163;
}
↓
double f(double x, double y, double z, double t) {
        double r34009164 = 1.0;
        double r34009165 = 3.0;
        double r34009166 = cbrt(r34009165);
        double r34009167 = r34009164 / r34009166;
        double r34009168 = x;
        double r34009169 = z;
        double r34009170 = r34009165 / r34009169;
        double r34009171 = r34009168 * r34009170;
        double r34009172 = 2.0;
        double r34009173 = y;
        double r34009174 = 27.0;
        double r34009175 = r34009173 * r34009174;
        double r34009176 = r34009172 * r34009175;
        double r34009177 = r34009171 / r34009176;
        double r34009178 = t;
        double r34009179 = sqrt(r34009178);
        double r34009180 = r34009177 * r34009179;
        double r34009181 = acos(r34009180);
        double r34009182 = r34009167 * r34009181;
        double r34009183 = 1.0;
        double r34009184 = r34009166 * r34009166;
        double r34009185 = r34009183 / r34009184;
        double r34009186 = r34009182 * r34009185;
        return r34009186;
}

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
Target1.2
Herbie0.4
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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 *-un-lft-identity0.3

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

  9. Applied sqrt-prod0.3

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

  10. Applied associate-*r*0.3

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019171 +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)))))