Average Error: 1.3 → 0.4
Time: 24.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\frac{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \cdot \sqrt{t}\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\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{\frac{x}{27 \cdot y} \cdot 3}{2 \cdot z} \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 r37928247 = 1.0;
        double r37928248 = 3.0;
        double r37928249 = r37928247 / r37928248;
        double r37928250 = x;
        double r37928251 = y;
        double r37928252 = 27.0;
        double r37928253 = r37928251 * r37928252;
        double r37928254 = r37928250 / r37928253;
        double r37928255 = r37928248 * r37928254;
        double r37928256 = z;
        double r37928257 = 2.0;
        double r37928258 = r37928256 * r37928257;
        double r37928259 = r37928255 / r37928258;
        double r37928260 = t;
        double r37928261 = sqrt(r37928260);
        double r37928262 = r37928259 * r37928261;
        double r37928263 = acos(r37928262);
        double r37928264 = r37928249 * r37928263;
        return r37928264;
}
↓
double f(double x, double y, double z, double t) {
        double r37928265 = 1.0;
        double r37928266 = 3.0;
        double r37928267 = cbrt(r37928266);
        double r37928268 = r37928265 / r37928267;
        double r37928269 = x;
        double r37928270 = 27.0;
        double r37928271 = y;
        double r37928272 = r37928270 * r37928271;
        double r37928273 = r37928269 / r37928272;
        double r37928274 = r37928273 * r37928266;
        double r37928275 = 2.0;
        double r37928276 = z;
        double r37928277 = r37928275 * r37928276;
        double r37928278 = r37928274 / r37928277;
        double r37928279 = t;
        double r37928280 = sqrt(r37928279);
        double r37928281 = r37928278 * r37928280;
        double r37928282 = acos(r37928281);
        double r37928283 = r37928268 * r37928282;
        double r37928284 = 1.0;
        double r37928285 = r37928267 * r37928267;
        double r37928286 = r37928284 / r37928285;
        double r37928287 = r37928283 * r37928286;
        return r37928287;
}

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

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-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 +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)))))