Average Error: 1.4 → 0.3
Time: 1.1m
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{1}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\sqrt{t} \cdot \left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\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(\sqrt{t} \cdot \left(\frac{x}{z \cdot y} \cdot 0.05555555555555555247160270937456516548991\right)\right)\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r37166322 = 1.0;
        double r37166323 = 3.0;
        double r37166324 = r37166322 / r37166323;
        double r37166325 = x;
        double r37166326 = y;
        double r37166327 = 27.0;
        double r37166328 = r37166326 * r37166327;
        double r37166329 = r37166325 / r37166328;
        double r37166330 = r37166323 * r37166329;
        double r37166331 = z;
        double r37166332 = 2.0;
        double r37166333 = r37166331 * r37166332;
        double r37166334 = r37166330 / r37166333;
        double r37166335 = t;
        double r37166336 = sqrt(r37166335);
        double r37166337 = r37166334 * r37166336;
        double r37166338 = acos(r37166337);
        double r37166339 = r37166324 * r37166338;
        return r37166339;
}
↓
double f(double x, double y, double z, double t) {
        double r37166340 = 1.0;
        double r37166341 = 3.0;
        double r37166342 = cbrt(r37166341);
        double r37166343 = r37166340 / r37166342;
        double r37166344 = t;
        double r37166345 = sqrt(r37166344);
        double r37166346 = x;
        double r37166347 = z;
        double r37166348 = y;
        double r37166349 = r37166347 * r37166348;
        double r37166350 = r37166346 / r37166349;
        double r37166351 = 0.05555555555555555;
        double r37166352 = r37166350 * r37166351;
        double r37166353 = r37166345 * r37166352;
        double r37166354 = acos(r37166353);
        double r37166355 = r37166343 * r37166354;
        double r37166356 = 1.0;
        double r37166357 = r37166342 * r37166342;
        double r37166358 = r37166356 / r37166357;
        double r37166359 = r37166355 * r37166358;
        return r37166359;
}

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

Original1.4
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 1.4

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

  3. Applied add-cube-cbrt1.4

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

  4. Applied *-un-lft-identity1.4

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

  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. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 
(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)))))