Average Error: 1.2 → 0.2
Time: 30.5s
Precision: 64
\[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]
\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)
double f(double x, double y, double z, double t) {
        double r34566335 = 1.0;
        double r34566336 = 3.0;
        double r34566337 = r34566335 / r34566336;
        double r34566338 = x;
        double r34566339 = y;
        double r34566340 = 27.0;
        double r34566341 = r34566339 * r34566340;
        double r34566342 = r34566338 / r34566341;
        double r34566343 = r34566336 * r34566342;
        double r34566344 = z;
        double r34566345 = 2.0;
        double r34566346 = r34566344 * r34566345;
        double r34566347 = r34566343 / r34566346;
        double r34566348 = t;
        double r34566349 = sqrt(r34566348);
        double r34566350 = r34566347 * r34566349;
        double r34566351 = acos(r34566350);
        double r34566352 = r34566337 * r34566351;
        return r34566352;
}
↓
double f(double x, double y, double z, double t) {
        double r34566353 = 1.0;
        double r34566354 = 3.0;
        double r34566355 = cbrt(r34566354);
        double r34566356 = r34566355 * r34566355;
        double r34566357 = r34566353 / r34566356;
        double r34566358 = 1.0;
        double r34566359 = x;
        double r34566360 = z;
        double r34566361 = y;
        double r34566362 = r34566360 * r34566361;
        double r34566363 = r34566359 / r34566362;
        double r34566364 = t;
        double r34566365 = sqrt(r34566364);
        double r34566366 = r34566363 * r34566365;
        double r34566367 = 0.05555555555555555;
        double r34566368 = r34566366 * r34566367;
        double r34566369 = acos(r34566368);
        double r34566370 = r34566369 / r34566355;
        double r34566371 = r34566358 * r34566370;
        double r34566372 = r34566357 * r34566371;
        return r34566372;
}

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

Original1.2
Target1.2
Herbie0.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.2

    \[\frac{1.0}{3.0} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.2

    \[\leadsto \frac{1.0}{\color{blue}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  4. Applied *-un-lft-identity1.2

    \[\leadsto \frac{\color{blue}{1 \cdot 1.0}}{\left(\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}\right) \cdot \sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  5. Applied times-frac0.3

    \[\leadsto \color{blue}{\left(\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \frac{1.0}{\sqrt[3]{3.0}}\right)} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\]

  6. Applied associate-*l*0.3

    \[\leadsto \color{blue}{\frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(\frac{1.0}{\sqrt[3]{3.0}} \cdot \cos^{-1} \left(\frac{3.0 \cdot \frac{x}{y \cdot 27.0}}{z \cdot 2.0} \cdot \sqrt{t}\right)\right)}\]

  7. Taylor expanded around 0 0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \color{blue}{\left(1.0 \cdot \frac{\cos^{-1} \left(0.05555555555555555 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3.0}}\right)}\]

  8. Final simplification0.2

    \[\leadsto \frac{1}{\sqrt[3]{3.0} \cdot \sqrt[3]{3.0}} \cdot \left(1.0 \cdot \frac{\cos^{-1} \left(\left(\frac{x}{z \cdot y} \cdot \sqrt{t}\right) \cdot 0.05555555555555555\right)}{\sqrt[3]{3.0}}\right)\]

Reproduce

herbie shell --seed 2019162 +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)))))