Average Error: 1.2 → 0.2
Time: 26.7s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\left(0.05555555555555555247160270937456516548991 \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)\right)
double f(double x, double y, double z, double t) {
        double r525345 = 1.0;
        double r525346 = 3.0;
        double r525347 = r525345 / r525346;
        double r525348 = x;
        double r525349 = y;
        double r525350 = 27.0;
        double r525351 = r525349 * r525350;
        double r525352 = r525348 / r525351;
        double r525353 = r525346 * r525352;
        double r525354 = z;
        double r525355 = 2.0;
        double r525356 = r525354 * r525355;
        double r525357 = r525353 / r525356;
        double r525358 = t;
        double r525359 = sqrt(r525358);
        double r525360 = r525357 * r525359;
        double r525361 = acos(r525360);
        double r525362 = r525347 * r525361;
        return r525362;
}
↓
double f(double x, double y, double z, double t) {
        double r525363 = 1.0;
        double r525364 = sqrt(r525363);
        double r525365 = 3.0;
        double r525366 = cbrt(r525365);
        double r525367 = r525366 * r525366;
        double r525368 = r525364 / r525367;
        double r525369 = r525364 / r525366;
        double r525370 = 0.05555555555555555;
        double r525371 = x;
        double r525372 = z;
        double r525373 = y;
        double r525374 = r525372 * r525373;
        double r525375 = r525371 / r525374;
        double r525376 = r525370 * r525375;
        double r525377 = t;
        double r525378 = sqrt(r525377);
        double r525379 = r525376 * r525378;
        double r525380 = acos(r525379);
        double r525381 = r525369 * r525380;
        double r525382 = r525368 * r525381;
        return r525382;
}

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

Derivation

  1. Initial program 1.2

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

  3. Applied add-cube-cbrt1.2

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

  4. Applied add-sqr-sqrt1.2

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Taylor expanded around 0 0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019322 +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)))))