Average Error: 1.3 → 0.3
Time: 17.2s
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 r14029350 = 1.0;
        double r14029351 = 3.0;
        double r14029352 = r14029350 / r14029351;
        double r14029353 = x;
        double r14029354 = y;
        double r14029355 = 27.0;
        double r14029356 = r14029354 * r14029355;
        double r14029357 = r14029353 / r14029356;
        double r14029358 = r14029351 * r14029357;
        double r14029359 = z;
        double r14029360 = 2.0;
        double r14029361 = r14029359 * r14029360;
        double r14029362 = r14029358 / r14029361;
        double r14029363 = t;
        double r14029364 = sqrt(r14029363);
        double r14029365 = r14029362 * r14029364;
        double r14029366 = acos(r14029365);
        double r14029367 = r14029352 * r14029366;
        return r14029367;
}
↓
double f(double x, double y, double z, double t) {
        double r14029368 = 1.0;
        double r14029369 = 3.0;
        double r14029370 = cbrt(r14029369);
        double r14029371 = r14029370 * r14029370;
        double r14029372 = r14029368 / r14029371;
        double r14029373 = 1.0;
        double r14029374 = x;
        double r14029375 = z;
        double r14029376 = y;
        double r14029377 = r14029375 * r14029376;
        double r14029378 = r14029374 / r14029377;
        double r14029379 = t;
        double r14029380 = sqrt(r14029379);
        double r14029381 = r14029378 * r14029380;
        double r14029382 = 0.05555555555555555;
        double r14029383 = r14029381 * r14029382;
        double r14029384 = acos(r14029383);
        double r14029385 = r14029384 / r14029370;
        double r14029386 = r14029373 * r14029385;
        double r14029387 = r14029372 * r14029386;
        return r14029387;
}

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
Herbie0.3
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27.0}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2.0}{3.0}}\right)}{3.0}\]

Derivation

  1. Initial program 1.3

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

  3. Applied add-cube-cbrt1.3

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

  4. Applied *-un-lft-identity1.3

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

  5. Applied times-frac0.4

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019156 
(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)))))