Average Error: 1.3 → 0.3
Time: 22.6s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\sqrt{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{\cos^{-1} \left(0.05555555555555555247160270937456516548991 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right) \cdot \sqrt{1}}{\sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r573369 = 1.0;
        double r573370 = 3.0;
        double r573371 = r573369 / r573370;
        double r573372 = x;
        double r573373 = y;
        double r573374 = 27.0;
        double r573375 = r573373 * r573374;
        double r573376 = r573372 / r573375;
        double r573377 = r573370 * r573376;
        double r573378 = z;
        double r573379 = 2.0;
        double r573380 = r573378 * r573379;
        double r573381 = r573377 / r573380;
        double r573382 = t;
        double r573383 = sqrt(r573382);
        double r573384 = r573381 * r573383;
        double r573385 = acos(r573384);
        double r573386 = r573371 * r573385;
        return r573386;
}
↓
double f(double x, double y, double z, double t) {
        double r573387 = 1.0;
        double r573388 = sqrt(r573387);
        double r573389 = 3.0;
        double r573390 = cbrt(r573389);
        double r573391 = r573390 * r573390;
        double r573392 = r573388 / r573391;
        double r573393 = 0.05555555555555555;
        double r573394 = t;
        double r573395 = sqrt(r573394);
        double r573396 = x;
        double r573397 = z;
        double r573398 = y;
        double r573399 = r573397 * r573398;
        double r573400 = r573396 / r573399;
        double r573401 = r573395 * r573400;
        double r573402 = r573393 * r573401;
        double r573403 = acos(r573402);
        double r573404 = r573403 * r573388;
        double r573405 = r573404 / r573390;
        double r573406 = r573392 * r573405;
        return r573406;
}

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

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 add-sqr-sqrt1.3

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

  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 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019303 
(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)))))