Average Error: 1.4 → 0.3
Time: 22.1s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt{3}} \cdot \frac{1 \cdot \cos^{-1} \left(\left(\frac{\frac{x}{z}}{y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)}{\sqrt{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{1}{\sqrt{3}} \cdot \frac{1 \cdot \cos^{-1} \left(\left(\frac{\frac{x}{z}}{y} \cdot 0.05555555555555555247160270937456516548991\right) \cdot \sqrt{t}\right)}{\sqrt{3}}
double f(double x, double y, double z, double t) {
        double r564492 = 1.0;
        double r564493 = 3.0;
        double r564494 = r564492 / r564493;
        double r564495 = x;
        double r564496 = y;
        double r564497 = 27.0;
        double r564498 = r564496 * r564497;
        double r564499 = r564495 / r564498;
        double r564500 = r564493 * r564499;
        double r564501 = z;
        double r564502 = 2.0;
        double r564503 = r564501 * r564502;
        double r564504 = r564500 / r564503;
        double r564505 = t;
        double r564506 = sqrt(r564505);
        double r564507 = r564504 * r564506;
        double r564508 = acos(r564507);
        double r564509 = r564494 * r564508;
        return r564509;
}
↓
double f(double x, double y, double z, double t) {
        double r564510 = 1.0;
        double r564511 = 3.0;
        double r564512 = sqrt(r564511);
        double r564513 = r564510 / r564512;
        double r564514 = 1.0;
        double r564515 = x;
        double r564516 = z;
        double r564517 = r564515 / r564516;
        double r564518 = y;
        double r564519 = r564517 / r564518;
        double r564520 = 0.05555555555555555;
        double r564521 = r564519 * r564520;
        double r564522 = t;
        double r564523 = sqrt(r564522);
        double r564524 = r564521 * r564523;
        double r564525 = acos(r564524);
        double r564526 = r564514 * r564525;
        double r564527 = r564526 / r564512;
        double r564528 = r564513 * r564527;
        return r564528;
}

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
Target1.3
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-sqr-sqrt0.4

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

  4. Applied *-un-lft-identity0.4

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

  5. Applied times-frac1.4

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

  6. Applied associate-*l*1.4

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

  7. Simplified0.4

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

  8. Taylor expanded around 0 0.3

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

  9. Simplified0.3

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

  10. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(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)))))