Average Error: 1.4 → 0.3
Time: 6.6s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}}{\sqrt[3]{\sqrt[3]{3}}}\right)\]
\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]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}}{\sqrt[3]{\sqrt[3]{3}}}\right)
double f(double x, double y, double z, double t) {
        double r740457 = 1.0;
        double r740458 = 3.0;
        double r740459 = r740457 / r740458;
        double r740460 = x;
        double r740461 = y;
        double r740462 = 27.0;
        double r740463 = r740461 * r740462;
        double r740464 = r740460 / r740463;
        double r740465 = r740458 * r740464;
        double r740466 = z;
        double r740467 = 2.0;
        double r740468 = r740466 * r740467;
        double r740469 = r740465 / r740468;
        double r740470 = t;
        double r740471 = sqrt(r740470);
        double r740472 = r740469 * r740471;
        double r740473 = acos(r740472);
        double r740474 = r740459 * r740473;
        return r740474;
}
↓
double f(double x, double y, double z, double t) {
        double r740475 = 1.0;
        double r740476 = 3.0;
        double r740477 = cbrt(r740476);
        double r740478 = r740477 * r740477;
        double r740479 = r740475 / r740478;
        double r740480 = 1.0;
        double r740481 = 0.05555555555555555;
        double r740482 = t;
        double r740483 = sqrt(r740482);
        double r740484 = x;
        double r740485 = z;
        double r740486 = y;
        double r740487 = r740485 * r740486;
        double r740488 = r740484 / r740487;
        double r740489 = r740483 * r740488;
        double r740490 = r740481 * r740489;
        double r740491 = acos(r740490);
        double r740492 = cbrt(r740478);
        double r740493 = r740491 / r740492;
        double r740494 = cbrt(r740477);
        double r740495 = r740493 / r740494;
        double r740496 = r740480 * r740495;
        double r740497 = r740479 * r740496;
        return r740497;
}

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
Target1.2
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 1.4

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

  3. Applied add-cube-cbrt1.4

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

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

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

  5. Applied times-frac0.5

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

  6. Applied associate-*l*0.4

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

  7. Taylor expanded around 0 0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \color{blue}{\left(1 \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{3}}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}}}}\right)\]

  10. Applied cbrt-prod0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\color{blue}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \sqrt[3]{\sqrt[3]{3}}}}\right)\]

  11. Applied associate-/r*0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \color{blue}{\frac{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}}{\sqrt[3]{\sqrt[3]{3}}}}\right)\]

  12. Final simplification0.3

    \[\leadsto \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(1 \cdot \frac{\frac{\cos^{-1} \left(0.055555555555555552 \cdot \left(\sqrt{t} \cdot \frac{x}{z \cdot y}\right)\right)}{\sqrt[3]{\sqrt[3]{3} \cdot \sqrt[3]{3}}}}{\sqrt[3]{\sqrt[3]{3}}}\right)\]

Reproduce

herbie shell --seed 2020039 
(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)))))