Average Error: 1.4 → 0.4
Time: 6.5s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\right)\right)\]
\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(\left(\sqrt[3]{\frac{1}{\sqrt[3]{3}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{3}}}\right) \cdot \left(\sqrt[3]{\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)\right)
double f(double x, double y, double z, double t) {
        double r923470 = 1.0;
        double r923471 = 3.0;
        double r923472 = r923470 / r923471;
        double r923473 = x;
        double r923474 = y;
        double r923475 = 27.0;
        double r923476 = r923474 * r923475;
        double r923477 = r923473 / r923476;
        double r923478 = r923471 * r923477;
        double r923479 = z;
        double r923480 = 2.0;
        double r923481 = r923479 * r923480;
        double r923482 = r923478 / r923481;
        double r923483 = t;
        double r923484 = sqrt(r923483);
        double r923485 = r923482 * r923484;
        double r923486 = acos(r923485);
        double r923487 = r923472 * r923486;
        return r923487;
}
↓
double f(double x, double y, double z, double t) {
        double r923488 = 1.0;
        double r923489 = 3.0;
        double r923490 = cbrt(r923489);
        double r923491 = r923490 * r923490;
        double r923492 = r923488 / r923491;
        double r923493 = 1.0;
        double r923494 = r923493 / r923490;
        double r923495 = cbrt(r923494);
        double r923496 = r923495 * r923495;
        double r923497 = x;
        double r923498 = y;
        double r923499 = 27.0;
        double r923500 = r923498 * r923499;
        double r923501 = r923497 / r923500;
        double r923502 = r923489 * r923501;
        double r923503 = z;
        double r923504 = 2.0;
        double r923505 = r923503 * r923504;
        double r923506 = r923502 / r923505;
        double r923507 = t;
        double r923508 = sqrt(r923507);
        double r923509 = r923506 * r923508;
        double r923510 = acos(r923509);
        double r923511 = r923495 * r923510;
        double r923512 = r923496 * r923511;
        double r923513 = r923492 * r923512;
        return r923513;
}

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
Herbie0.4
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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-frac0.4

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

  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. Using strategy rm

  8. Applied add-cube-cbrt1.4

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

  9. Applied associate-*l*0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019362 
(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)))))