Average Error: 1.3 → 1.2
Time: 19.5s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\frac{\cos^{-1} \left(\left(\frac{2001599834386887}{36028797018963968} \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}{\frac{3}{1}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\frac{\cos^{-1} \left(\left(\frac{2001599834386887}{36028797018963968} \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}{\frac{3}{1}}
double f(double x, double y, double z, double t) {
        double r506472 = 1.0;
        double r506473 = 3.0;
        double r506474 = r506472 / r506473;
        double r506475 = x;
        double r506476 = y;
        double r506477 = 27.0;
        double r506478 = r506476 * r506477;
        double r506479 = r506475 / r506478;
        double r506480 = r506473 * r506479;
        double r506481 = z;
        double r506482 = 2.0;
        double r506483 = r506481 * r506482;
        double r506484 = r506480 / r506483;
        double r506485 = t;
        double r506486 = sqrt(r506485);
        double r506487 = r506484 * r506486;
        double r506488 = acos(r506487);
        double r506489 = r506474 * r506488;
        return r506489;
}
↓
double f(double x, double y, double z, double t) {
        double r506490 = 2001599834386887.0;
        double r506491 = 3.602879701896397e+16;
        double r506492 = r506490 / r506491;
        double r506493 = x;
        double r506494 = z;
        double r506495 = y;
        double r506496 = r506494 * r506495;
        double r506497 = r506493 / r506496;
        double r506498 = r506492 * r506497;
        double r506499 = t;
        double r506500 = sqrt(r506499);
        double r506501 = r506498 * r506500;
        double r506502 = acos(r506501);
        double r506503 = 3.0;
        double r506504 = 1.0;
        double r506505 = r506503 / r506504;
        double r506506 = r506502 / r506505;
        return r506506;
}

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
Herbie1.2
\[\frac{\cos^{-1} \left(\frac{\frac{x}{27}}{y \cdot z} \cdot \frac{\sqrt{t}}{\frac{2}{3}}\right)}{3}\]

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-cube-cbrt1.3

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

  5. Applied times-frac0.3

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

  6. Applied associate-*l*0.3

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

  7. Taylor expanded around 0 0.3

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

  8. Simplified0.3

    \[\leadsto \frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \left(\frac{\sqrt[3]{1}}{\sqrt[3]{3}} \cdot \cos^{-1} \left(\color{blue}{\left(\frac{2001599834386887}{36028797018963968} \cdot \frac{x}{z \cdot y}\right)} \cdot \sqrt{t}\right)\right)\]

  9. Final simplification1.2

    \[\leadsto \frac{\cos^{-1} \left(\left(\frac{2001599834386887}{36028797018963968} \cdot \frac{x}{z \cdot y}\right) \cdot \sqrt{t}\right)}{\frac{3}{1}}\]

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)))))