Average Error: 1.4 → 0.3
Time: 52.2s
Precision: 64
\[\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)\]
↓
\[\left(\frac{\cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555247160270937456516548991\right)}{\sqrt[3]{3}} \cdot 1\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}\]
\frac{1}{3} \cdot \cos^{-1} \left(\frac{3 \cdot \frac{x}{y \cdot 27}}{z \cdot 2} \cdot \sqrt{t}\right)
↓
\left(\frac{\cos^{-1} \left(\left(\frac{x}{y \cdot z} \cdot \sqrt{t}\right) \cdot 0.05555555555555555247160270937456516548991\right)}{\sqrt[3]{3}} \cdot 1\right) \cdot \frac{1}{\sqrt[3]{3} \cdot \sqrt[3]{3}}
double f(double x, double y, double z, double t) {
        double r19412696 = 1.0;
        double r19412697 = 3.0;
        double r19412698 = r19412696 / r19412697;
        double r19412699 = x;
        double r19412700 = y;
        double r19412701 = 27.0;
        double r19412702 = r19412700 * r19412701;
        double r19412703 = r19412699 / r19412702;
        double r19412704 = r19412697 * r19412703;
        double r19412705 = z;
        double r19412706 = 2.0;
        double r19412707 = r19412705 * r19412706;
        double r19412708 = r19412704 / r19412707;
        double r19412709 = t;
        double r19412710 = sqrt(r19412709);
        double r19412711 = r19412708 * r19412710;
        double r19412712 = acos(r19412711);
        double r19412713 = r19412698 * r19412712;
        return r19412713;
}
↓
double f(double x, double y, double z, double t) {
        double r19412714 = x;
        double r19412715 = y;
        double r19412716 = z;
        double r19412717 = r19412715 * r19412716;
        double r19412718 = r19412714 / r19412717;
        double r19412719 = t;
        double r19412720 = sqrt(r19412719);
        double r19412721 = r19412718 * r19412720;
        double r19412722 = 0.05555555555555555;
        double r19412723 = r19412721 * r19412722;
        double r19412724 = acos(r19412723);
        double r19412725 = 3.0;
        double r19412726 = cbrt(r19412725);
        double r19412727 = r19412724 / r19412726;
        double r19412728 = 1.0;
        double r19412729 = r19412727 * r19412728;
        double r19412730 = 1.0;
        double r19412731 = r19412726 * r19412726;
        double r19412732 = r19412730 / r19412731;
        double r19412733 = r19412729 * r19412732;
        return r19412733;
}

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-frac0.4

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

  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. Taylor expanded around 0 0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(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)))))