Diagrams.Solve.Polynomial:cubForm from diagrams-solve-0.1, D

Average Error: 1.2 → 0.3

Time: 8.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r708723 = 1.0;
        double r708724 = 3.0;
        double r708725 = r708723 / r708724;
        double r708726 = x;
        double r708727 = y;
        double r708728 = 27.0;
        double r708729 = r708727 * r708728;
        double r708730 = r708726 / r708729;
        double r708731 = r708724 * r708730;
        double r708732 = z;
        double r708733 = 2.0;
        double r708734 = r708732 * r708733;
        double r708735 = r708731 / r708734;
        double r708736 = t;
        double r708737 = sqrt(r708736);
        double r708738 = r708735 * r708737;
        double r708739 = acos(r708738);
        double r708740 = r708725 * r708739;
        return r708740;
}

↓

double f(double x, double y, double z, double t) {
        double r708741 = 1.0;
        double r708742 = 3.0;
        double r708743 = cbrt(r708742);
        double r708744 = r708743 * r708743;
        double r708745 = r708741 / r708744;
        double r708746 = 1.0;
        double r708747 = r708746 / r708743;
        double r708748 = x;
        double r708749 = y;
        double r708750 = 27.0;
        double r708751 = r708749 * r708750;
        double r708752 = r708748 / r708751;
        double r708753 = r708742 * r708752;
        double r708754 = z;
        double r708755 = 2.0;
        double r708756 = r708754 * r708755;
        double r708757 = r708753 / r708756;
        double r708758 = t;
        double r708759 = sqrt(r708758);
        double r708760 = r708757 * r708759;
        double r708761 = acos(r708760);
        double r708762 = r708747 * r708761;
        double r708763 = r708745 * r708762;
        return r708763;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	1.2
Target	1.2
Herbie	0.3

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

Derivation

1. Initial program 1.2

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

3. Applied add-cube-cbrt 1.2

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

4. Applied *-un-lft-identity 1.2

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

5. Applied times-frac 0.3

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

6. Applied associate-*l* 0.3

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

7. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(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)))))