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

Average Error: 20.3 → 17.8

Time: 15.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)} \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r759977 = 2.0;
        double r759978 = x;
        double r759979 = sqrt(r759978);
        double r759980 = r759977 * r759979;
        double r759981 = y;
        double r759982 = z;
        double r759983 = t;
        double r759984 = r759982 * r759983;
        double r759985 = 3.0;
        double r759986 = r759984 / r759985;
        double r759987 = r759981 - r759986;
        double r759988 = cos(r759987);
        double r759989 = r759980 * r759988;
        double r759990 = a;
        double r759991 = b;
        double r759992 = r759991 * r759985;
        double r759993 = r759990 / r759992;
        double r759994 = r759989 - r759993;
        return r759994;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r759995 = y;
        double r759996 = z;
        double r759997 = t;
        double r759998 = r759996 * r759997;
        double r759999 = 3.0;
        double r760000 = r759998 / r759999;
        double r760001 = r759995 - r760000;
        double r760002 = cos(r760001);
        double r760003 = 0.9999071837214661;
        bool r760004 = r760002 <= r760003;
        double r760005 = cos(r759995);
        double r760006 = 0.3333333333333333;
        double r760007 = r759997 * r759996;
        double r760008 = r760006 * r760007;
        double r760009 = cos(r760008);
        double r760010 = r760005 * r760009;
        double r760011 = 2.0;
        double r760012 = x;
        double r760013 = sqrt(r760012);
        double r760014 = r760011 * r760013;
        double r760015 = r760010 * r760014;
        double r760016 = sin(r760008);
        double r760017 = sin(r759995);
        double r760018 = r760016 * r760017;
        double r760019 = r760018 * r760014;
        double r760020 = cbrt(r760019);
        double r760021 = r760020 * r760020;
        double r760022 = r760021 * r760020;
        double r760023 = r760015 + r760022;
        double r760024 = a;
        double r760025 = b;
        double r760026 = r760025 * r759999;
        double r760027 = r760024 / r760026;
        double r760028 = r760023 - r760027;
        double r760029 = 1.0;
        double r760030 = 0.5;
        double r760031 = 2.0;
        double r760032 = pow(r759995, r760031);
        double r760033 = r760030 * r760032;
        double r760034 = r760029 - r760033;
        double r760035 = r760014 * r760034;
        double r760036 = r760035 - r760027;
        double r760037 = r760004 ? r760028 : r760036;
        return r760037;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	20.3
Target	18.4
Herbie	17.8

\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999071837214661

   1. Initial program 19.9

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

   3. Applied cos-diff 19.4

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3}\]

   4. Applied distribute-lft-in 19.4

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3}\]

   5. Simplified 19.4

      \[\leadsto \left(\color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)} + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]

   6. Simplified 19.4

      \[\leadsto \left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \color{blue}{\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) - \frac{a}{b \cdot 3}\]

   7. Taylor expanded around inf 19.4

      \[\leadsto \left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \color{blue}{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right)} \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\]

   8. Taylor expanded around inf 19.3

      \[\leadsto \left(\left(\cos y \cdot \color{blue}{\cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 19.3

       \[\leadsto \left(\left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \color{blue}{\left(\sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)} \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}}\right) - \frac{a}{b \cdot 3}\]

   if 0.9999071837214661 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 20.9

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

   2. Taylor expanded around 0 15.2

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]
3. Recombined 2 regimes into one program.

4. Final simplification 17.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99990718372146614:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \left(\sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)} \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) \cdot \sqrt[3]{\left(\sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) \cdot \sin y\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.379333748723514e+129) (- (* (* 2 (sqrt x)) (cos (- (/ 1 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2) (cos (- y (* (/ t 3) z)))) (/ (/ a 3) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2 (sqrt x))) (/ (/ a b) 3))))

  (- (* (* 2 (sqrt x)) (cos (- y (/ (* z t) 3)))) (/ a (* b 3))))