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

Average Error: 20.5 → 17.7

Time: 10.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999996888221743:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r731915 = 2.0;
        double r731916 = x;
        double r731917 = sqrt(r731916);
        double r731918 = r731915 * r731917;
        double r731919 = y;
        double r731920 = z;
        double r731921 = t;
        double r731922 = r731920 * r731921;
        double r731923 = 3.0;
        double r731924 = r731922 / r731923;
        double r731925 = r731919 - r731924;
        double r731926 = cos(r731925);
        double r731927 = r731918 * r731926;
        double r731928 = a;
        double r731929 = b;
        double r731930 = r731929 * r731923;
        double r731931 = r731928 / r731930;
        double r731932 = r731927 - r731931;
        return r731932;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r731933 = y;
        double r731934 = z;
        double r731935 = t;
        double r731936 = r731934 * r731935;
        double r731937 = 3.0;
        double r731938 = r731936 / r731937;
        double r731939 = r731933 - r731938;
        double r731940 = cos(r731939);
        double r731941 = 0.9999999688822174;
        bool r731942 = r731940 <= r731941;
        double r731943 = 2.0;
        double r731944 = x;
        double r731945 = sqrt(r731944);
        double r731946 = r731943 * r731945;
        double r731947 = cos(r731933);
        double r731948 = 0.3333333333333333;
        double r731949 = r731935 * r731934;
        double r731950 = r731948 * r731949;
        double r731951 = cos(r731950);
        double r731952 = r731947 * r731951;
        double r731953 = r731946 * r731952;
        double r731954 = sin(r731933);
        double r731955 = sin(r731950);
        double r731956 = r731954 * r731955;
        double r731957 = r731946 * r731956;
        double r731958 = r731953 + r731957;
        double r731959 = a;
        double r731960 = b;
        double r731961 = r731959 / r731960;
        double r731962 = r731961 / r731937;
        double r731963 = r731958 - r731962;
        double r731964 = 1.0;
        double r731965 = 0.5;
        double r731966 = 2.0;
        double r731967 = pow(r731933, r731966);
        double r731968 = r731965 * r731967;
        double r731969 = r731964 - r731968;
        double r731970 = r731946 * r731969;
        double r731971 = r731960 * r731937;
        double r731972 = r731959 / r731971;
        double r731973 = r731970 - r731972;
        double r731974 = r731942 ? r731963 : r731973;
        return r731974;
}

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	20.5
Target	18.4
Herbie	17.7

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

Derivation

1. Split input into 2 regimes

2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999688822174

   1. Initial program 19.4

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

   3. Applied cos-diff 18.9

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

   4. Applied distribute-lft-in 18.9

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

   5. Taylor expanded around inf 18.9

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

   6. Taylor expanded around inf 18.9

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

   8. Applied associate-/r* 18.9

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

   if 0.9999999688822174 < (cos (- y (/ (* z t) 3.0)))

   1. Initial program 22.4

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

   2. Taylor expanded around 0 15.7

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

4. Final simplification 17.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999996888221743:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right)\right)\right) - \frac{\frac{a}{b}}{3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 
(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))))