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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999972455529012593800075592298526316881:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

\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 0.9999972455529012593800075592298526316881:\\
\;\;\;\;\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \log \left(e^{\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}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\

\end{array}

double f(double x, double y, double z, double t, double a, double b) {
        double r595884 = 2.0;
        double r595885 = x;
        double r595886 = sqrt(r595885);
        double r595887 = r595884 * r595886;
        double r595888 = y;
        double r595889 = z;
        double r595890 = t;
        double r595891 = r595889 * r595890;
        double r595892 = 3.0;
        double r595893 = r595891 / r595892;
        double r595894 = r595888 - r595893;
        double r595895 = cos(r595894);
        double r595896 = r595887 * r595895;
        double r595897 = a;
        double r595898 = b;
        double r595899 = r595898 * r595892;
        double r595900 = r595897 / r595899;
        double r595901 = r595896 - r595900;
        return r595901;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r595902 = y;
        double r595903 = z;
        double r595904 = t;
        double r595905 = r595903 * r595904;
        double r595906 = 3.0;
        double r595907 = r595905 / r595906;
        double r595908 = r595902 - r595907;
        double r595909 = cos(r595908);
        double r595910 = 0.9999972455529013;
        bool r595911 = r595909 <= r595910;
        double r595912 = cos(r595902);
        double r595913 = cos(r595907);
        double r595914 = r595912 * r595913;
        double r595915 = 2.0;
        double r595916 = x;
        double r595917 = sqrt(r595916);
        double r595918 = r595915 * r595917;
        double r595919 = r595914 * r595918;
        double r595920 = sin(r595902);
        double r595921 = sin(r595907);
        double r595922 = r595920 * r595921;
        double r595923 = exp(r595922);
        double r595924 = log(r595923);
        double r595925 = r595924 * r595918;
        double r595926 = r595919 + r595925;
        double r595927 = a;
        double r595928 = b;
        double r595929 = r595928 * r595906;
        double r595930 = r595927 / r595929;
        double r595931 = r595926 - r595930;
        double r595932 = -0.5;
        double r595933 = 2.0;
        double r595934 = pow(r595902, r595933);
        double r595935 = 1.0;
        double r595936 = fma(r595932, r595934, r595935);
        double r595937 = r595918 * r595936;
        double r595938 = r595937 - r595930;
        double r595939 = r595911 ? r595931 : r595938;
        return r595939;
}

Original	20.5
Target	18.8
Herbie	18.0

Derivation

Split input into 2 regimes
if (cos (- y (/ (* z t) 3.0))) < 0.9999972455529013
1. Initial program 20.1
  \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
2. Using strategy rm
3. Applied cos-diff19.5
  \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3}\]
4. Applied distribute-lft-in19.5
  \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3}\]
5. Simplified19.5
  \[\leadsto \left(\color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)} + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right) - \frac{a}{b \cdot 3}\]
6. Simplified19.5
  \[\leadsto \left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \color{blue}{\left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right)}\right) - \frac{a}{b \cdot 3}\]
7. Using strategy rm
8. Applied add-log-exp19.5
  \[\leadsto \left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \color{blue}{\log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right)} \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\]
if 0.9999972455529013 < (cos (- y (/ (* z t) 3.0)))
1. Initial program 21.3
  \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
2. Taylor expanded around 0 15.4
  \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]
3. Simplified15.4
  \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right)} - \frac{a}{b \cdot 3}\]
Recombined 2 regimes into one program.
Final simplification18.0
\[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9999972455529012593800075592298526316881:\\ \;\;\;\;\left(\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) \cdot \left(2 \cdot \sqrt{x}\right) + \log \left(e^{\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)}\right) \cdot \left(2 \cdot \sqrt{x}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, {y}^{2}, 1\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(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))))

Error

Target

Derivation

`if (cos (- y (/ (* z t) 3.0))) < 0.9999972455529013`

`if 0.9999972455529013 < (cos (- y (/ (* z t) 3.0)))`

Reproduce