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

↓

\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9981936348282990367764000438910443335772:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

\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 0.9981936348282990367764000438910443335772:\\
\;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{z \cdot t}{3}\right)\right)\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}

double f(double x, double y, double z, double t, double a, double b) {
        double r301206 = 2.0;
        double r301207 = x;
        double r301208 = sqrt(r301207);
        double r301209 = r301206 * r301208;
        double r301210 = y;
        double r301211 = z;
        double r301212 = t;
        double r301213 = r301211 * r301212;
        double r301214 = 3.0;
        double r301215 = r301213 / r301214;
        double r301216 = r301210 - r301215;
        double r301217 = cos(r301216);
        double r301218 = r301209 * r301217;
        double r301219 = a;
        double r301220 = b;
        double r301221 = r301220 * r301214;
        double r301222 = r301219 / r301221;
        double r301223 = r301218 - r301222;
        return r301223;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r301224 = y;
        double r301225 = z;
        double r301226 = t;
        double r301227 = r301225 * r301226;
        double r301228 = 3.0;
        double r301229 = r301227 / r301228;
        double r301230 = r301224 - r301229;
        double r301231 = cos(r301230);
        double r301232 = 0.998193634828299;
        bool r301233 = r301231 <= r301232;
        double r301234 = 2.0;
        double r301235 = x;
        double r301236 = sqrt(r301235);
        double r301237 = r301234 * r301236;
        double r301238 = cos(r301224);
        double r301239 = cos(r301229);
        double r301240 = r301238 * r301239;
        double r301241 = r301237 * r301240;
        double r301242 = sin(r301224);
        double r301243 = r301237 * r301242;
        double r301244 = sin(r301229);
        double r301245 = log1p(r301244);
        double r301246 = expm1(r301245);
        double r301247 = r301243 * r301246;
        double r301248 = r301241 + r301247;
        double r301249 = a;
        double r301250 = b;
        double r301251 = r301250 * r301228;
        double r301252 = r301249 / r301251;
        double r301253 = r301248 - r301252;
        double r301254 = 1.0;
        double r301255 = 0.5;
        double r301256 = 2.0;
        double r301257 = pow(r301224, r301256);
        double r301258 = r301255 * r301257;
        double r301259 = r301254 - r301258;
        double r301260 = r301237 * r301259;
        double r301261 = r301260 - r301252;
        double r301262 = r301233 ? r301253 : r301261;
        return r301262;
}

Original	20.5
Target	18.4
Herbie	18.1

Derivation

Split input into 2 regimes
if (cos (- y (/ (* z t) 3.0))) < 0.998193634828299
1. Initial program 19.8
  \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
2. Using strategy rm
3. Applied cos-diff19.2
  \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3}\]
4. Applied distribute-lft-in19.2
  \[\leadsto \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right)\right)} - \frac{a}{b \cdot 3}\]
5. Using strategy rm
6. Applied expm1-log1p-u19.2
  \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(2 \cdot \sqrt{x}\right) \cdot \left(\sin y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right)\right) - \frac{a}{b \cdot 3}\]
7. Using strategy rm
8. Applied associate-*r*19.2
  \[\leadsto \left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \color{blue}{\left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{z \cdot t}{3}\right)\right)\right)}\right) - \frac{a}{b \cdot 3}\]
if 0.998193634828299 < (cos (- y (/ (* z t) 3.0)))
1. Initial program 21.6
  \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
2. Taylor expanded around 0 16.5
  \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(1 - \frac{1}{2} \cdot {y}^{2}\right)} - \frac{a}{b \cdot 3}\]
Recombined 2 regimes into one program.
Final simplification18.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.9981936348282990367764000438910443335772:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\frac{z \cdot t}{3}\right)\right) + \left(\left(2 \cdot \sqrt{x}\right) \cdot \sin y\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{z \cdot t}{3}\right)\right)\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 +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

Try it out

Target

Derivation

`if (cos (- y (/ (* z t) 3.0))) < 0.998193634828299`

`if 0.998193634828299 < (cos (- y (/ (* z t) 3.0)))`

Reproduce

In
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