Average Error: 20.8 → 17.1
Time: 14.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot t \le -9.3073940399080106 \cdot 10^{231} \lor \neg \left(z \cdot t \le 1.104365485840529 \cdot 10^{270}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{expm1}\left(\log 2 - \frac{1}{4} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\frac{\sqrt[3]{3}}{t}}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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}\;z \cdot t \le -9.3073940399080106 \cdot 10^{231} \lor \neg \left(z \cdot t \le 1.104365485840529 \cdot 10^{270}\right):\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{expm1}\left(\log 2 - \frac{1}{4} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\frac{\sqrt[3]{3}}{t}}\right) - \frac{a}{b \cdot 3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r709258 = 2.0;
        double r709259 = x;
        double r709260 = sqrt(r709259);
        double r709261 = r709258 * r709260;
        double r709262 = y;
        double r709263 = z;
        double r709264 = t;
        double r709265 = r709263 * r709264;
        double r709266 = 3.0;
        double r709267 = r709265 / r709266;
        double r709268 = r709262 - r709267;
        double r709269 = cos(r709268);
        double r709270 = r709261 * r709269;
        double r709271 = a;
        double r709272 = b;
        double r709273 = r709272 * r709266;
        double r709274 = r709271 / r709273;
        double r709275 = r709270 - r709274;
        return r709275;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r709276 = z;
        double r709277 = t;
        double r709278 = r709276 * r709277;
        double r709279 = -9.30739403990801e+231;
        bool r709280 = r709278 <= r709279;
        double r709281 = 1.1043654858405285e+270;
        bool r709282 = r709278 <= r709281;
        double r709283 = !r709282;
        bool r709284 = r709280 || r709283;
        double r709285 = 2.0;
        double r709286 = x;
        double r709287 = sqrt(r709286);
        double r709288 = r709285 * r709287;
        double r709289 = 2.0;
        double r709290 = log(r709289);
        double r709291 = 0.25;
        double r709292 = y;
        double r709293 = pow(r709292, r709289);
        double r709294 = r709291 * r709293;
        double r709295 = r709290 - r709294;
        double r709296 = expm1(r709295);
        double r709297 = r709288 * r709296;
        double r709298 = a;
        double r709299 = b;
        double r709300 = 3.0;
        double r709301 = r709299 * r709300;
        double r709302 = r709298 / r709301;
        double r709303 = r709297 - r709302;
        double r709304 = cbrt(r709300);
        double r709305 = r709304 * r709304;
        double r709306 = r709276 / r709305;
        double r709307 = 1.0;
        double r709308 = r709304 / r709277;
        double r709309 = r709307 / r709308;
        double r709310 = r709306 * r709309;
        double r709311 = r709292 - r709310;
        double r709312 = cos(r709311);
        double r709313 = r709288 * r709312;
        double r709314 = r709313 - r709302;
        double r709315 = r709284 ? r709303 : r709314;
        return r709315;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original20.8
Target18.9
Herbie17.1
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z t) < -9.30739403990801e+231 or 1.1043654858405285e+270 < (* z t)

    1. Initial program 56.2

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

    3. Applied add-cube-cbrt56.2

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

    4. Applied times-frac56.1

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

    6. Applied expm1-log1p-u56.1

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{t}{\sqrt[3]{3}}\right)\right)\right)} - \frac{a}{b \cdot 3}\]

    7. Taylor expanded around 0 33.6

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \mathsf{expm1}\left(\color{blue}{\log 2 - \frac{1}{4} \cdot {y}^{2}}\right) - \frac{a}{b \cdot 3}\]

    if -9.30739403990801e+231 < (* z t) < 1.1043654858405285e+270

    1. Initial program 13.7

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

    3. Applied add-cube-cbrt13.7

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

    4. Applied times-frac13.7

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

    6. Applied clear-num13.7

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

  4. Final simplification17.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \le -9.3073940399080106 \cdot 10^{231} \lor \neg \left(z \cdot t \le 1.104365485840529 \cdot 10^{270}\right):\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \mathsf{expm1}\left(\log 2 - \frac{1}{4} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt[3]{3} \cdot \sqrt[3]{3}} \cdot \frac{1}{\frac{\sqrt[3]{3}}{t}}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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))))