Average Error: 20.9 → 17.9
Time: 21.4s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999998507955556:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) + \sin y \cdot \sin \left({\left(\sqrt[3]{\frac{z \cdot t}{3}}\right)}^{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.99999998507955556:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) + \sin y \cdot \sin \left({\left(\sqrt[3]{\frac{z \cdot t}{3}}\right)}^{3}\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 r843154 = 2.0;
        double r843155 = x;
        double r843156 = sqrt(r843155);
        double r843157 = r843154 * r843156;
        double r843158 = y;
        double r843159 = z;
        double r843160 = t;
        double r843161 = r843159 * r843160;
        double r843162 = 3.0;
        double r843163 = r843161 / r843162;
        double r843164 = r843158 - r843163;
        double r843165 = cos(r843164);
        double r843166 = r843157 * r843165;
        double r843167 = a;
        double r843168 = b;
        double r843169 = r843168 * r843162;
        double r843170 = r843167 / r843169;
        double r843171 = r843166 - r843170;
        return r843171;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r843172 = y;
        double r843173 = z;
        double r843174 = t;
        double r843175 = r843173 * r843174;
        double r843176 = 3.0;
        double r843177 = r843175 / r843176;
        double r843178 = r843172 - r843177;
        double r843179 = cos(r843178);
        double r843180 = 0.9999999850795556;
        bool r843181 = r843179 <= r843180;
        double r843182 = 2.0;
        double r843183 = x;
        double r843184 = sqrt(r843183);
        double r843185 = r843182 * r843184;
        double r843186 = cos(r843172);
        double r843187 = cbrt(r843177);
        double r843188 = r843187 * r843187;
        double r843189 = r843188 * r843187;
        double r843190 = cos(r843189);
        double r843191 = r843186 * r843190;
        double r843192 = sin(r843172);
        double r843193 = 3.0;
        double r843194 = pow(r843187, r843193);
        double r843195 = sin(r843194);
        double r843196 = r843192 * r843195;
        double r843197 = r843191 + r843196;
        double r843198 = r843185 * r843197;
        double r843199 = a;
        double r843200 = b;
        double r843201 = r843200 * r843176;
        double r843202 = r843199 / r843201;
        double r843203 = r843198 - r843202;
        double r843204 = 1.0;
        double r843205 = 0.5;
        double r843206 = 2.0;
        double r843207 = pow(r843172, r843206);
        double r843208 = r843205 * r843207;
        double r843209 = r843204 - r843208;
        double r843210 = r843185 * r843209;
        double r843211 = r843210 - r843202;
        double r843212 = r843181 ? r843203 : r843211;
        return r843212;
}

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

Original20.9
Target18.7
Herbie17.9
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999850795556

    1. Initial program 20.2

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

    3. Applied add-cube-cbrt20.2

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

    5. Applied cos-diff19.5

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

    6. Simplified19.5

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

    if 0.9999999850795556 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.0

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

    2. Taylor expanded around 0 15.3

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.99999998507955556:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(\left(\sqrt[3]{\frac{z \cdot t}{3}} \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) \cdot \sqrt[3]{\frac{z \cdot t}{3}}\right) + \sin y \cdot \sin \left({\left(\sqrt[3]{\frac{z \cdot t}{3}}\right)}^{3}\right)\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 +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))))