Average Error: 20.5 → 18.8
Time: 12.7s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.711112522981125:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array}\]
\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}\;y \le -2.711112522981125:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\

\mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r846105 = 2.0;
        double r846106 = x;
        double r846107 = sqrt(r846106);
        double r846108 = r846105 * r846107;
        double r846109 = y;
        double r846110 = z;
        double r846111 = t;
        double r846112 = r846110 * r846111;
        double r846113 = 3.0;
        double r846114 = r846112 / r846113;
        double r846115 = r846109 - r846114;
        double r846116 = cos(r846115);
        double r846117 = r846108 * r846116;
        double r846118 = a;
        double r846119 = b;
        double r846120 = r846119 * r846113;
        double r846121 = r846118 / r846120;
        double r846122 = r846117 - r846121;
        return r846122;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r846123 = y;
        double r846124 = -2.711112522981125;
        bool r846125 = r846123 <= r846124;
        double r846126 = 2.0;
        double r846127 = x;
        double r846128 = sqrt(r846127);
        double r846129 = r846126 * r846128;
        double r846130 = z;
        double r846131 = 3.0;
        double r846132 = sqrt(r846131);
        double r846133 = r846130 / r846132;
        double r846134 = t;
        double r846135 = r846134 / r846132;
        double r846136 = r846133 * r846135;
        double r846137 = r846123 - r846136;
        double r846138 = cos(r846137);
        double r846139 = r846129 * r846138;
        double r846140 = a;
        double r846141 = b;
        double r846142 = r846141 * r846131;
        double r846143 = r846140 / r846142;
        double r846144 = r846139 - r846143;
        double r846145 = 2.4348734333899417e-18;
        bool r846146 = r846123 <= r846145;
        double r846147 = 1.0;
        double r846148 = 0.5;
        double r846149 = 2.0;
        double r846150 = pow(r846123, r846149);
        double r846151 = r846148 * r846150;
        double r846152 = r846147 - r846151;
        double r846153 = r846129 * r846152;
        double r846154 = r846153 - r846143;
        double r846155 = cos(r846123);
        double r846156 = 0.3333333333333333;
        double r846157 = r846134 * r846130;
        double r846158 = r846156 * r846157;
        double r846159 = cos(r846158);
        double r846160 = r846155 * r846159;
        double r846161 = sin(r846123);
        double r846162 = r846130 * r846134;
        double r846163 = r846162 / r846131;
        double r846164 = sin(r846163);
        double r846165 = r846161 * r846164;
        double r846166 = r846160 + r846165;
        double r846167 = r846129 * r846166;
        double r846168 = r846147 / r846141;
        double r846169 = r846140 / r846131;
        double r846170 = r846168 * r846169;
        double r846171 = r846167 - r846170;
        double r846172 = r846146 ? r846154 : r846171;
        double r846173 = r846125 ? r846144 : r846172;
        return r846173;
}

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

Original20.5
Target18.5
Herbie18.8
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.333333333333333315}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.51629061355598715 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.333333333333333315}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -2.711112522981125

    1. Initial program 21.1

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

    3. Applied add-sqr-sqrt21.0

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

    4. Applied times-frac21.1

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

    if -2.711112522981125 < y < 2.4348734333899417e-18

    1. Initial program 19.6

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

    2. Taylor expanded around 0 16.6

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

    if 2.4348734333899417e-18 < y

    1. Initial program 21.5

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

    3. Applied cos-diff20.7

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

    4. Taylor expanded around inf 20.8

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

    6. Applied *-un-lft-identity20.8

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{\color{blue}{1 \cdot a}}{b \cdot 3}\]

    7. Applied times-frac20.8

      \[\leadsto \left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \color{blue}{\frac{1}{b} \cdot \frac{a}{3}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification18.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -2.711112522981125:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))