Average Error: 20.6 → 17.7
Time: 31.8s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999999999999017452623206736461725085974:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(\frac{t \cdot z}{3}\right)}\right) + \sin y \cdot \sqrt[3]{\sin \left(\frac{t \cdot z}{3}\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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{t \cdot z}{3}\right) \le 0.9999999999999017452623206736461725085974:\\
\;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(\frac{t \cdot z}{3}\right)}\right) + \sin y \cdot \sqrt[3]{\sin \left(\frac{t \cdot z}{3}\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\

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

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r36783182 = 2.0;
        double r36783183 = x;
        double r36783184 = sqrt(r36783183);
        double r36783185 = r36783182 * r36783184;
        double r36783186 = y;
        double r36783187 = z;
        double r36783188 = t;
        double r36783189 = r36783187 * r36783188;
        double r36783190 = 3.0;
        double r36783191 = r36783189 / r36783190;
        double r36783192 = r36783186 - r36783191;
        double r36783193 = cos(r36783192);
        double r36783194 = r36783185 * r36783193;
        double r36783195 = a;
        double r36783196 = b;
        double r36783197 = r36783196 * r36783190;
        double r36783198 = r36783195 / r36783197;
        double r36783199 = r36783194 - r36783198;
        return r36783199;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r36783200 = y;
        double r36783201 = t;
        double r36783202 = z;
        double r36783203 = r36783201 * r36783202;
        double r36783204 = 3.0;
        double r36783205 = r36783203 / r36783204;
        double r36783206 = r36783200 - r36783205;
        double r36783207 = cos(r36783206);
        double r36783208 = 0.9999999999999017;
        bool r36783209 = r36783207 <= r36783208;
        double r36783210 = x;
        double r36783211 = sqrt(r36783210);
        double r36783212 = 2.0;
        double r36783213 = r36783211 * r36783212;
        double r36783214 = cos(r36783200);
        double r36783215 = cos(r36783205);
        double r36783216 = exp(r36783215);
        double r36783217 = log(r36783216);
        double r36783218 = r36783214 * r36783217;
        double r36783219 = sin(r36783200);
        double r36783220 = sin(r36783205);
        double r36783221 = r36783220 * r36783220;
        double r36783222 = r36783220 * r36783221;
        double r36783223 = cbrt(r36783222);
        double r36783224 = r36783219 * r36783223;
        double r36783225 = r36783218 + r36783224;
        double r36783226 = r36783213 * r36783225;
        double r36783227 = a;
        double r36783228 = b;
        double r36783229 = r36783228 * r36783204;
        double r36783230 = r36783227 / r36783229;
        double r36783231 = r36783226 - r36783230;
        double r36783232 = 1.0;
        double r36783233 = 0.5;
        double r36783234 = r36783200 * r36783200;
        double r36783235 = r36783233 * r36783234;
        double r36783236 = r36783232 - r36783235;
        double r36783237 = r36783236 * r36783213;
        double r36783238 = r36783237 - r36783230;
        double r36783239 = r36783209 ? r36783231 : r36783238;
        return r36783239;
}

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

Original20.6
Target18.3
Herbie17.7
\[\begin{array}{l} \mathbf{if}\;z \lt -1.379333748723514136852843173740882251575 \cdot 10^{129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z \lt 3.516290613555987147199887107423758623887 \cdot 10^{106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333148296162562473909929395}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (cos (- y (/ (* z t) 3.0))) < 0.9999999999999017

    1. Initial program 19.2

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

    3. Applied cos-diff18.5

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

    5. Applied add-log-exp18.5

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

    7. Applied add-cbrt-cube18.5

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

    if 0.9999999999999017 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 22.8

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

    2. Taylor expanded around 0 16.3

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

    3. Simplified16.3

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

  4. Final simplification17.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999999999999017452623206736461725085974:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\cos y \cdot \log \left(e^{\cos \left(\frac{t \cdot z}{3}\right)}\right) + \sin y \cdot \sqrt[3]{\sin \left(\frac{t \cdot z}{3}\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))