Average Error: 20.6 → 17.7
Time: 31.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{t \cdot z}{3}\right) \le 0.9999999999999017452623206736461725085974:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \left(\sqrt[3]{\sin \left(\frac{t \cdot z}{3}\right) \cdot \left(\sin \left(\frac{t \cdot z}{3}\right) \cdot \sin \left(\frac{t \cdot z}{3}\right)\right)} \cdot \sin y + \cos y \cdot \sqrt[3]{\cos \left(\frac{t \cdot z}{3}\right) \cdot \left(\cos \left(\frac{t \cdot z}{3}\right) \cdot \cos \left(\frac{t \cdot z}{3}\right)\right)}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \frac{1}{2} \cdot \left(y \cdot y\right)\right) \cdot \left(\sqrt{x} \cdot 2\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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(\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)} \cdot \sin y + \cos y \cdot \sqrt[3]{\cos \left(\frac{t \cdot z}{3}\right) \cdot \left(\cos \left(\frac{t \cdot z}{3}\right) \cdot \cos \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 r37878385 = 2.0;
        double r37878386 = x;
        double r37878387 = sqrt(r37878386);
        double r37878388 = r37878385 * r37878387;
        double r37878389 = y;
        double r37878390 = z;
        double r37878391 = t;
        double r37878392 = r37878390 * r37878391;
        double r37878393 = 3.0;
        double r37878394 = r37878392 / r37878393;
        double r37878395 = r37878389 - r37878394;
        double r37878396 = cos(r37878395);
        double r37878397 = r37878388 * r37878396;
        double r37878398 = a;
        double r37878399 = b;
        double r37878400 = r37878399 * r37878393;
        double r37878401 = r37878398 / r37878400;
        double r37878402 = r37878397 - r37878401;
        return r37878402;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r37878403 = y;
        double r37878404 = t;
        double r37878405 = z;
        double r37878406 = r37878404 * r37878405;
        double r37878407 = 3.0;
        double r37878408 = r37878406 / r37878407;
        double r37878409 = r37878403 - r37878408;
        double r37878410 = cos(r37878409);
        double r37878411 = 0.9999999999999017;
        bool r37878412 = r37878410 <= r37878411;
        double r37878413 = x;
        double r37878414 = sqrt(r37878413);
        double r37878415 = 2.0;
        double r37878416 = r37878414 * r37878415;
        double r37878417 = sin(r37878408);
        double r37878418 = r37878417 * r37878417;
        double r37878419 = r37878417 * r37878418;
        double r37878420 = cbrt(r37878419);
        double r37878421 = sin(r37878403);
        double r37878422 = r37878420 * r37878421;
        double r37878423 = cos(r37878403);
        double r37878424 = cos(r37878408);
        double r37878425 = r37878424 * r37878424;
        double r37878426 = r37878424 * r37878425;
        double r37878427 = cbrt(r37878426);
        double r37878428 = r37878423 * r37878427;
        double r37878429 = r37878422 + r37878428;
        double r37878430 = r37878416 * r37878429;
        double r37878431 = a;
        double r37878432 = b;
        double r37878433 = r37878432 * r37878407;
        double r37878434 = r37878431 / r37878433;
        double r37878435 = r37878430 - r37878434;
        double r37878436 = 1.0;
        double r37878437 = 0.5;
        double r37878438 = r37878403 * r37878403;
        double r37878439 = r37878437 * r37878438;
        double r37878440 = r37878436 - r37878439;
        double r37878441 = r37878440 * r37878416;
        double r37878442 = r37878441 - r37878434;
        double r37878443 = r37878412 ? r37878435 : r37878442;
        return r37878443;
}

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-cbrt-cube18.5

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

    7. Applied add-cbrt-cube18.5

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

    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

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

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))))