Average Error: 20.5 → 18.8
Time: 14.0s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -2.711112522981125:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z}{\sqrt{3}} \cdot \frac{t}{\sqrt{3}}\right) - \frac{a}{b \cdot 3}\\ \mathbf{elif}\;y \le 2.4348734333899417 \cdot 10^{-18}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \cos \left(0.333333333333333315 \cdot \left(t \cdot z\right)\right) + \sin y \cdot \sin \left(\frac{z \cdot t}{3}\right)\right) - \frac{1}{b} \cdot \frac{a}{3}\\ \end{array}\]
\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 r767520 = 2.0;
        double r767521 = x;
        double r767522 = sqrt(r767521);
        double r767523 = r767520 * r767522;
        double r767524 = y;
        double r767525 = z;
        double r767526 = t;
        double r767527 = r767525 * r767526;
        double r767528 = 3.0;
        double r767529 = r767527 / r767528;
        double r767530 = r767524 - r767529;
        double r767531 = cos(r767530);
        double r767532 = r767523 * r767531;
        double r767533 = a;
        double r767534 = b;
        double r767535 = r767534 * r767528;
        double r767536 = r767533 / r767535;
        double r767537 = r767532 - r767536;
        return r767537;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r767538 = y;
        double r767539 = -2.711112522981125;
        bool r767540 = r767538 <= r767539;
        double r767541 = 2.0;
        double r767542 = x;
        double r767543 = sqrt(r767542);
        double r767544 = r767541 * r767543;
        double r767545 = z;
        double r767546 = 3.0;
        double r767547 = sqrt(r767546);
        double r767548 = r767545 / r767547;
        double r767549 = t;
        double r767550 = r767549 / r767547;
        double r767551 = r767548 * r767550;
        double r767552 = r767538 - r767551;
        double r767553 = cos(r767552);
        double r767554 = r767544 * r767553;
        double r767555 = a;
        double r767556 = b;
        double r767557 = r767556 * r767546;
        double r767558 = r767555 / r767557;
        double r767559 = r767554 - r767558;
        double r767560 = 2.4348734333899417e-18;
        bool r767561 = r767538 <= r767560;
        double r767562 = 1.0;
        double r767563 = 0.5;
        double r767564 = 2.0;
        double r767565 = pow(r767538, r767564);
        double r767566 = r767563 * r767565;
        double r767567 = r767562 - r767566;
        double r767568 = r767544 * r767567;
        double r767569 = r767568 - r767558;
        double r767570 = cos(r767538);
        double r767571 = 0.3333333333333333;
        double r767572 = r767549 * r767545;
        double r767573 = r767571 * r767572;
        double r767574 = cos(r767573);
        double r767575 = r767570 * r767574;
        double r767576 = sin(r767538);
        double r767577 = r767545 * r767549;
        double r767578 = r767577 / r767546;
        double r767579 = sin(r767578);
        double r767580 = r767576 * r767579;
        double r767581 = r767575 + r767580;
        double r767582 = r767544 * r767581;
        double r767583 = r767562 / r767556;
        double r767584 = r767555 / r767546;
        double r767585 = r767583 * r767584;
        double r767586 = r767582 - r767585;
        double r767587 = r767561 ? r767569 : r767586;
        double r767588 = r767540 ? r767559 : r767587;
        return r767588;
}

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))))