Average Error: 20.8 → 17.9
Time: 14.4s
Precision: 64
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}\]
↓
\[\begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin \left(\frac{z \cdot t}{3}\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]
\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 0.999999999481199109:\\
\;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin \left(\frac{z \cdot t}{3}\right) \cdot \sin y\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 r760644 = 2.0;
        double r760645 = x;
        double r760646 = sqrt(r760645);
        double r760647 = r760644 * r760646;
        double r760648 = y;
        double r760649 = z;
        double r760650 = t;
        double r760651 = r760649 * r760650;
        double r760652 = 3.0;
        double r760653 = r760651 / r760652;
        double r760654 = r760648 - r760653;
        double r760655 = cos(r760654);
        double r760656 = r760647 * r760655;
        double r760657 = a;
        double r760658 = b;
        double r760659 = r760658 * r760652;
        double r760660 = r760657 / r760659;
        double r760661 = r760656 - r760660;
        return r760661;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r760662 = y;
        double r760663 = z;
        double r760664 = t;
        double r760665 = r760663 * r760664;
        double r760666 = 3.0;
        double r760667 = r760665 / r760666;
        double r760668 = r760662 - r760667;
        double r760669 = cos(r760668);
        double r760670 = 0.9999999994811991;
        bool r760671 = r760669 <= r760670;
        double r760672 = 2.0;
        double r760673 = x;
        double r760674 = sqrt(r760673);
        double r760675 = r760672 * r760674;
        double r760676 = cos(r760662);
        double r760677 = cos(r760667);
        double r760678 = 3.0;
        double r760679 = pow(r760677, r760678);
        double r760680 = cbrt(r760679);
        double r760681 = cbrt(r760680);
        double r760682 = r760681 * r760681;
        double r760683 = cbrt(r760677);
        double r760684 = r760682 * r760683;
        double r760685 = r760676 * r760684;
        double r760686 = sin(r760667);
        double r760687 = sin(r760662);
        double r760688 = r760686 * r760687;
        double r760689 = r760685 + r760688;
        double r760690 = r760675 * r760689;
        double r760691 = a;
        double r760692 = b;
        double r760693 = r760692 * r760666;
        double r760694 = r760691 / r760693;
        double r760695 = r760690 - r760694;
        double r760696 = 1.0;
        double r760697 = 0.5;
        double r760698 = 2.0;
        double r760699 = pow(r760662, r760698);
        double r760700 = r760697 * r760699;
        double r760701 = r760696 - r760700;
        double r760702 = r760675 * r760701;
        double r760703 = r760702 - r760694;
        double r760704 = r760671 ? r760695 : r760703;
        return r760704;
}

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

Original20.8
Target18.6
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))) < 0.9999999994811991

    1. Initial program 20.3

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

    3. Applied cos-diff19.5

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

    4. Simplified19.5

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

    6. Applied add-cube-cbrt19.5

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

    8. Applied add-cbrt-cube19.5

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

    9. Simplified19.5

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

    11. Applied add-cbrt-cube19.5

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

    12. Simplified19.5

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

    if 0.9999999994811991 < (cos (- y (/ (* z t) 3.0)))

    1. Initial program 21.7

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

    2. Taylor expanded around 0 15.2

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

  4. Final simplification17.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(y - \frac{z \cdot t}{3}\right) \le 0.999999999481199109:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(\cos y \cdot \left(\left(\sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(\cos \left(\frac{z \cdot t}{3}\right)\right)}^{3}}}\right) \cdot \sqrt[3]{\cos \left(\frac{z \cdot t}{3}\right)}\right) + \sin \left(\frac{z \cdot t}{3}\right) \cdot \sin y\right) - \frac{a}{b \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \left(1 - \frac{1}{2} \cdot {y}^{2}\right) - \frac{a}{b \cdot 3}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))