Average Error: 46.7 → 45.2
Time: 11.7s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -1.1779311528106351 \cdot 10^{-256}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}{16}\right)\\ \mathbf{elif}\;z \le 2.21843112193388165 \cdot 10^{139}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)
↓
\begin{array}{l}
\mathbf{if}\;z \le -1.1779311528106351 \cdot 10^{-256}:\\
\;\;\;\;\left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}{16}\right)\\

\mathbf{elif}\;z \le 2.21843112193388165 \cdot 10^{139}:\\
\;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\\

\mathbf{else}:\\
\;\;\;\;x\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r1074690 = x;
        double r1074691 = y;
        double r1074692 = 2.0;
        double r1074693 = r1074691 * r1074692;
        double r1074694 = 1.0;
        double r1074695 = r1074693 + r1074694;
        double r1074696 = z;
        double r1074697 = r1074695 * r1074696;
        double r1074698 = t;
        double r1074699 = r1074697 * r1074698;
        double r1074700 = 16.0;
        double r1074701 = r1074699 / r1074700;
        double r1074702 = cos(r1074701);
        double r1074703 = r1074690 * r1074702;
        double r1074704 = a;
        double r1074705 = r1074704 * r1074692;
        double r1074706 = r1074705 + r1074694;
        double r1074707 = b;
        double r1074708 = r1074706 * r1074707;
        double r1074709 = r1074708 * r1074698;
        double r1074710 = r1074709 / r1074700;
        double r1074711 = cos(r1074710);
        double r1074712 = r1074703 * r1074711;
        return r1074712;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r1074713 = z;
        double r1074714 = -1.1779311528106351e-256;
        bool r1074715 = r1074713 <= r1074714;
        double r1074716 = x;
        double r1074717 = 1.0;
        double r1074718 = r1074716 * r1074717;
        double r1074719 = a;
        double r1074720 = 2.0;
        double r1074721 = r1074719 * r1074720;
        double r1074722 = 1.0;
        double r1074723 = r1074721 + r1074722;
        double r1074724 = cbrt(r1074723);
        double r1074725 = r1074724 * r1074724;
        double r1074726 = b;
        double r1074727 = t;
        double r1074728 = r1074726 * r1074727;
        double r1074729 = r1074724 * r1074728;
        double r1074730 = r1074725 * r1074729;
        double r1074731 = 16.0;
        double r1074732 = r1074730 / r1074731;
        double r1074733 = cos(r1074732);
        double r1074734 = r1074718 * r1074733;
        double r1074735 = 2.2184311219338817e+139;
        bool r1074736 = r1074713 <= r1074735;
        double r1074737 = y;
        double r1074738 = r1074737 * r1074720;
        double r1074739 = r1074738 + r1074722;
        double r1074740 = r1074739 * r1074713;
        double r1074741 = r1074740 * r1074727;
        double r1074742 = r1074741 / r1074731;
        double r1074743 = cos(r1074742);
        double r1074744 = r1074716 * r1074743;
        double r1074745 = 0.0;
        double r1074746 = r1074745 / r1074731;
        double r1074747 = cos(r1074746);
        double r1074748 = r1074744 * r1074747;
        double r1074749 = r1074736 ? r1074748 : r1074716;
        double r1074750 = r1074715 ? r1074734 : r1074749;
        return r1074750;
}

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

Original46.7
Target45.0
Herbie45.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Split input into 3 regimes

  2. if z < -1.1779311528106351e-256

    1. Initial program 47.1

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    2. Taylor expanded around 0 46.3

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
    3. Using strategy rm

    4. Applied associate-*l*46.0

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\color{blue}{\left(a \cdot 2 + 1\right) \cdot \left(b \cdot t\right)}}{16}\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt46.0

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\color{blue}{\left(\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \sqrt[3]{a \cdot 2 + 1}\right)} \cdot \left(b \cdot t\right)}{16}\right)\]

    7. Applied associate-*l*46.0

      \[\leadsto \left(x \cdot 1\right) \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}}{16}\right)\]

    if -1.1779311528106351e-256 < z < 2.2184311219338817e+139

    1. Initial program 42.3

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    2. Taylor expanded around 0 41.5

      \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\]

    if 2.2184311219338817e+139 < z

    1. Initial program 57.8

      \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    2. Taylor expanded around 0 55.4

      \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

    3. Taylor expanded around 0 53.7

      \[\leadsto \color{blue}{x}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification45.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.1779311528106351 \cdot 10^{-256}:\\ \;\;\;\;\left(x \cdot 1\right) \cdot \cos \left(\frac{\left(\sqrt[3]{a \cdot 2 + 1} \cdot \sqrt[3]{a \cdot 2 + 1}\right) \cdot \left(\sqrt[3]{a \cdot 2 + 1} \cdot \left(b \cdot t\right)\right)}{16}\right)\\ \mathbf{elif}\;z \le 2.21843112193388165 \cdot 10^{139}:\\ \;\;\;\;\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* x (cos (* (/ b 16) (/ t (+ (- 1 (* a 2)) (pow (* a 2) 2))))))

  (* (* x (cos (/ (* (* (+ (* y 2) 1) z) t) 16))) (cos (/ (* (* (+ (* a 2) 1) b) t) 16))))