Average Error: 45.8 → 43.7
Time: 39.9s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -4.114840475589630641652677177845850777519 \cdot 10^{-283} \lor \neg \left(t \le 1067984773110579986432\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{t}{16} \cdot b\right)\right) \cdot \left(x \cdot \cos \left(\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\frac{16}{1 + 2 \cdot y}}}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\frac{16}{1 + 2 \cdot y}} \cdot \sqrt[3]{\frac{16}{1 + 2 \cdot y}}}\right)\right)\right)\right)\\ \end{array}\]
\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}\;t \le -4.114840475589630641652677177845850777519 \cdot 10^{-283} \lor \neg \left(t \le 1067984773110579986432\right):\\
\;\;\;\;x\\

\mathbf{else}:\\
\;\;\;\;\cos \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{t}{16} \cdot b\right)\right) \cdot \left(x \cdot \cos \left(\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\frac{16}{1 + 2 \cdot y}}}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\frac{16}{1 + 2 \cdot y}} \cdot \sqrt[3]{\frac{16}{1 + 2 \cdot y}}}\right)\right)\right)\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r696080 = x;
        double r696081 = y;
        double r696082 = 2.0;
        double r696083 = r696081 * r696082;
        double r696084 = 1.0;
        double r696085 = r696083 + r696084;
        double r696086 = z;
        double r696087 = r696085 * r696086;
        double r696088 = t;
        double r696089 = r696087 * r696088;
        double r696090 = 16.0;
        double r696091 = r696089 / r696090;
        double r696092 = cos(r696091);
        double r696093 = r696080 * r696092;
        double r696094 = a;
        double r696095 = r696094 * r696082;
        double r696096 = r696095 + r696084;
        double r696097 = b;
        double r696098 = r696096 * r696097;
        double r696099 = r696098 * r696088;
        double r696100 = r696099 / r696090;
        double r696101 = cos(r696100);
        double r696102 = r696093 * r696101;
        return r696102;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r696103 = t;
        double r696104 = -4.1148404755896306e-283;
        bool r696105 = r696103 <= r696104;
        double r696106 = 1.06798477311058e+21;
        bool r696107 = r696103 <= r696106;
        double r696108 = !r696107;
        bool r696109 = r696105 || r696108;
        double r696110 = x;
        double r696111 = 1.0;
        double r696112 = 2.0;
        double r696113 = a;
        double r696114 = r696112 * r696113;
        double r696115 = r696111 + r696114;
        double r696116 = 16.0;
        double r696117 = r696103 / r696116;
        double r696118 = b;
        double r696119 = r696117 * r696118;
        double r696120 = r696115 * r696119;
        double r696121 = cos(r696120);
        double r696122 = cbrt(r696103);
        double r696123 = y;
        double r696124 = r696112 * r696123;
        double r696125 = r696111 + r696124;
        double r696126 = r696116 / r696125;
        double r696127 = cbrt(r696126);
        double r696128 = z;
        double r696129 = cbrt(r696128);
        double r696130 = r696127 / r696129;
        double r696131 = r696122 / r696130;
        double r696132 = r696122 * r696122;
        double r696133 = r696127 * r696127;
        double r696134 = r696132 / r696133;
        double r696135 = r696129 * r696134;
        double r696136 = r696129 * r696135;
        double r696137 = r696131 * r696136;
        double r696138 = cos(r696137);
        double r696139 = r696110 * r696138;
        double r696140 = r696121 * r696139;
        double r696141 = r696109 ? r696110 : r696140;
        return r696141;
}

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

Original45.8
Target44.1
Herbie43.7
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if t < -4.1148404755896306e-283 or 1.06798477311058e+21 < t

    1. Initial program 51.0

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

    2. Simplified50.8

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

    3. Taylor expanded around 0 50.0

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

    4. Taylor expanded around 0 48.5

      \[\leadsto \left(\cos 0 \cdot x\right) \cdot \color{blue}{1}\]

    if -4.1148404755896306e-283 < t < 1.06798477311058e+21

    1. Initial program 33.0

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

    2. Simplified32.5

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

    4. Applied add-cube-cbrt32.5

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

    5. Applied add-cube-cbrt32.5

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

    6. Applied times-frac32.5

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

    7. Applied add-cube-cbrt32.5

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

    8. Applied times-frac32.0

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

    9. Simplified32.1

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

    10. Simplified32.1

      \[\leadsto \left(\cos \left(\left(\left(\frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\frac{16}{y \cdot 2 + 1}} \cdot \sqrt[3]{\frac{16}{y \cdot 2 + 1}}} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right) \cdot \color{blue}{\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\frac{16}{y \cdot 2 + 1}}}{\sqrt[3]{z}}}}\right) \cdot x\right) \cdot \cos \left(\left(\frac{t}{16} \cdot b\right) \cdot \left(1 + 2 \cdot a\right)\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification43.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -4.114840475589630641652677177845850777519 \cdot 10^{-283} \lor \neg \left(t \le 1067984773110579986432\right):\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(1 + 2 \cdot a\right) \cdot \left(\frac{t}{16} \cdot b\right)\right) \cdot \left(x \cdot \cos \left(\frac{\sqrt[3]{t}}{\frac{\sqrt[3]{\frac{16}{1 + 2 \cdot y}}}{\sqrt[3]{z}}} \cdot \left(\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \frac{\sqrt[3]{t} \cdot \sqrt[3]{t}}{\sqrt[3]{\frac{16}{1 + 2 \cdot y}} \cdot \sqrt[3]{\frac{16}{1 + 2 \cdot y}}}\right)\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))