Average Error: 46.1 → 44.1
Time: 14.0s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot x\]
\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)
↓
\cos \left(\frac{0}{16}\right) \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r826169 = x;
        double r826170 = y;
        double r826171 = 2.0;
        double r826172 = r826170 * r826171;
        double r826173 = 1.0;
        double r826174 = r826172 + r826173;
        double r826175 = z;
        double r826176 = r826174 * r826175;
        double r826177 = t;
        double r826178 = r826176 * r826177;
        double r826179 = 16.0;
        double r826180 = r826178 / r826179;
        double r826181 = cos(r826180);
        double r826182 = r826169 * r826181;
        double r826183 = a;
        double r826184 = r826183 * r826171;
        double r826185 = r826184 + r826173;
        double r826186 = b;
        double r826187 = r826185 * r826186;
        double r826188 = r826187 * r826177;
        double r826189 = r826188 / r826179;
        double r826190 = cos(r826189);
        double r826191 = r826182 * r826190;
        return r826191;
}
↓
double f(double x, double __attribute__((unused)) y, double __attribute__((unused)) z, double __attribute__((unused)) t, double __attribute__((unused)) a, double __attribute__((unused)) b) {
        double r826192 = 0.0;
        double r826193 = 16.0;
        double r826194 = r826192 / r826193;
        double r826195 = cos(r826194);
        double r826196 = x;
        double r826197 = r826195 * r826196;
        return r826197;
}

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

Original46.1
Target44.4
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.4

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

  3. Taylor expanded around 0 44.1

    \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{0}{16}\right)\]

  4. Final simplification44.1

    \[\leadsto \cos \left(\frac{0}{16}\right) \cdot x\]

Reproduce

herbie shell --seed 2020081 
(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))))