Average Error: 46.1 → 44.1
Time: 10.7s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r982236 = x;
        double r982237 = y;
        double r982238 = 2.0;
        double r982239 = r982237 * r982238;
        double r982240 = 1.0;
        double r982241 = r982239 + r982240;
        double r982242 = z;
        double r982243 = r982241 * r982242;
        double r982244 = t;
        double r982245 = r982243 * r982244;
        double r982246 = 16.0;
        double r982247 = r982245 / r982246;
        double r982248 = cos(r982247);
        double r982249 = r982236 * r982248;
        double r982250 = a;
        double r982251 = r982250 * r982238;
        double r982252 = r982251 + r982240;
        double r982253 = b;
        double r982254 = r982252 * r982253;
        double r982255 = r982254 * r982244;
        double r982256 = r982255 / r982246;
        double r982257 = cos(r982256);
        double r982258 = r982249 * r982257;
        return r982258;
}
↓
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 r982259 = x;
        return r982259;
}

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
Target44.3
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 45.5

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020060 
(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))))