Average Error: 45.7 → 43.8
Time: 34.3s
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 r1288921 = x;
        double r1288922 = y;
        double r1288923 = 2.0;
        double r1288924 = r1288922 * r1288923;
        double r1288925 = 1.0;
        double r1288926 = r1288924 + r1288925;
        double r1288927 = z;
        double r1288928 = r1288926 * r1288927;
        double r1288929 = t;
        double r1288930 = r1288928 * r1288929;
        double r1288931 = 16.0;
        double r1288932 = r1288930 / r1288931;
        double r1288933 = cos(r1288932);
        double r1288934 = r1288921 * r1288933;
        double r1288935 = a;
        double r1288936 = r1288935 * r1288923;
        double r1288937 = r1288936 + r1288925;
        double r1288938 = b;
        double r1288939 = r1288937 * r1288938;
        double r1288940 = r1288939 * r1288929;
        double r1288941 = r1288940 / r1288931;
        double r1288942 = cos(r1288941);
        double r1288943 = r1288934 * r1288942;
        return r1288943;
}
↓
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 r1288944 = x;
        return r1288944;
}

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

Original45.7
Target44.0
Herbie43.8
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.7

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

  2. Simplified45.5

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

  3. Taylor expanded around 0 45.1

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

  4. Taylor expanded around 0 43.8

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

  5. Final simplification43.8

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019179 
(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))))