Average Error: 46.2 → 44.2
Time: 11.7s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot x\]
\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 r1057892 = x;
        double r1057893 = y;
        double r1057894 = 2.0;
        double r1057895 = r1057893 * r1057894;
        double r1057896 = 1.0;
        double r1057897 = r1057895 + r1057896;
        double r1057898 = z;
        double r1057899 = r1057897 * r1057898;
        double r1057900 = t;
        double r1057901 = r1057899 * r1057900;
        double r1057902 = 16.0;
        double r1057903 = r1057901 / r1057902;
        double r1057904 = cos(r1057903);
        double r1057905 = r1057892 * r1057904;
        double r1057906 = a;
        double r1057907 = r1057906 * r1057894;
        double r1057908 = r1057907 + r1057896;
        double r1057909 = b;
        double r1057910 = r1057908 * r1057909;
        double r1057911 = r1057910 * r1057900;
        double r1057912 = r1057911 / r1057902;
        double r1057913 = cos(r1057912);
        double r1057914 = r1057905 * r1057913;
        return r1057914;
}
↓
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 r1057915 = 0.0;
        double r1057916 = 16.0;
        double r1057917 = r1057915 / r1057916;
        double r1057918 = cos(r1057917);
        double r1057919 = x;
        double r1057920 = r1057918 * r1057919;
        return r1057920;
}

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

Original46.2
Target44.5
Herbie44.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.2

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

  2. Taylor expanded around 0 45.6

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2020100 
(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))))