Average Error: 46.6 → 44.6
Time: 13.2s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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 \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r996957 = x;
        double r996958 = y;
        double r996959 = 2.0;
        double r996960 = r996958 * r996959;
        double r996961 = 1.0;
        double r996962 = r996960 + r996961;
        double r996963 = z;
        double r996964 = r996962 * r996963;
        double r996965 = t;
        double r996966 = r996964 * r996965;
        double r996967 = 16.0;
        double r996968 = r996966 / r996967;
        double r996969 = cos(r996968);
        double r996970 = r996957 * r996969;
        double r996971 = a;
        double r996972 = r996971 * r996959;
        double r996973 = r996972 + r996961;
        double r996974 = b;
        double r996975 = r996973 * r996974;
        double r996976 = r996975 * r996965;
        double r996977 = r996976 / r996967;
        double r996978 = cos(r996977);
        double r996979 = r996970 * r996978;
        return r996979;
}
↓
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 r996980 = x;
        double r996981 = 0.0;
        double r996982 = 16.0;
        double r996983 = r996981 / r996982;
        double r996984 = cos(r996983);
        double r996985 = r996980 * r996984;
        return r996985;
}

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

Original46.6
Target44.8
Herbie44.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.6

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.6

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

  4. Final simplification44.6

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

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(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))))