Average Error: 46.1 → 44.2
Time: 16.6s
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 r1529961 = x;
        double r1529962 = y;
        double r1529963 = 2.0;
        double r1529964 = r1529962 * r1529963;
        double r1529965 = 1.0;
        double r1529966 = r1529964 + r1529965;
        double r1529967 = z;
        double r1529968 = r1529966 * r1529967;
        double r1529969 = t;
        double r1529970 = r1529968 * r1529969;
        double r1529971 = 16.0;
        double r1529972 = r1529970 / r1529971;
        double r1529973 = cos(r1529972);
        double r1529974 = r1529961 * r1529973;
        double r1529975 = a;
        double r1529976 = r1529975 * r1529963;
        double r1529977 = r1529976 + r1529965;
        double r1529978 = b;
        double r1529979 = r1529977 * r1529978;
        double r1529980 = r1529979 * r1529969;
        double r1529981 = r1529980 / r1529971;
        double r1529982 = cos(r1529981);
        double r1529983 = r1529974 * r1529982;
        return r1529983;
}
↓
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 r1529984 = x;
        double r1529985 = 0.0;
        double r1529986 = 16.0;
        double r1529987 = r1529985 / r1529986;
        double r1529988 = cos(r1529987);
        double r1529989 = r1529984 * r1529988;
        return r1529989;
}

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
Target44.5
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 46.1

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

  2. Taylor expanded around 0 45.3

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2020047 
(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))))