Average Error: 46.5 → 44.4
Time: 25.5s
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 r559877 = x;
        double r559878 = y;
        double r559879 = 2.0;
        double r559880 = r559878 * r559879;
        double r559881 = 1.0;
        double r559882 = r559880 + r559881;
        double r559883 = z;
        double r559884 = r559882 * r559883;
        double r559885 = t;
        double r559886 = r559884 * r559885;
        double r559887 = 16.0;
        double r559888 = r559886 / r559887;
        double r559889 = cos(r559888);
        double r559890 = r559877 * r559889;
        double r559891 = a;
        double r559892 = r559891 * r559879;
        double r559893 = r559892 + r559881;
        double r559894 = b;
        double r559895 = r559893 * r559894;
        double r559896 = r559895 * r559885;
        double r559897 = r559896 / r559887;
        double r559898 = cos(r559897);
        double r559899 = r559890 * r559898;
        return r559899;
}
↓
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 r559900 = x;
        return r559900;
}

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

Original46.5
Target44.7
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.5

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

  2. Taylor expanded around 0 45.8

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\]

  3. Taylor expanded around 0 44.4

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

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019323 
(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))))