Average Error: 46.2 → 44.2
Time: 19.4s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{0}{16}\right)\right)\]
\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)
↓
\cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{0}{16}\right)\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r793852 = x;
        double r793853 = y;
        double r793854 = 2.0;
        double r793855 = r793853 * r793854;
        double r793856 = 1.0;
        double r793857 = r793855 + r793856;
        double r793858 = z;
        double r793859 = r793857 * r793858;
        double r793860 = t;
        double r793861 = r793859 * r793860;
        double r793862 = 16.0;
        double r793863 = r793861 / r793862;
        double r793864 = cos(r793863);
        double r793865 = r793852 * r793864;
        double r793866 = a;
        double r793867 = r793866 * r793854;
        double r793868 = r793867 + r793856;
        double r793869 = b;
        double r793870 = r793868 * r793869;
        double r793871 = r793870 * r793860;
        double r793872 = r793871 / r793862;
        double r793873 = cos(r793872);
        double r793874 = r793865 * r793873;
        return r793874;
}
↓
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 r793875 = 0.0;
        double r793876 = 16.0;
        double r793877 = r793875 / r793876;
        double r793878 = cos(r793877);
        double r793879 = x;
        double r793880 = r793879 * r793878;
        double r793881 = r793878 * r793880;
        return r793881;
}

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
Target44.4
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. Simplified46.2

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

  3. Taylor expanded around 0 45.5

    \[\leadsto \cos \left(\frac{\left(\mathsf{fma}\left(a, 2, 1\right) \cdot b\right) \cdot t}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right)\]

  4. Taylor expanded around 0 44.2

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

  5. Final simplification44.2

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

Reproduce

herbie shell --seed 2019351 +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))))