Average Error: 46.3 → 44.2
Time: 11.2s
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 r1052851 = x;
        double r1052852 = y;
        double r1052853 = 2.0;
        double r1052854 = r1052852 * r1052853;
        double r1052855 = 1.0;
        double r1052856 = r1052854 + r1052855;
        double r1052857 = z;
        double r1052858 = r1052856 * r1052857;
        double r1052859 = t;
        double r1052860 = r1052858 * r1052859;
        double r1052861 = 16.0;
        double r1052862 = r1052860 / r1052861;
        double r1052863 = cos(r1052862);
        double r1052864 = r1052851 * r1052863;
        double r1052865 = a;
        double r1052866 = r1052865 * r1052853;
        double r1052867 = r1052866 + r1052855;
        double r1052868 = b;
        double r1052869 = r1052867 * r1052868;
        double r1052870 = r1052869 * r1052859;
        double r1052871 = r1052870 / r1052861;
        double r1052872 = cos(r1052871);
        double r1052873 = r1052864 * r1052872;
        return r1052873;
}
↓
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 r1052874 = x;
        double r1052875 = 0.0;
        double r1052876 = 16.0;
        double r1052877 = r1052875 / r1052876;
        double r1052878 = cos(r1052877);
        double r1052879 = r1052874 * r1052878;
        return r1052879;
}

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

Original46.3
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 46.3

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

  2. Taylor expanded around 0 45.5

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2019353 
(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))))