Average Error: 46.0 → 44.0
Time: 1.1m
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 r138984779 = x;
        double r138984780 = y;
        double r138984781 = 2.0;
        double r138984782 = r138984780 * r138984781;
        double r138984783 = 1.0;
        double r138984784 = r138984782 + r138984783;
        double r138984785 = z;
        double r138984786 = r138984784 * r138984785;
        double r138984787 = t;
        double r138984788 = r138984786 * r138984787;
        double r138984789 = 16.0;
        double r138984790 = r138984788 / r138984789;
        double r138984791 = cos(r138984790);
        double r138984792 = r138984779 * r138984791;
        double r138984793 = a;
        double r138984794 = r138984793 * r138984781;
        double r138984795 = r138984794 + r138984783;
        double r138984796 = b;
        double r138984797 = r138984795 * r138984796;
        double r138984798 = r138984797 * r138984787;
        double r138984799 = r138984798 / r138984789;
        double r138984800 = cos(r138984799);
        double r138984801 = r138984792 * r138984800;
        return r138984801;
}
↓
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 r138984802 = x;
        return r138984802;
}

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

Original46.0
Target44.2
Herbie44.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

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

  2. Taylor expanded around 0 45.4

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

  3. Taylor expanded around 0 44.0

    \[\leadsto \color{blue}{x}\]

  4. Final simplification44.0

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019173 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))