Average Error: 46.3 → 44.2
Time: 26.8s
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 r551686 = x;
        double r551687 = y;
        double r551688 = 2.0;
        double r551689 = r551687 * r551688;
        double r551690 = 1.0;
        double r551691 = r551689 + r551690;
        double r551692 = z;
        double r551693 = r551691 * r551692;
        double r551694 = t;
        double r551695 = r551693 * r551694;
        double r551696 = 16.0;
        double r551697 = r551695 / r551696;
        double r551698 = cos(r551697);
        double r551699 = r551686 * r551698;
        double r551700 = a;
        double r551701 = r551700 * r551688;
        double r551702 = r551701 + r551690;
        double r551703 = b;
        double r551704 = r551702 * r551703;
        double r551705 = r551704 * r551694;
        double r551706 = r551705 / r551696;
        double r551707 = cos(r551706);
        double r551708 = r551699 * r551707;
        return r551708;
}
↓
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 r551709 = x;
        return r551709;
}

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
Target44.4
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 46.3

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

  2. Taylor expanded around 0 45.4

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

  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

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019326 
(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))))