Average Error: 46.1 → 44.1
Time: 10.7s
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 r958770 = x;
        double r958771 = y;
        double r958772 = 2.0;
        double r958773 = r958771 * r958772;
        double r958774 = 1.0;
        double r958775 = r958773 + r958774;
        double r958776 = z;
        double r958777 = r958775 * r958776;
        double r958778 = t;
        double r958779 = r958777 * r958778;
        double r958780 = 16.0;
        double r958781 = r958779 / r958780;
        double r958782 = cos(r958781);
        double r958783 = r958770 * r958782;
        double r958784 = a;
        double r958785 = r958784 * r958772;
        double r958786 = r958785 + r958774;
        double r958787 = b;
        double r958788 = r958786 * r958787;
        double r958789 = r958788 * r958778;
        double r958790 = r958789 / r958780;
        double r958791 = cos(r958790);
        double r958792 = r958783 * r958791;
        return r958792;
}
↓
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 r958793 = x;
        double r958794 = 0.0;
        double r958795 = 16.0;
        double r958796 = r958794 / r958795;
        double r958797 = cos(r958796);
        double r958798 = r958793 * r958797;
        return r958798;
}

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

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

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.3

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

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

Reproduce

herbie shell --seed 2020021 
(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))))