Average Error: 46.6 → 44.6
Time: 11.7s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\]
\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)
↓
\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r939769 = x;
        double r939770 = y;
        double r939771 = 2.0;
        double r939772 = r939770 * r939771;
        double r939773 = 1.0;
        double r939774 = r939772 + r939773;
        double r939775 = z;
        double r939776 = r939774 * r939775;
        double r939777 = t;
        double r939778 = r939776 * r939777;
        double r939779 = 16.0;
        double r939780 = r939778 / r939779;
        double r939781 = cos(r939780);
        double r939782 = r939769 * r939781;
        double r939783 = a;
        double r939784 = r939783 * r939771;
        double r939785 = r939784 + r939773;
        double r939786 = b;
        double r939787 = r939785 * r939786;
        double r939788 = r939787 * r939777;
        double r939789 = r939788 / r939779;
        double r939790 = cos(r939789);
        double r939791 = r939782 * r939790;
        return r939791;
}
↓
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 r939792 = x;
        double r939793 = 0.0;
        double r939794 = 16.0;
        double r939795 = r939793 / r939794;
        double r939796 = cos(r939795);
        double r939797 = r939792 * r939796;
        double r939798 = r939797 * r939796;
        return r939798;
}

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

Original46.6
Target44.8
Herbie44.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.6

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.6

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

  4. Final simplification44.6

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

Reproduce

herbie shell --seed 2020018 
(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))))