Average Error: 46.0 → 44.5
Time: 58.6s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r40063733 = x;
        double r40063734 = y;
        double r40063735 = 2.0;
        double r40063736 = r40063734 * r40063735;
        double r40063737 = 1.0;
        double r40063738 = r40063736 + r40063737;
        double r40063739 = z;
        double r40063740 = r40063738 * r40063739;
        double r40063741 = t;
        double r40063742 = r40063740 * r40063741;
        double r40063743 = 16.0;
        double r40063744 = r40063742 / r40063743;
        double r40063745 = cos(r40063744);
        double r40063746 = r40063733 * r40063745;
        double r40063747 = a;
        double r40063748 = r40063747 * r40063735;
        double r40063749 = r40063748 + r40063737;
        double r40063750 = b;
        double r40063751 = r40063749 * r40063750;
        double r40063752 = r40063751 * r40063741;
        double r40063753 = r40063752 / r40063743;
        double r40063754 = cos(r40063753);
        double r40063755 = r40063746 * r40063754;
        return r40063755;
}
↓
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 r40063756 = x;
        return r40063756;
}

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
Target44.6
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Taylor expanded around 0 45.4

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \color{blue}{1}\]

  3. Taylor expanded around 0 44.5

    \[\leadsto \color{blue}{x} \cdot 1\]

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019162 
(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))))))

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