Average Error: 46.9 → 44.7
Time: 23.6s
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 r653688 = x;
        double r653689 = y;
        double r653690 = 2.0;
        double r653691 = r653689 * r653690;
        double r653692 = 1.0;
        double r653693 = r653691 + r653692;
        double r653694 = z;
        double r653695 = r653693 * r653694;
        double r653696 = t;
        double r653697 = r653695 * r653696;
        double r653698 = 16.0;
        double r653699 = r653697 / r653698;
        double r653700 = cos(r653699);
        double r653701 = r653688 * r653700;
        double r653702 = a;
        double r653703 = r653702 * r653690;
        double r653704 = r653703 + r653692;
        double r653705 = b;
        double r653706 = r653704 * r653705;
        double r653707 = r653706 * r653696;
        double r653708 = r653707 / r653698;
        double r653709 = cos(r653708);
        double r653710 = r653701 * r653709;
        return r653710;
}
↓
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 r653711 = x;
        return r653711;
}

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

Original46.9
Target45.0
Herbie44.7
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.9

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

  2. Taylor expanded around 0 46.1

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

  3. Taylor expanded around 0 44.7

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

  4. Final simplification44.7

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019294 
(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))))