Average Error: 46.4 → 44.3
Time: 17.8s
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 r616400 = x;
        double r616401 = y;
        double r616402 = 2.0;
        double r616403 = r616401 * r616402;
        double r616404 = 1.0;
        double r616405 = r616403 + r616404;
        double r616406 = z;
        double r616407 = r616405 * r616406;
        double r616408 = t;
        double r616409 = r616407 * r616408;
        double r616410 = 16.0;
        double r616411 = r616409 / r616410;
        double r616412 = cos(r616411);
        double r616413 = r616400 * r616412;
        double r616414 = a;
        double r616415 = r616414 * r616402;
        double r616416 = r616415 + r616404;
        double r616417 = b;
        double r616418 = r616416 * r616417;
        double r616419 = r616418 * r616408;
        double r616420 = r616419 / r616410;
        double r616421 = cos(r616420);
        double r616422 = r616413 * r616421;
        return r616422;
}
↓
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 r616423 = x;
        return r616423;
}

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

Original46.4
Target44.7
Herbie44.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.3

    \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1}\]

  4. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 1978988140 
(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))))