Average Error: 46.0 → 44.5
Time: 37.5s
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 r43828407 = x;
        double r43828408 = y;
        double r43828409 = 2.0;
        double r43828410 = r43828408 * r43828409;
        double r43828411 = 1.0;
        double r43828412 = r43828410 + r43828411;
        double r43828413 = z;
        double r43828414 = r43828412 * r43828413;
        double r43828415 = t;
        double r43828416 = r43828414 * r43828415;
        double r43828417 = 16.0;
        double r43828418 = r43828416 / r43828417;
        double r43828419 = cos(r43828418);
        double r43828420 = r43828407 * r43828419;
        double r43828421 = a;
        double r43828422 = r43828421 * r43828409;
        double r43828423 = r43828422 + r43828411;
        double r43828424 = b;
        double r43828425 = r43828423 * r43828424;
        double r43828426 = r43828425 * r43828415;
        double r43828427 = r43828426 / r43828417;
        double r43828428 = cos(r43828427);
        double r43828429 = r43828420 * r43828428;
        return r43828429;
}
↓
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 r43828430 = x;
        return r43828430;
}

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
Target44.7
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

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

  3. Taylor expanded around 0 44.5

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

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019163 
(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))))