Average Error: 46.6 → 44.6
Time: 1.6m
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 r42346350 = x;
        double r42346351 = y;
        double r42346352 = 2.0;
        double r42346353 = r42346351 * r42346352;
        double r42346354 = 1.0;
        double r42346355 = r42346353 + r42346354;
        double r42346356 = z;
        double r42346357 = r42346355 * r42346356;
        double r42346358 = t;
        double r42346359 = r42346357 * r42346358;
        double r42346360 = 16.0;
        double r42346361 = r42346359 / r42346360;
        double r42346362 = cos(r42346361);
        double r42346363 = r42346350 * r42346362;
        double r42346364 = a;
        double r42346365 = r42346364 * r42346352;
        double r42346366 = r42346365 + r42346354;
        double r42346367 = b;
        double r42346368 = r42346366 * r42346367;
        double r42346369 = r42346368 * r42346358;
        double r42346370 = r42346369 / r42346360;
        double r42346371 = cos(r42346370);
        double r42346372 = r42346363 * r42346371;
        return r42346372;
}
↓
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 r42346373 = x;
        return r42346373;
}

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. Using strategy rm

  3. Applied associate-*l*46.3

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

  4. Taylor expanded around 0 45.4

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

  5. Taylor expanded around 0 44.6

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

  6. Final simplification44.6

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019168 
(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) 2.0))))))

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