Average Error: 46.6 → 44.5
Time: 29.3s
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 r40021386 = x;
        double r40021387 = y;
        double r40021388 = 2.0;
        double r40021389 = r40021387 * r40021388;
        double r40021390 = 1.0;
        double r40021391 = r40021389 + r40021390;
        double r40021392 = z;
        double r40021393 = r40021391 * r40021392;
        double r40021394 = t;
        double r40021395 = r40021393 * r40021394;
        double r40021396 = 16.0;
        double r40021397 = r40021395 / r40021396;
        double r40021398 = cos(r40021397);
        double r40021399 = r40021386 * r40021398;
        double r40021400 = a;
        double r40021401 = r40021400 * r40021388;
        double r40021402 = r40021401 + r40021390;
        double r40021403 = b;
        double r40021404 = r40021402 * r40021403;
        double r40021405 = r40021404 * r40021394;
        double r40021406 = r40021405 / r40021396;
        double r40021407 = cos(r40021406);
        double r40021408 = r40021399 * r40021407;
        return r40021408;
}
↓
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 r40021409 = x;
        return r40021409;
}

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
Target44.7
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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. Simplified46.0

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

  3. Taylor expanded around 0 45.4

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

  4. Taylor expanded around 0 44.5

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

  5. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(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))))