Average Error: 46.4 → 44.3
Time: 33.1s
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 r533442 = x;
        double r533443 = y;
        double r533444 = 2.0;
        double r533445 = r533443 * r533444;
        double r533446 = 1.0;
        double r533447 = r533445 + r533446;
        double r533448 = z;
        double r533449 = r533447 * r533448;
        double r533450 = t;
        double r533451 = r533449 * r533450;
        double r533452 = 16.0;
        double r533453 = r533451 / r533452;
        double r533454 = cos(r533453);
        double r533455 = r533442 * r533454;
        double r533456 = a;
        double r533457 = r533456 * r533444;
        double r533458 = r533457 + r533446;
        double r533459 = b;
        double r533460 = r533458 * r533459;
        double r533461 = r533460 * r533450;
        double r533462 = r533461 / r533452;
        double r533463 = cos(r533462);
        double r533464 = r533455 * r533463;
        return r533464;
}
↓
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 r533465 = x;
        return r533465;
}

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
Target44.6
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. Simplified46.4

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

  3. Taylor expanded around 0 45.6

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

  4. Taylor expanded around 0 44.3

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

  5. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019199 +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))))