Average Error: 46.1 → 44.2
Time: 20.1s
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 r885333 = x;
        double r885334 = y;
        double r885335 = 2.0;
        double r885336 = r885334 * r885335;
        double r885337 = 1.0;
        double r885338 = r885336 + r885337;
        double r885339 = z;
        double r885340 = r885338 * r885339;
        double r885341 = t;
        double r885342 = r885340 * r885341;
        double r885343 = 16.0;
        double r885344 = r885342 / r885343;
        double r885345 = cos(r885344);
        double r885346 = r885333 * r885345;
        double r885347 = a;
        double r885348 = r885347 * r885335;
        double r885349 = r885348 + r885337;
        double r885350 = b;
        double r885351 = r885349 * r885350;
        double r885352 = r885351 * r885341;
        double r885353 = r885352 / r885343;
        double r885354 = cos(r885353);
        double r885355 = r885346 * r885354;
        return r885355;
}
↓
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 r885356 = x;
        return r885356;
}

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

Original46.1
Target44.5
Herbie44.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Simplified46.1

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

  3. Taylor expanded around 0 45.3

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

  4. Taylor expanded around 0 44.2

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

  5. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(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))))