Average Error: 46.3 → 44.2
Time: 53.4s
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 r45161384 = x;
        double r45161385 = y;
        double r45161386 = 2.0;
        double r45161387 = r45161385 * r45161386;
        double r45161388 = 1.0;
        double r45161389 = r45161387 + r45161388;
        double r45161390 = z;
        double r45161391 = r45161389 * r45161390;
        double r45161392 = t;
        double r45161393 = r45161391 * r45161392;
        double r45161394 = 16.0;
        double r45161395 = r45161393 / r45161394;
        double r45161396 = cos(r45161395);
        double r45161397 = r45161384 * r45161396;
        double r45161398 = a;
        double r45161399 = r45161398 * r45161386;
        double r45161400 = r45161399 + r45161388;
        double r45161401 = b;
        double r45161402 = r45161400 * r45161401;
        double r45161403 = r45161402 * r45161392;
        double r45161404 = r45161403 / r45161394;
        double r45161405 = cos(r45161404);
        double r45161406 = r45161397 * r45161405;
        return r45161406;
}
↓
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 r45161407 = x;
        return r45161407;
}

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

Original46.3
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 46.3

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

  2. Taylor expanded around 0 45.6

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019169 
(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))))