Average Error: 45.8 → 43.9
Time: 23.4s
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 r569613 = x;
        double r569614 = y;
        double r569615 = 2.0;
        double r569616 = r569614 * r569615;
        double r569617 = 1.0;
        double r569618 = r569616 + r569617;
        double r569619 = z;
        double r569620 = r569618 * r569619;
        double r569621 = t;
        double r569622 = r569620 * r569621;
        double r569623 = 16.0;
        double r569624 = r569622 / r569623;
        double r569625 = cos(r569624);
        double r569626 = r569613 * r569625;
        double r569627 = a;
        double r569628 = r569627 * r569615;
        double r569629 = r569628 + r569617;
        double r569630 = b;
        double r569631 = r569629 * r569630;
        double r569632 = r569631 * r569621;
        double r569633 = r569632 / r569623;
        double r569634 = cos(r569633);
        double r569635 = r569626 * r569634;
        return r569635;
}
↓
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 r569636 = x;
        return r569636;
}

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

Original45.8
Target44.2
Herbie43.9
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.8

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 43.9

    \[\leadsto \color{blue}{x} \cdot 1\]

  4. Final simplification43.9

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019303 
(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))))