Average Error: 45.5 → 44.1
Time: 1.1m
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r43426673 = x;
        double r43426674 = y;
        double r43426675 = 2.0;
        double r43426676 = r43426674 * r43426675;
        double r43426677 = 1.0;
        double r43426678 = r43426676 + r43426677;
        double r43426679 = z;
        double r43426680 = r43426678 * r43426679;
        double r43426681 = t;
        double r43426682 = r43426680 * r43426681;
        double r43426683 = 16.0;
        double r43426684 = r43426682 / r43426683;
        double r43426685 = cos(r43426684);
        double r43426686 = r43426673 * r43426685;
        double r43426687 = a;
        double r43426688 = r43426687 * r43426675;
        double r43426689 = r43426688 + r43426677;
        double r43426690 = b;
        double r43426691 = r43426689 * r43426690;
        double r43426692 = r43426691 * r43426681;
        double r43426693 = r43426692 / r43426683;
        double r43426694 = cos(r43426693);
        double r43426695 = r43426686 * r43426694;
        return r43426695;
}
↓
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 r43426696 = x;
        return r43426696;
}

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

Original45.5
Target44.3
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.5

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Simplified45.3

    \[\leadsto \color{blue}{\left(\cos \left(\frac{t}{\frac{\frac{16.0}{z}}{\mathsf{fma}\left(2.0, y, 1.0\right)}}\right) \cdot x\right) \cdot \cos \left(\frac{b}{\frac{\frac{16.0}{t}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)}\]

  3. Taylor expanded around 0 44.9

    \[\leadsto \left(\cos \left(\frac{t}{\frac{\frac{16.0}{z}}{\mathsf{fma}\left(2.0, y, 1.0\right)}}\right) \cdot x\right) \cdot \color{blue}{1}\]

  4. Taylor expanded around 0 44.1

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

  5. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019158 +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))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))