Average Error: 46.0 → 44.5
Time: 35.5s
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 r35754146 = x;
        double r35754147 = y;
        double r35754148 = 2.0;
        double r35754149 = r35754147 * r35754148;
        double r35754150 = 1.0;
        double r35754151 = r35754149 + r35754150;
        double r35754152 = z;
        double r35754153 = r35754151 * r35754152;
        double r35754154 = t;
        double r35754155 = r35754153 * r35754154;
        double r35754156 = 16.0;
        double r35754157 = r35754155 / r35754156;
        double r35754158 = cos(r35754157);
        double r35754159 = r35754146 * r35754158;
        double r35754160 = a;
        double r35754161 = r35754160 * r35754148;
        double r35754162 = r35754161 + r35754150;
        double r35754163 = b;
        double r35754164 = r35754162 * r35754163;
        double r35754165 = r35754164 * r35754154;
        double r35754166 = r35754165 / r35754156;
        double r35754167 = cos(r35754166);
        double r35754168 = r35754159 * r35754167;
        return r35754168;
}
↓
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 r35754169 = x;
        return r35754169;
}

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

Original46.0
Target44.7
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

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

  2. Simplified45.8

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

  3. Taylor expanded around 0 45.2

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

  4. Taylor expanded around 0 44.5

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

  5. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019163 +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))))