Average Error: 46.8 → 44.7
Time: 29.6s
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 r35296699 = x;
        double r35296700 = y;
        double r35296701 = 2.0;
        double r35296702 = r35296700 * r35296701;
        double r35296703 = 1.0;
        double r35296704 = r35296702 + r35296703;
        double r35296705 = z;
        double r35296706 = r35296704 * r35296705;
        double r35296707 = t;
        double r35296708 = r35296706 * r35296707;
        double r35296709 = 16.0;
        double r35296710 = r35296708 / r35296709;
        double r35296711 = cos(r35296710);
        double r35296712 = r35296699 * r35296711;
        double r35296713 = a;
        double r35296714 = r35296713 * r35296701;
        double r35296715 = r35296714 + r35296703;
        double r35296716 = b;
        double r35296717 = r35296715 * r35296716;
        double r35296718 = r35296717 * r35296707;
        double r35296719 = r35296718 / r35296709;
        double r35296720 = cos(r35296719);
        double r35296721 = r35296712 * r35296720;
        return r35296721;
}
↓
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 r35296722 = x;
        return r35296722;
}

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

Original46.8
Target45.0
Herbie44.7
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.8

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

  2. Simplified46.2

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

  3. Taylor expanded around 0 45.7

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

  4. Taylor expanded around 0 44.7

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

  5. Final simplification44.7

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019171 +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) 2.0))))))

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