Average Error: 46.4 → 44.3
Time: 31.5s
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 r844148 = x;
        double r844149 = y;
        double r844150 = 2.0;
        double r844151 = r844149 * r844150;
        double r844152 = 1.0;
        double r844153 = r844151 + r844152;
        double r844154 = z;
        double r844155 = r844153 * r844154;
        double r844156 = t;
        double r844157 = r844155 * r844156;
        double r844158 = 16.0;
        double r844159 = r844157 / r844158;
        double r844160 = cos(r844159);
        double r844161 = r844148 * r844160;
        double r844162 = a;
        double r844163 = r844162 * r844150;
        double r844164 = r844163 + r844152;
        double r844165 = b;
        double r844166 = r844164 * r844165;
        double r844167 = r844166 * r844156;
        double r844168 = r844167 / r844158;
        double r844169 = cos(r844168);
        double r844170 = r844161 * r844169;
        return r844170;
}
↓
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 r844171 = x;
        return r844171;
}

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

Original46.4
Target44.5
Herbie44.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  2. Simplified45.8

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

  3. Taylor expanded around 0 45.3

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

  4. Taylor expanded around 0 44.3

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

  5. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019196 +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))))