Average Error: 45.6 → 43.6
Time: 24.5s
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 r595219 = x;
        double r595220 = y;
        double r595221 = 2.0;
        double r595222 = r595220 * r595221;
        double r595223 = 1.0;
        double r595224 = r595222 + r595223;
        double r595225 = z;
        double r595226 = r595224 * r595225;
        double r595227 = t;
        double r595228 = r595226 * r595227;
        double r595229 = 16.0;
        double r595230 = r595228 / r595229;
        double r595231 = cos(r595230);
        double r595232 = r595219 * r595231;
        double r595233 = a;
        double r595234 = r595233 * r595221;
        double r595235 = r595234 + r595223;
        double r595236 = b;
        double r595237 = r595235 * r595236;
        double r595238 = r595237 * r595227;
        double r595239 = r595238 / r595229;
        double r595240 = cos(r595239);
        double r595241 = r595232 * r595240;
        return r595241;
}
↓
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 r595242 = x;
        return r595242;
}

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

Original45.6
Target43.9
Herbie43.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.6

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

  2. Simplified45.6

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

  3. Taylor expanded around 0 44.8

    \[\leadsto \cos \left(\frac{\color{blue}{0}}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)}{16}\right)\right)\]

  4. Taylor expanded around 0 43.6

    \[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right)\]

  5. Final simplification43.6

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(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))))