Average Error: 47.2 → 45.0
Time: 29.0s
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 r45784197 = x;
        double r45784198 = y;
        double r45784199 = 2.0;
        double r45784200 = r45784198 * r45784199;
        double r45784201 = 1.0;
        double r45784202 = r45784200 + r45784201;
        double r45784203 = z;
        double r45784204 = r45784202 * r45784203;
        double r45784205 = t;
        double r45784206 = r45784204 * r45784205;
        double r45784207 = 16.0;
        double r45784208 = r45784206 / r45784207;
        double r45784209 = cos(r45784208);
        double r45784210 = r45784197 * r45784209;
        double r45784211 = a;
        double r45784212 = r45784211 * r45784199;
        double r45784213 = r45784212 + r45784201;
        double r45784214 = b;
        double r45784215 = r45784213 * r45784214;
        double r45784216 = r45784215 * r45784205;
        double r45784217 = r45784216 / r45784207;
        double r45784218 = cos(r45784217);
        double r45784219 = r45784210 * r45784218;
        return r45784219;
}
↓
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 r45784220 = x;
        return r45784220;
}

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

Original47.2
Target45.2
Herbie45.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 47.2

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

  2. Simplified47.2

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

  3. Taylor expanded around 0 46.5

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

  4. Taylor expanded around 0 45.0

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

  5. Final simplification45.0

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019174 +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))))