Average Error: 45.8 → 43.9
Time: 24.6s
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 r613317 = x;
        double r613318 = y;
        double r613319 = 2.0;
        double r613320 = r613318 * r613319;
        double r613321 = 1.0;
        double r613322 = r613320 + r613321;
        double r613323 = z;
        double r613324 = r613322 * r613323;
        double r613325 = t;
        double r613326 = r613324 * r613325;
        double r613327 = 16.0;
        double r613328 = r613326 / r613327;
        double r613329 = cos(r613328);
        double r613330 = r613317 * r613329;
        double r613331 = a;
        double r613332 = r613331 * r613319;
        double r613333 = r613332 + r613321;
        double r613334 = b;
        double r613335 = r613333 * r613334;
        double r613336 = r613335 * r613325;
        double r613337 = r613336 / r613327;
        double r613338 = cos(r613337);
        double r613339 = r613330 * r613338;
        return r613339;
}
↓
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 r613340 = x;
        return r613340;
}

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

Original45.8
Target44.2
Herbie43.9
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.8

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 43.9

    \[\leadsto \color{blue}{x} \cdot 1\]

  4. Final simplification43.9

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019303 
(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))))