Average Error: 46.8 → 44.7
Time: 14.6s
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 r917271 = x;
        double r917272 = y;
        double r917273 = 2.0;
        double r917274 = r917272 * r917273;
        double r917275 = 1.0;
        double r917276 = r917274 + r917275;
        double r917277 = z;
        double r917278 = r917276 * r917277;
        double r917279 = t;
        double r917280 = r917278 * r917279;
        double r917281 = 16.0;
        double r917282 = r917280 / r917281;
        double r917283 = cos(r917282);
        double r917284 = r917271 * r917283;
        double r917285 = a;
        double r917286 = r917285 * r917273;
        double r917287 = r917286 + r917275;
        double r917288 = b;
        double r917289 = r917287 * r917288;
        double r917290 = r917289 * r917279;
        double r917291 = r917290 / r917281;
        double r917292 = cos(r917291);
        double r917293 = r917284 * r917292;
        return r917293;
}
↓
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 r917294 = x;
        return r917294;
}

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. Taylor expanded around 0 46.0

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

  3. Taylor expanded around 0 44.7

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

  4. Final simplification44.7

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019322 
(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))))