Average Error: 46.5 → 44.3
Time: 26.4s
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 r546247 = x;
        double r546248 = y;
        double r546249 = 2.0;
        double r546250 = r546248 * r546249;
        double r546251 = 1.0;
        double r546252 = r546250 + r546251;
        double r546253 = z;
        double r546254 = r546252 * r546253;
        double r546255 = t;
        double r546256 = r546254 * r546255;
        double r546257 = 16.0;
        double r546258 = r546256 / r546257;
        double r546259 = cos(r546258);
        double r546260 = r546247 * r546259;
        double r546261 = a;
        double r546262 = r546261 * r546249;
        double r546263 = r546262 + r546251;
        double r546264 = b;
        double r546265 = r546263 * r546264;
        double r546266 = r546265 * r546255;
        double r546267 = r546266 / r546257;
        double r546268 = cos(r546267);
        double r546269 = r546260 * r546268;
        return r546269;
}
↓
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 r546270 = x;
        return r546270;
}

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

Original46.5
Target44.6
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 46.5

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.3

    \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{0}{16}\right)\]

  4. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019305 
(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))))