Average Error: 46.1 → 44.6
Time: 31.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r44413225 = x;
        double r44413226 = y;
        double r44413227 = 2.0;
        double r44413228 = r44413226 * r44413227;
        double r44413229 = 1.0;
        double r44413230 = r44413228 + r44413229;
        double r44413231 = z;
        double r44413232 = r44413230 * r44413231;
        double r44413233 = t;
        double r44413234 = r44413232 * r44413233;
        double r44413235 = 16.0;
        double r44413236 = r44413234 / r44413235;
        double r44413237 = cos(r44413236);
        double r44413238 = r44413225 * r44413237;
        double r44413239 = a;
        double r44413240 = r44413239 * r44413227;
        double r44413241 = r44413240 + r44413229;
        double r44413242 = b;
        double r44413243 = r44413241 * r44413242;
        double r44413244 = r44413243 * r44413233;
        double r44413245 = r44413244 / r44413235;
        double r44413246 = cos(r44413245);
        double r44413247 = r44413238 * r44413246;
        return r44413247;
}
↓
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 r44413248 = x;
        return r44413248;
}

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

Original46.1
Target44.8
Herbie44.6
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Taylor expanded around 0 45.6

    \[\leadsto \left(x \cdot \color{blue}{1}\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  3. Taylor expanded around 0 44.6

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

  4. Final simplification44.6

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019165 
(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))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))