Average Error: 46.6 → 44.5
Time: 10.6s
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 r952937 = x;
        double r952938 = y;
        double r952939 = 2.0;
        double r952940 = r952938 * r952939;
        double r952941 = 1.0;
        double r952942 = r952940 + r952941;
        double r952943 = z;
        double r952944 = r952942 * r952943;
        double r952945 = t;
        double r952946 = r952944 * r952945;
        double r952947 = 16.0;
        double r952948 = r952946 / r952947;
        double r952949 = cos(r952948);
        double r952950 = r952937 * r952949;
        double r952951 = a;
        double r952952 = r952951 * r952939;
        double r952953 = r952952 + r952941;
        double r952954 = b;
        double r952955 = r952953 * r952954;
        double r952956 = r952955 * r952945;
        double r952957 = r952956 / r952947;
        double r952958 = cos(r952957);
        double r952959 = r952950 * r952958;
        return r952959;
}
↓
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 r952960 = x;
        return r952960;
}

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

Original46.6
Target44.8
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.6

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

  2. Taylor expanded around 0 45.8

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

  3. Taylor expanded around 0 44.5

    \[\leadsto \left(x \cdot 1\right) \cdot \color{blue}{1}\]

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020001 
(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))))