Average Error: 46.1 → 44.1
Time: 10.0s
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 r919932 = x;
        double r919933 = y;
        double r919934 = 2.0;
        double r919935 = r919933 * r919934;
        double r919936 = 1.0;
        double r919937 = r919935 + r919936;
        double r919938 = z;
        double r919939 = r919937 * r919938;
        double r919940 = t;
        double r919941 = r919939 * r919940;
        double r919942 = 16.0;
        double r919943 = r919941 / r919942;
        double r919944 = cos(r919943);
        double r919945 = r919932 * r919944;
        double r919946 = a;
        double r919947 = r919946 * r919934;
        double r919948 = r919947 + r919936;
        double r919949 = b;
        double r919950 = r919948 * r919949;
        double r919951 = r919950 * r919940;
        double r919952 = r919951 / r919942;
        double r919953 = cos(r919952);
        double r919954 = r919945 * r919953;
        return r919954;
}
↓
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 r919955 = x;
        return r919955;
}

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
Target44.3
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.4

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020062 
(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))))