Average Error: 46.4 → 44.2
Time: 16.2s
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 r1037981 = x;
        double r1037982 = y;
        double r1037983 = 2.0;
        double r1037984 = r1037982 * r1037983;
        double r1037985 = 1.0;
        double r1037986 = r1037984 + r1037985;
        double r1037987 = z;
        double r1037988 = r1037986 * r1037987;
        double r1037989 = t;
        double r1037990 = r1037988 * r1037989;
        double r1037991 = 16.0;
        double r1037992 = r1037990 / r1037991;
        double r1037993 = cos(r1037992);
        double r1037994 = r1037981 * r1037993;
        double r1037995 = a;
        double r1037996 = r1037995 * r1037983;
        double r1037997 = r1037996 + r1037985;
        double r1037998 = b;
        double r1037999 = r1037997 * r1037998;
        double r1038000 = r1037999 * r1037989;
        double r1038001 = r1038000 / r1037991;
        double r1038002 = cos(r1038001);
        double r1038003 = r1037994 * r1038002;
        return r1038003;
}
↓
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 r1038004 = x;
        return r1038004;
}

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

Original46.4
Target44.4
Herbie44.2
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.2

    \[\leadsto \color{blue}{x} \cdot 1\]

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020046 
(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))))