Average Error: 46.1 → 44.1
Time: 2.4m
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 r43204025 = x;
        double r43204026 = y;
        double r43204027 = 2.0;
        double r43204028 = r43204026 * r43204027;
        double r43204029 = 1.0;
        double r43204030 = r43204028 + r43204029;
        double r43204031 = z;
        double r43204032 = r43204030 * r43204031;
        double r43204033 = t;
        double r43204034 = r43204032 * r43204033;
        double r43204035 = 16.0;
        double r43204036 = r43204034 / r43204035;
        double r43204037 = cos(r43204036);
        double r43204038 = r43204025 * r43204037;
        double r43204039 = a;
        double r43204040 = r43204039 * r43204027;
        double r43204041 = r43204040 + r43204029;
        double r43204042 = b;
        double r43204043 = r43204041 * r43204042;
        double r43204044 = r43204043 * r43204033;
        double r43204045 = r43204044 / r43204035;
        double r43204046 = cos(r43204045);
        double r43204047 = r43204038 * r43204046;
        return r43204047;
}
↓
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 r43204048 = x;
        return r43204048;
}

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
Target44.4
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

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019200 
(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) 2.0))))))

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