Average Error: 45.7 → 43.8
Time: 34.5s
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 r24629904 = x;
        double r24629905 = y;
        double r24629906 = 2.0;
        double r24629907 = r24629905 * r24629906;
        double r24629908 = 1.0;
        double r24629909 = r24629907 + r24629908;
        double r24629910 = z;
        double r24629911 = r24629909 * r24629910;
        double r24629912 = t;
        double r24629913 = r24629911 * r24629912;
        double r24629914 = 16.0;
        double r24629915 = r24629913 / r24629914;
        double r24629916 = cos(r24629915);
        double r24629917 = r24629904 * r24629916;
        double r24629918 = a;
        double r24629919 = r24629918 * r24629906;
        double r24629920 = r24629919 + r24629908;
        double r24629921 = b;
        double r24629922 = r24629920 * r24629921;
        double r24629923 = r24629922 * r24629912;
        double r24629924 = r24629923 / r24629914;
        double r24629925 = cos(r24629924);
        double r24629926 = r24629917 * r24629925;
        return r24629926;
}
↓
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 r24629927 = x;
        return r24629927;
}

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

Original45.7
Target44.0
Herbie43.8
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.7

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

  2. Simplified45.2

    \[\leadsto \color{blue}{\left(\cos \left(\left(\frac{z}{16} \cdot t\right) \cdot \mathsf{fma}\left(2, y, 1\right)\right) \cdot x\right) \cdot \cos \left(\left(\frac{b}{16} \cdot t\right) \cdot \mathsf{fma}\left(a, 2, 1\right)\right)}\]

  3. Taylor expanded around 0 44.8

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

  4. Taylor expanded around 0 43.8

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

  5. Final simplification43.8

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))))