Average Error: 45.6 → 43.6
Time: 22.0s
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 r567936 = x;
        double r567937 = y;
        double r567938 = 2.0;
        double r567939 = r567937 * r567938;
        double r567940 = 1.0;
        double r567941 = r567939 + r567940;
        double r567942 = z;
        double r567943 = r567941 * r567942;
        double r567944 = t;
        double r567945 = r567943 * r567944;
        double r567946 = 16.0;
        double r567947 = r567945 / r567946;
        double r567948 = cos(r567947);
        double r567949 = r567936 * r567948;
        double r567950 = a;
        double r567951 = r567950 * r567938;
        double r567952 = r567951 + r567940;
        double r567953 = b;
        double r567954 = r567952 * r567953;
        double r567955 = r567954 * r567944;
        double r567956 = r567955 / r567946;
        double r567957 = cos(r567956);
        double r567958 = r567949 * r567957;
        return r567958;
}
↓
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 r567959 = x;
        return r567959;
}

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

Original45.6
Target43.9
Herbie43.6
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.6

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

  2. Taylor expanded around 0 44.8

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

  3. Taylor expanded around 0 43.6

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

  4. Final simplification43.6

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019208 
(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))))