Average Error: 46.5 → 44.4
Time: 16.5s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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 \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r877001 = x;
        double r877002 = y;
        double r877003 = 2.0;
        double r877004 = r877002 * r877003;
        double r877005 = 1.0;
        double r877006 = r877004 + r877005;
        double r877007 = z;
        double r877008 = r877006 * r877007;
        double r877009 = t;
        double r877010 = r877008 * r877009;
        double r877011 = 16.0;
        double r877012 = r877010 / r877011;
        double r877013 = cos(r877012);
        double r877014 = r877001 * r877013;
        double r877015 = a;
        double r877016 = r877015 * r877003;
        double r877017 = r877016 + r877005;
        double r877018 = b;
        double r877019 = r877017 * r877018;
        double r877020 = r877019 * r877009;
        double r877021 = r877020 / r877011;
        double r877022 = cos(r877021);
        double r877023 = r877014 * r877022;
        return r877023;
}
↓
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 r877024 = x;
        double r877025 = 0.0;
        double r877026 = 16.0;
        double r877027 = r877025 / r877026;
        double r877028 = cos(r877027);
        double r877029 = r877024 * r877028;
        return r877029;
}

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

Original46.5
Target44.7
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.5

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

  2. Taylor expanded around 0 45.6

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

  3. Taylor expanded around 0 44.4

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot 1\]

  4. Final simplification44.4

    \[\leadsto x \cdot \cos \left(\frac{0}{16}\right)\]

Reproduce

herbie shell --seed 2019350 
(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))))