Average Error: 46.4 → 44.4
Time: 49.2s
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 r43599046 = x;
        double r43599047 = y;
        double r43599048 = 2.0;
        double r43599049 = r43599047 * r43599048;
        double r43599050 = 1.0;
        double r43599051 = r43599049 + r43599050;
        double r43599052 = z;
        double r43599053 = r43599051 * r43599052;
        double r43599054 = t;
        double r43599055 = r43599053 * r43599054;
        double r43599056 = 16.0;
        double r43599057 = r43599055 / r43599056;
        double r43599058 = cos(r43599057);
        double r43599059 = r43599046 * r43599058;
        double r43599060 = a;
        double r43599061 = r43599060 * r43599048;
        double r43599062 = r43599061 + r43599050;
        double r43599063 = b;
        double r43599064 = r43599062 * r43599063;
        double r43599065 = r43599064 * r43599054;
        double r43599066 = r43599065 / r43599056;
        double r43599067 = cos(r43599066);
        double r43599068 = r43599059 * r43599067;
        return r43599068;
}
↓
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 r43599069 = x;
        return r43599069;
}

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
Target44.7
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 46.4

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

  2. Taylor expanded around 0 45.6

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

  3. Taylor expanded around 0 44.4

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

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019170 
(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))))