Average Error: 46.0 → 44.5
Time: 52.4s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r41761815 = x;
        double r41761816 = y;
        double r41761817 = 2.0;
        double r41761818 = r41761816 * r41761817;
        double r41761819 = 1.0;
        double r41761820 = r41761818 + r41761819;
        double r41761821 = z;
        double r41761822 = r41761820 * r41761821;
        double r41761823 = t;
        double r41761824 = r41761822 * r41761823;
        double r41761825 = 16.0;
        double r41761826 = r41761824 / r41761825;
        double r41761827 = cos(r41761826);
        double r41761828 = r41761815 * r41761827;
        double r41761829 = a;
        double r41761830 = r41761829 * r41761817;
        double r41761831 = r41761830 + r41761819;
        double r41761832 = b;
        double r41761833 = r41761831 * r41761832;
        double r41761834 = r41761833 * r41761823;
        double r41761835 = r41761834 / r41761825;
        double r41761836 = cos(r41761835);
        double r41761837 = r41761828 * r41761836;
        return r41761837;
}
↓
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 r41761838 = x;
        return r41761838;
}

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

Original46.0
Target44.6
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Taylor expanded around 0 45.4

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \color{blue}{1}\]

  3. Taylor expanded around 0 44.5

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

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019162 
(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))))))

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