Average Error: 46.1 → 44.1
Time: 13.2s
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 r914838 = x;
        double r914839 = y;
        double r914840 = 2.0;
        double r914841 = r914839 * r914840;
        double r914842 = 1.0;
        double r914843 = r914841 + r914842;
        double r914844 = z;
        double r914845 = r914843 * r914844;
        double r914846 = t;
        double r914847 = r914845 * r914846;
        double r914848 = 16.0;
        double r914849 = r914847 / r914848;
        double r914850 = cos(r914849);
        double r914851 = r914838 * r914850;
        double r914852 = a;
        double r914853 = r914852 * r914840;
        double r914854 = r914853 + r914842;
        double r914855 = b;
        double r914856 = r914854 * r914855;
        double r914857 = r914856 * r914846;
        double r914858 = r914857 / r914848;
        double r914859 = cos(r914858);
        double r914860 = r914851 * r914859;
        return r914860;
}
↓
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 r914861 = x;
        return r914861;
}

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

Original46.1
Target44.3
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.4

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(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))))