Average Error: 46.4 → 44.5
Time: 11.5s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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 r854665 = x;
        double r854666 = y;
        double r854667 = 2.0;
        double r854668 = r854666 * r854667;
        double r854669 = 1.0;
        double r854670 = r854668 + r854669;
        double r854671 = z;
        double r854672 = r854670 * r854671;
        double r854673 = t;
        double r854674 = r854672 * r854673;
        double r854675 = 16.0;
        double r854676 = r854674 / r854675;
        double r854677 = cos(r854676);
        double r854678 = r854665 * r854677;
        double r854679 = a;
        double r854680 = r854679 * r854667;
        double r854681 = r854680 + r854669;
        double r854682 = b;
        double r854683 = r854681 * r854682;
        double r854684 = r854683 * r854673;
        double r854685 = r854684 / r854675;
        double r854686 = cos(r854685);
        double r854687 = r854678 * r854686;
        return r854687;
}
↓
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 r854688 = x;
        return r854688;
}

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

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 45.7

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

  3. Taylor expanded around 0 44.5

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

  4. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020083 
(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))))