Average Error: 46.4 → 44.3
Time: 30.4s
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 r606548 = x;
        double r606549 = y;
        double r606550 = 2.0;
        double r606551 = r606549 * r606550;
        double r606552 = 1.0;
        double r606553 = r606551 + r606552;
        double r606554 = z;
        double r606555 = r606553 * r606554;
        double r606556 = t;
        double r606557 = r606555 * r606556;
        double r606558 = 16.0;
        double r606559 = r606557 / r606558;
        double r606560 = cos(r606559);
        double r606561 = r606548 * r606560;
        double r606562 = a;
        double r606563 = r606562 * r606550;
        double r606564 = r606563 + r606552;
        double r606565 = b;
        double r606566 = r606564 * r606565;
        double r606567 = r606566 * r606556;
        double r606568 = r606567 / r606558;
        double r606569 = cos(r606568);
        double r606570 = r606561 * r606569;
        return r606570;
}
↓
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 r606571 = x;
        return r606571;
}

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
Target44.5
Herbie44.3
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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. Simplified46.1

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

  3. Taylor expanded around 0 45.3

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

  4. Taylor expanded around 0 44.3

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

  5. Final simplification44.3

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019196 
(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))))