Average Error: 46.1 → 44.2
Time: 18.4s
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 r900386 = x;
        double r900387 = y;
        double r900388 = 2.0;
        double r900389 = r900387 * r900388;
        double r900390 = 1.0;
        double r900391 = r900389 + r900390;
        double r900392 = z;
        double r900393 = r900391 * r900392;
        double r900394 = t;
        double r900395 = r900393 * r900394;
        double r900396 = 16.0;
        double r900397 = r900395 / r900396;
        double r900398 = cos(r900397);
        double r900399 = r900386 * r900398;
        double r900400 = a;
        double r900401 = r900400 * r900388;
        double r900402 = r900401 + r900390;
        double r900403 = b;
        double r900404 = r900402 * r900403;
        double r900405 = r900404 * r900394;
        double r900406 = r900405 / r900396;
        double r900407 = cos(r900406);
        double r900408 = r900399 * r900407;
        return r900408;
}
↓
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 r900409 = x;
        return r900409;
}

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

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 45.3

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020047 +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))))