Average Error: 46.4 → 44.4
Time: 12.2s
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 r983538 = x;
        double r983539 = y;
        double r983540 = 2.0;
        double r983541 = r983539 * r983540;
        double r983542 = 1.0;
        double r983543 = r983541 + r983542;
        double r983544 = z;
        double r983545 = r983543 * r983544;
        double r983546 = t;
        double r983547 = r983545 * r983546;
        double r983548 = 16.0;
        double r983549 = r983547 / r983548;
        double r983550 = cos(r983549);
        double r983551 = r983538 * r983550;
        double r983552 = a;
        double r983553 = r983552 * r983540;
        double r983554 = r983553 + r983542;
        double r983555 = b;
        double r983556 = r983554 * r983555;
        double r983557 = r983556 * r983546;
        double r983558 = r983557 / r983548;
        double r983559 = cos(r983558);
        double r983560 = r983551 * r983559;
        return r983560;
}
↓
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 r983561 = x;
        return r983561;
}

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
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

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

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

  3. Taylor expanded around 0 44.4

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

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020027 
(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))))