Average Error: 46.3 → 44.4
Time: 11.6s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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 \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r950486 = x;
        double r950487 = y;
        double r950488 = 2.0;
        double r950489 = r950487 * r950488;
        double r950490 = 1.0;
        double r950491 = r950489 + r950490;
        double r950492 = z;
        double r950493 = r950491 * r950492;
        double r950494 = t;
        double r950495 = r950493 * r950494;
        double r950496 = 16.0;
        double r950497 = r950495 / r950496;
        double r950498 = cos(r950497);
        double r950499 = r950486 * r950498;
        double r950500 = a;
        double r950501 = r950500 * r950488;
        double r950502 = r950501 + r950490;
        double r950503 = b;
        double r950504 = r950502 * r950503;
        double r950505 = r950504 * r950494;
        double r950506 = r950505 / r950496;
        double r950507 = cos(r950506);
        double r950508 = r950499 * r950507;
        return r950508;
}
↓
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 r950509 = x;
        double r950510 = 0.0;
        double r950511 = 16.0;
        double r950512 = r950510 / r950511;
        double r950513 = cos(r950512);
        double r950514 = r950509 * r950513;
        return r950514;
}

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

Original46.3
Target44.6
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 46.3

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

  2. Taylor expanded around 0 45.6

    \[\leadsto \left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\]

  3. Taylor expanded around 0 44.4

    \[\leadsto \color{blue}{x} \cdot \cos \left(\frac{0}{16}\right)\]

  4. Final simplification44.4

    \[\leadsto x \cdot \cos \left(\frac{0}{16}\right)\]

Reproduce

herbie shell --seed 2020065 
(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))))