Average Error: 46.1 → 44.2
Time: 17.4s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x \cdot \cos \left(\frac{0}{16}\right)\]
\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 r995587 = x;
        double r995588 = y;
        double r995589 = 2.0;
        double r995590 = r995588 * r995589;
        double r995591 = 1.0;
        double r995592 = r995590 + r995591;
        double r995593 = z;
        double r995594 = r995592 * r995593;
        double r995595 = t;
        double r995596 = r995594 * r995595;
        double r995597 = 16.0;
        double r995598 = r995596 / r995597;
        double r995599 = cos(r995598);
        double r995600 = r995587 * r995599;
        double r995601 = a;
        double r995602 = r995601 * r995589;
        double r995603 = r995602 + r995591;
        double r995604 = b;
        double r995605 = r995603 * r995604;
        double r995606 = r995605 * r995595;
        double r995607 = r995606 / r995597;
        double r995608 = cos(r995607);
        double r995609 = r995600 * r995608;
        return r995609;
}
↓
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 r995610 = x;
        double r995611 = 0.0;
        double r995612 = 16.0;
        double r995613 = r995611 / r995612;
        double r995614 = cos(r995613);
        double r995615 = r995610 * r995614;
        return r995615;
}

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

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2020047 
(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))))