Average Error: 46.1 → 44.1
Time: 10.4s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\cos \left(\frac{0}{16}\right) \cdot \left(1 \cdot x\right)\]
\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)
↓
\cos \left(\frac{0}{16}\right) \cdot \left(1 \cdot x\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r789675 = x;
        double r789676 = y;
        double r789677 = 2.0;
        double r789678 = r789676 * r789677;
        double r789679 = 1.0;
        double r789680 = r789678 + r789679;
        double r789681 = z;
        double r789682 = r789680 * r789681;
        double r789683 = t;
        double r789684 = r789682 * r789683;
        double r789685 = 16.0;
        double r789686 = r789684 / r789685;
        double r789687 = cos(r789686);
        double r789688 = r789675 * r789687;
        double r789689 = a;
        double r789690 = r789689 * r789677;
        double r789691 = r789690 + r789679;
        double r789692 = b;
        double r789693 = r789691 * r789692;
        double r789694 = r789693 * r789683;
        double r789695 = r789694 / r789685;
        double r789696 = cos(r789695);
        double r789697 = r789688 * r789696;
        return r789697;
}
↓
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 r789698 = 0.0;
        double r789699 = 16.0;
        double r789700 = r789698 / r789699;
        double r789701 = cos(r789700);
        double r789702 = 1.0;
        double r789703 = x;
        double r789704 = r789702 * r789703;
        double r789705 = r789701 * r789704;
        return r789705;
}

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

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 45.4

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(1 \cdot x\right)\]

Reproduce

herbie shell --seed 2020062 
(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))))