Average Error: 46.2 → 44.2
Time: 17.4s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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 r1109797 = x;
        double r1109798 = y;
        double r1109799 = 2.0;
        double r1109800 = r1109798 * r1109799;
        double r1109801 = 1.0;
        double r1109802 = r1109800 + r1109801;
        double r1109803 = z;
        double r1109804 = r1109802 * r1109803;
        double r1109805 = t;
        double r1109806 = r1109804 * r1109805;
        double r1109807 = 16.0;
        double r1109808 = r1109806 / r1109807;
        double r1109809 = cos(r1109808);
        double r1109810 = r1109797 * r1109809;
        double r1109811 = a;
        double r1109812 = r1109811 * r1109799;
        double r1109813 = r1109812 + r1109801;
        double r1109814 = b;
        double r1109815 = r1109813 * r1109814;
        double r1109816 = r1109815 * r1109805;
        double r1109817 = r1109816 / r1109807;
        double r1109818 = cos(r1109817);
        double r1109819 = r1109810 * r1109818;
        return r1109819;
}
↓
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 r1109820 = x;
        return r1109820;
}

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

Original46.2
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 46.2

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

  2. Taylor expanded around 0 45.5

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020045 +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))))