Average Error: 46.3 → 44.2
Time: 14.8s
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 r833707 = x;
        double r833708 = y;
        double r833709 = 2.0;
        double r833710 = r833708 * r833709;
        double r833711 = 1.0;
        double r833712 = r833710 + r833711;
        double r833713 = z;
        double r833714 = r833712 * r833713;
        double r833715 = t;
        double r833716 = r833714 * r833715;
        double r833717 = 16.0;
        double r833718 = r833716 / r833717;
        double r833719 = cos(r833718);
        double r833720 = r833707 * r833719;
        double r833721 = a;
        double r833722 = r833721 * r833709;
        double r833723 = r833722 + r833711;
        double r833724 = b;
        double r833725 = r833723 * r833724;
        double r833726 = r833725 * r833715;
        double r833727 = r833726 / r833717;
        double r833728 = cos(r833727);
        double r833729 = r833720 * r833728;
        return r833729;
}
↓
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 r833730 = x;
        double r833731 = 0.0;
        double r833732 = 16.0;
        double r833733 = r833731 / r833732;
        double r833734 = cos(r833733);
        double r833735 = r833730 * r833734;
        return r833735;
}

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
Target44.5
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 46.3

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

  2. Taylor expanded around 0 45.5

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

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

Reproduce

herbie shell --seed 2019353 +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))))