Average Error: 46.2 → 44.2
Time: 12.8s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[x\]
\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 r939719 = x;
        double r939720 = y;
        double r939721 = 2.0;
        double r939722 = r939720 * r939721;
        double r939723 = 1.0;
        double r939724 = r939722 + r939723;
        double r939725 = z;
        double r939726 = r939724 * r939725;
        double r939727 = t;
        double r939728 = r939726 * r939727;
        double r939729 = 16.0;
        double r939730 = r939728 / r939729;
        double r939731 = cos(r939730);
        double r939732 = r939719 * r939731;
        double r939733 = a;
        double r939734 = r939733 * r939721;
        double r939735 = r939734 + r939723;
        double r939736 = b;
        double r939737 = r939735 * r939736;
        double r939738 = r939737 * r939727;
        double r939739 = r939738 / r939729;
        double r939740 = cos(r939739);
        double r939741 = r939732 * r939740;
        return r939741;
}
↓
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 r939742 = x;
        return r939742;
}

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 45.6

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

  3. Taylor expanded around 0 44.2

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

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020100 +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))))