Average Error: 47.2 → 45.0
Time: 28.1s
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 r63261690 = x;
        double r63261691 = y;
        double r63261692 = 2.0;
        double r63261693 = r63261691 * r63261692;
        double r63261694 = 1.0;
        double r63261695 = r63261693 + r63261694;
        double r63261696 = z;
        double r63261697 = r63261695 * r63261696;
        double r63261698 = t;
        double r63261699 = r63261697 * r63261698;
        double r63261700 = 16.0;
        double r63261701 = r63261699 / r63261700;
        double r63261702 = cos(r63261701);
        double r63261703 = r63261690 * r63261702;
        double r63261704 = a;
        double r63261705 = r63261704 * r63261692;
        double r63261706 = r63261705 + r63261694;
        double r63261707 = b;
        double r63261708 = r63261706 * r63261707;
        double r63261709 = r63261708 * r63261698;
        double r63261710 = r63261709 / r63261700;
        double r63261711 = cos(r63261710);
        double r63261712 = r63261703 * r63261711;
        return r63261712;
}
↓
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 r63261713 = x;
        return r63261713;
}

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

Original47.2
Target45.2
Herbie45.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 47.2

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

  2. Taylor expanded around 0 46.5

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

  3. Taylor expanded around 0 45.0

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

  4. Final simplification45.0

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))