Average Error: 45.8 → 43.9
Time: 25.0s
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 r542990 = x;
        double r542991 = y;
        double r542992 = 2.0;
        double r542993 = r542991 * r542992;
        double r542994 = 1.0;
        double r542995 = r542993 + r542994;
        double r542996 = z;
        double r542997 = r542995 * r542996;
        double r542998 = t;
        double r542999 = r542997 * r542998;
        double r543000 = 16.0;
        double r543001 = r542999 / r543000;
        double r543002 = cos(r543001);
        double r543003 = r542990 * r543002;
        double r543004 = a;
        double r543005 = r543004 * r542992;
        double r543006 = r543005 + r542994;
        double r543007 = b;
        double r543008 = r543006 * r543007;
        double r543009 = r543008 * r542998;
        double r543010 = r543009 / r543000;
        double r543011 = cos(r543010);
        double r543012 = r543003 * r543011;
        return r543012;
}
↓
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 r543013 = x;
        return r543013;
}

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

Original45.8
Target44.2
Herbie43.9
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.8

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 43.9

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

  4. Final simplification43.9

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019303 
(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))))