Average Error: 46.0 → 43.8
Time: 11.9s
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 r1025300 = x;
        double r1025301 = y;
        double r1025302 = 2.0;
        double r1025303 = r1025301 * r1025302;
        double r1025304 = 1.0;
        double r1025305 = r1025303 + r1025304;
        double r1025306 = z;
        double r1025307 = r1025305 * r1025306;
        double r1025308 = t;
        double r1025309 = r1025307 * r1025308;
        double r1025310 = 16.0;
        double r1025311 = r1025309 / r1025310;
        double r1025312 = cos(r1025311);
        double r1025313 = r1025300 * r1025312;
        double r1025314 = a;
        double r1025315 = r1025314 * r1025302;
        double r1025316 = r1025315 + r1025304;
        double r1025317 = b;
        double r1025318 = r1025316 * r1025317;
        double r1025319 = r1025318 * r1025308;
        double r1025320 = r1025319 / r1025310;
        double r1025321 = cos(r1025320);
        double r1025322 = r1025313 * r1025321;
        return r1025322;
}
↓
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 r1025323 = x;
        return r1025323;
}

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

Original46.0
Target44.1
Herbie43.8
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

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

  2. Taylor expanded around 0 45.2

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

  3. Taylor expanded around 0 43.8

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

  4. Final simplification43.8

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020089 
(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))))