Average Error: 46.4 → 44.4
Time: 39.1s
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 r5368 = x;
        double r5369 = y;
        double r5370 = 2.0;
        double r5371 = r5369 * r5370;
        double r5372 = 1.0;
        double r5373 = r5371 + r5372;
        double r5374 = z;
        double r5375 = r5373 * r5374;
        double r5376 = t;
        double r5377 = r5375 * r5376;
        double r5378 = 16.0;
        double r5379 = r5377 / r5378;
        double r5380 = cos(r5379);
        double r5381 = r5368 * r5380;
        double r5382 = a;
        double r5383 = r5382 * r5370;
        double r5384 = r5383 + r5372;
        double r5385 = b;
        double r5386 = r5384 * r5385;
        double r5387 = r5386 * r5376;
        double r5388 = r5387 / r5378;
        double r5389 = cos(r5388);
        double r5390 = r5381 * r5389;
        return r5390;
}
↓
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 r5391 = x;
        return r5391;
}

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

Original46.4
Target44.7
Herbie44.4
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.4

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.4

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

  4. Final simplification44.4

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020025 +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))))