Average Error: 46.4 → 44.2
Time: 13.8s
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 r912510 = x;
        double r912511 = y;
        double r912512 = 2.0;
        double r912513 = r912511 * r912512;
        double r912514 = 1.0;
        double r912515 = r912513 + r912514;
        double r912516 = z;
        double r912517 = r912515 * r912516;
        double r912518 = t;
        double r912519 = r912517 * r912518;
        double r912520 = 16.0;
        double r912521 = r912519 / r912520;
        double r912522 = cos(r912521);
        double r912523 = r912510 * r912522;
        double r912524 = a;
        double r912525 = r912524 * r912512;
        double r912526 = r912525 + r912514;
        double r912527 = b;
        double r912528 = r912526 * r912527;
        double r912529 = r912528 * r912518;
        double r912530 = r912529 / r912520;
        double r912531 = cos(r912530);
        double r912532 = r912523 * r912531;
        return r912532;
}
↓
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 r912533 = x;
        return r912533;
}

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
Target44.4
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 46.4

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

  2. Taylor expanded around 0 45.7

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

  3. Taylor expanded around 0 44.2

    \[\leadsto \color{blue}{x} \cdot 1\]

  4. Final simplification44.2

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020046 +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))))