Average Error: 46.1 → 44.1
Time: 10.9s
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 r835581 = x;
        double r835582 = y;
        double r835583 = 2.0;
        double r835584 = r835582 * r835583;
        double r835585 = 1.0;
        double r835586 = r835584 + r835585;
        double r835587 = z;
        double r835588 = r835586 * r835587;
        double r835589 = t;
        double r835590 = r835588 * r835589;
        double r835591 = 16.0;
        double r835592 = r835590 / r835591;
        double r835593 = cos(r835592);
        double r835594 = r835581 * r835593;
        double r835595 = a;
        double r835596 = r835595 * r835583;
        double r835597 = r835596 + r835585;
        double r835598 = b;
        double r835599 = r835597 * r835598;
        double r835600 = r835599 * r835589;
        double r835601 = r835600 / r835591;
        double r835602 = cos(r835601);
        double r835603 = r835594 * r835602;
        return r835603;
}
↓
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 r835604 = x;
        return r835604;
}

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

Original46.1
Target44.3
Herbie44.1
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.1

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

  2. Taylor expanded around 0 45.5

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

  3. Taylor expanded around 0 44.1

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

  4. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2020060 
(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))))