Average Error: 46.1 → 44.1
Time: 2.0m
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 r40423536 = x;
        double r40423537 = y;
        double r40423538 = 2.0;
        double r40423539 = r40423537 * r40423538;
        double r40423540 = 1.0;
        double r40423541 = r40423539 + r40423540;
        double r40423542 = z;
        double r40423543 = r40423541 * r40423542;
        double r40423544 = t;
        double r40423545 = r40423543 * r40423544;
        double r40423546 = 16.0;
        double r40423547 = r40423545 / r40423546;
        double r40423548 = cos(r40423547);
        double r40423549 = r40423536 * r40423548;
        double r40423550 = a;
        double r40423551 = r40423550 * r40423538;
        double r40423552 = r40423551 + r40423540;
        double r40423553 = b;
        double r40423554 = r40423552 * r40423553;
        double r40423555 = r40423554 * r40423544;
        double r40423556 = r40423555 / r40423546;
        double r40423557 = cos(r40423556);
        double r40423558 = r40423549 * r40423557;
        return r40423558;
}
↓
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 r40423559 = x;
        return r40423559;
}

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
Target44.4
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. Simplified45.7

    \[\leadsto \color{blue}{\cos \left(\frac{b}{16} \cdot \left(\mathsf{fma}\left(a, 2, 1\right) \cdot t\right)\right) \cdot \left(x \cdot \cos \left(\mathsf{fma}\left(y, 2, 1\right) \cdot \frac{t \cdot z}{16}\right)\right)}\]

  3. Taylor expanded around 0 45.0

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

  4. Taylor expanded around 0 44.1

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

  5. Final simplification44.1

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x y z t a b)
  :name "Codec.Picture.Jpg.FastDct:referenceDct from JuicyPixels-3.2.6.1"

  :herbie-target
  (* x (cos (* (/ b 16.0) (/ t (+ (- 1.0 (* a 2.0)) (pow (* a 2.0) 2.0))))))

  (* (* x (cos (/ (* (* (+ (* y 2.0) 1.0) z) t) 16.0))) (cos (/ (* (* (+ (* a 2.0) 1.0) b) t) 16.0))))