Average Error: 45.9 → 44.0
Time: 27.1s
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 r628584 = x;
        double r628585 = y;
        double r628586 = 2.0;
        double r628587 = r628585 * r628586;
        double r628588 = 1.0;
        double r628589 = r628587 + r628588;
        double r628590 = z;
        double r628591 = r628589 * r628590;
        double r628592 = t;
        double r628593 = r628591 * r628592;
        double r628594 = 16.0;
        double r628595 = r628593 / r628594;
        double r628596 = cos(r628595);
        double r628597 = r628584 * r628596;
        double r628598 = a;
        double r628599 = r628598 * r628586;
        double r628600 = r628599 + r628588;
        double r628601 = b;
        double r628602 = r628600 * r628601;
        double r628603 = r628602 * r628592;
        double r628604 = r628603 / r628594;
        double r628605 = cos(r628604);
        double r628606 = r628597 * r628605;
        return r628606;
}
↓
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 r628607 = x;
        return r628607;
}

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

Original45.9
Target44.3
Herbie44.0
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 45.9

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

  2. Simplified45.9

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

  3. Taylor expanded around 0 45.2

    \[\leadsto \cos \left(\frac{\color{blue}{0}}{16}\right) \cdot \left(x \cdot \cos \left(\frac{t \cdot \left(\mathsf{fma}\left(y, 2, 1\right) \cdot z\right)}{16}\right)\right)\]

  4. Taylor expanded around 0 44.0

    \[\leadsto \cos \left(\frac{0}{16}\right) \cdot \left(x \cdot \color{blue}{1}\right)\]

  5. Final simplification44.0

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019325 +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))))