Average Error: 46.0 → 44.5
Time: 1.1m
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]
↓
\[x\]
\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)
↓
x
double f(double x, double y, double z, double t, double a, double b) {
        double r40332572 = x;
        double r40332573 = y;
        double r40332574 = 2.0;
        double r40332575 = r40332573 * r40332574;
        double r40332576 = 1.0;
        double r40332577 = r40332575 + r40332576;
        double r40332578 = z;
        double r40332579 = r40332577 * r40332578;
        double r40332580 = t;
        double r40332581 = r40332579 * r40332580;
        double r40332582 = 16.0;
        double r40332583 = r40332581 / r40332582;
        double r40332584 = cos(r40332583);
        double r40332585 = r40332572 * r40332584;
        double r40332586 = a;
        double r40332587 = r40332586 * r40332574;
        double r40332588 = r40332587 + r40332576;
        double r40332589 = b;
        double r40332590 = r40332588 * r40332589;
        double r40332591 = r40332590 * r40332580;
        double r40332592 = r40332591 / r40332582;
        double r40332593 = cos(r40332592);
        double r40332594 = r40332585 * r40332593;
        return r40332594;
}
↓
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 r40332595 = x;
        return r40332595;
}

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

Original46.0
Target44.7
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16.0} \cdot \frac{t}{\left(1.0 - a \cdot 2.0\right) + {\left(a \cdot 2.0\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.0

    \[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2.0 + 1.0\right) \cdot z\right) \cdot t}{16.0}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2.0 + 1.0\right) \cdot b\right) \cdot t}{16.0}\right)\]

  2. Simplified45.9

    \[\leadsto \color{blue}{\left(\cos \left(\frac{t}{\frac{\frac{16.0}{z}}{\mathsf{fma}\left(2.0, y, 1.0\right)}}\right) \cdot x\right) \cdot \cos \left(\frac{b}{\frac{\frac{16.0}{t}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)}\]

  3. Taylor expanded around 0 45.4

    \[\leadsto \left(\color{blue}{1} \cdot x\right) \cdot \cos \left(\frac{b}{\frac{\frac{16.0}{t}}{\mathsf{fma}\left(a, 2.0, 1.0\right)}}\right)\]

  4. Taylor expanded around 0 44.5

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

  5. Final simplification44.5

    \[\leadsto x\]

Reproduce

herbie shell --seed 2019163 +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))))))

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