Average Error: 46.7 → 44.5
Time: 19.5s
Precision: 64
\[\left(x \cdot \cos \left(\frac{\left(\left(y \cdot 2 + 1\right) \cdot z\right) \cdot t}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]
↓
\[\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)\]
\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)
↓
\left(x \cdot \cos \left(\frac{0}{16}\right)\right) \cdot \cos \left(\frac{0}{16}\right)
double f(double x, double y, double z, double t, double a, double b) {
        double r578569 = x;
        double r578570 = y;
        double r578571 = 2.0;
        double r578572 = r578570 * r578571;
        double r578573 = 1.0;
        double r578574 = r578572 + r578573;
        double r578575 = z;
        double r578576 = r578574 * r578575;
        double r578577 = t;
        double r578578 = r578576 * r578577;
        double r578579 = 16.0;
        double r578580 = r578578 / r578579;
        double r578581 = cos(r578580);
        double r578582 = r578569 * r578581;
        double r578583 = a;
        double r578584 = r578583 * r578571;
        double r578585 = r578584 + r578573;
        double r578586 = b;
        double r578587 = r578585 * r578586;
        double r578588 = r578587 * r578577;
        double r578589 = r578588 / r578579;
        double r578590 = cos(r578589);
        double r578591 = r578582 * r578590;
        return r578591;
}
↓
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 r578592 = x;
        double r578593 = 0.0;
        double r578594 = 16.0;
        double r578595 = r578593 / r578594;
        double r578596 = cos(r578595);
        double r578597 = r578592 * r578596;
        double r578598 = r578597 * r578596;
        return r578598;
}

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

Original46.7
Target44.8
Herbie44.5
\[x \cdot \cos \left(\frac{b}{16} \cdot \frac{t}{\left(1 - a \cdot 2\right) + {\left(a \cdot 2\right)}^{2}}\right)\]

Derivation

  1. Initial program 46.7

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

  2. Taylor expanded around 0 45.8

    \[\leadsto \left(x \cdot \cos \left(\frac{\color{blue}{0}}{16}\right)\right) \cdot \cos \left(\frac{\left(\left(a \cdot 2 + 1\right) \cdot b\right) \cdot t}{16}\right)\]

  3. Taylor expanded around 0 44.5

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

  4. Final simplification44.5

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

Reproduce

herbie shell --seed 2020042 +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))))