Average Error: 45.9 → 44.0
Time: 12.5s
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 r910634 = x;
        double r910635 = y;
        double r910636 = 2.0;
        double r910637 = r910635 * r910636;
        double r910638 = 1.0;
        double r910639 = r910637 + r910638;
        double r910640 = z;
        double r910641 = r910639 * r910640;
        double r910642 = t;
        double r910643 = r910641 * r910642;
        double r910644 = 16.0;
        double r910645 = r910643 / r910644;
        double r910646 = cos(r910645);
        double r910647 = r910634 * r910646;
        double r910648 = a;
        double r910649 = r910648 * r910636;
        double r910650 = r910649 + r910638;
        double r910651 = b;
        double r910652 = r910650 * r910651;
        double r910653 = r910652 * r910642;
        double r910654 = r910653 / r910644;
        double r910655 = cos(r910654);
        double r910656 = r910647 * r910655;
        return r910656;
}
↓
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 r910657 = x;
        double r910658 = 0.0;
        double r910659 = 16.0;
        double r910660 = r910658 / r910659;
        double r910661 = cos(r910660);
        double r910662 = r910657 * r910661;
        double r910663 = r910662 * r910661;
        return r910663;
}

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
Target44.2
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. Taylor expanded around 0 45.3

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

  3. Taylor expanded around 0 44.0

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

  4. Final simplification44.0

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

Reproduce

herbie shell --seed 2020024 +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))))