Average Error: 0.1 → 0.0
Time: 5.6s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r198750 = x;
        double r198751 = y;
        double r198752 = r198750 * r198751;
        double r198753 = z;
        double r198754 = t;
        double r198755 = r198753 * r198754;
        double r198756 = 16.0;
        double r198757 = r198755 / r198756;
        double r198758 = r198752 + r198757;
        double r198759 = a;
        double r198760 = b;
        double r198761 = r198759 * r198760;
        double r198762 = 4.0;
        double r198763 = r198761 / r198762;
        double r198764 = r198758 - r198763;
        double r198765 = c;
        double r198766 = r198764 + r198765;
        return r198766;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r198767 = z;
        double r198768 = t;
        double r198769 = 16.0;
        double r198770 = r198768 / r198769;
        double r198771 = y;
        double r198772 = x;
        double r198773 = a;
        double r198774 = 4.0;
        double r198775 = r198773 / r198774;
        double r198776 = -r198775;
        double r198777 = b;
        double r198778 = c;
        double r198779 = fma(r198776, r198777, r198778);
        double r198780 = fma(r198771, r198772, r198779);
        double r198781 = fma(r198767, r198770, r198780);
        return r198781;
}

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

Bits error versus c

Derivation

  1. Initial program 0.1

    \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)}\]

  3. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, \mathsf{fma}\left(y, x, \mathsf{fma}\left(-\frac{a}{4}, b, c\right)\right)\right)\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16)) (/ (* a b) 4)) c))