Average Error: 0.1 → 0.0
Time: 1.5s
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 r165853 = x;
        double r165854 = y;
        double r165855 = r165853 * r165854;
        double r165856 = z;
        double r165857 = t;
        double r165858 = r165856 * r165857;
        double r165859 = 16.0;
        double r165860 = r165858 / r165859;
        double r165861 = r165855 + r165860;
        double r165862 = a;
        double r165863 = b;
        double r165864 = r165862 * r165863;
        double r165865 = 4.0;
        double r165866 = r165864 / r165865;
        double r165867 = r165861 - r165866;
        double r165868 = c;
        double r165869 = r165867 + r165868;
        return r165869;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r165870 = z;
        double r165871 = t;
        double r165872 = 16.0;
        double r165873 = r165871 / r165872;
        double r165874 = y;
        double r165875 = x;
        double r165876 = a;
        double r165877 = 4.0;
        double r165878 = r165876 / r165877;
        double r165879 = -r165878;
        double r165880 = b;
        double r165881 = c;
        double r165882 = fma(r165879, r165880, r165881);
        double r165883 = fma(r165874, r165875, r165882);
        double r165884 = fma(r165870, r165873, r165883);
        return r165884;
}

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 2020020 +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))