Average Error: 0.1 → 0.0
Time: 1.4s
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 r207115 = x;
        double r207116 = y;
        double r207117 = r207115 * r207116;
        double r207118 = z;
        double r207119 = t;
        double r207120 = r207118 * r207119;
        double r207121 = 16.0;
        double r207122 = r207120 / r207121;
        double r207123 = r207117 + r207122;
        double r207124 = a;
        double r207125 = b;
        double r207126 = r207124 * r207125;
        double r207127 = 4.0;
        double r207128 = r207126 / r207127;
        double r207129 = r207123 - r207128;
        double r207130 = c;
        double r207131 = r207129 + r207130;
        return r207131;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r207132 = z;
        double r207133 = t;
        double r207134 = 16.0;
        double r207135 = r207133 / r207134;
        double r207136 = y;
        double r207137 = x;
        double r207138 = a;
        double r207139 = 4.0;
        double r207140 = r207138 / r207139;
        double r207141 = -r207140;
        double r207142 = b;
        double r207143 = c;
        double r207144 = fma(r207141, r207142, r207143);
        double r207145 = fma(r207136, r207137, r207144);
        double r207146 = fma(r207132, r207135, r207145);
        return r207146;
}

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 2020062 +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))