Average Error: 0.1 → 0.0
Time: 2.3s
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 r193331 = x;
        double r193332 = y;
        double r193333 = r193331 * r193332;
        double r193334 = z;
        double r193335 = t;
        double r193336 = r193334 * r193335;
        double r193337 = 16.0;
        double r193338 = r193336 / r193337;
        double r193339 = r193333 + r193338;
        double r193340 = a;
        double r193341 = b;
        double r193342 = r193340 * r193341;
        double r193343 = 4.0;
        double r193344 = r193342 / r193343;
        double r193345 = r193339 - r193344;
        double r193346 = c;
        double r193347 = r193345 + r193346;
        return r193347;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r193348 = z;
        double r193349 = t;
        double r193350 = 16.0;
        double r193351 = r193349 / r193350;
        double r193352 = y;
        double r193353 = x;
        double r193354 = a;
        double r193355 = 4.0;
        double r193356 = r193354 / r193355;
        double r193357 = -r193356;
        double r193358 = b;
        double r193359 = c;
        double r193360 = fma(r193357, r193358, r193359);
        double r193361 = fma(r193352, r193353, r193360);
        double r193362 = fma(r193348, r193351, r193361);
        return r193362;
}

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 2020035 +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))