Average Error: 0.1 → 0.0
Time: 6.3s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(\frac{a}{4}, -b, c\right)\right)\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, \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 r123271 = x;
        double r123272 = y;
        double r123273 = r123271 * r123272;
        double r123274 = z;
        double r123275 = t;
        double r123276 = r123274 * r123275;
        double r123277 = 16.0;
        double r123278 = r123276 / r123277;
        double r123279 = r123273 + r123278;
        double r123280 = a;
        double r123281 = b;
        double r123282 = r123280 * r123281;
        double r123283 = 4.0;
        double r123284 = r123282 / r123283;
        double r123285 = r123279 - r123284;
        double r123286 = c;
        double r123287 = r123285 + r123286;
        return r123287;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r123288 = z;
        double r123289 = 16.0;
        double r123290 = r123288 / r123289;
        double r123291 = t;
        double r123292 = x;
        double r123293 = y;
        double r123294 = a;
        double r123295 = 4.0;
        double r123296 = r123294 / r123295;
        double r123297 = b;
        double r123298 = -r123297;
        double r123299 = c;
        double r123300 = fma(r123296, r123298, r123299);
        double r123301 = fma(r123292, r123293, r123300);
        double r123302 = fma(r123290, r123291, r123301);
        return r123302;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))