Average Error: 0.1 → 0.0
Time: 12.6s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right) - 0.25 \cdot \left(a \cdot b\right)\right)\]
\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, c\right) - 0.25 \cdot \left(a \cdot b\right)\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r236253 = x;
        double r236254 = y;
        double r236255 = r236253 * r236254;
        double r236256 = z;
        double r236257 = t;
        double r236258 = r236256 * r236257;
        double r236259 = 16.0;
        double r236260 = r236258 / r236259;
        double r236261 = r236255 + r236260;
        double r236262 = a;
        double r236263 = b;
        double r236264 = r236262 * r236263;
        double r236265 = 4.0;
        double r236266 = r236264 / r236265;
        double r236267 = r236261 - r236266;
        double r236268 = c;
        double r236269 = r236267 + r236268;
        return r236269;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r236270 = z;
        double r236271 = 16.0;
        double r236272 = r236270 / r236271;
        double r236273 = t;
        double r236274 = x;
        double r236275 = y;
        double r236276 = c;
        double r236277 = fma(r236274, r236275, r236276);
        double r236278 = 0.25;
        double r236279 = a;
        double r236280 = b;
        double r236281 = r236279 * r236280;
        double r236282 = r236278 * r236281;
        double r236283 = r236277 - r236282;
        double r236284 = fma(r236272, r236273, r236283);
        return r236284;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \color{blue}{\left(x \cdot y + c\right) - 0.25 \cdot \left(a \cdot b\right)}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \color{blue}{\mathsf{fma}\left(x, y, c\right) - 0.25 \cdot \left(a \cdot b\right)}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, c\right) - 0.25 \cdot \left(a \cdot b\right)\right)\]

Reproduce

herbie shell --seed 2019326 +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))