Average Error: 0.2 → 0.0
Time: 4.2s
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 r194260 = x;
        double r194261 = y;
        double r194262 = r194260 * r194261;
        double r194263 = z;
        double r194264 = t;
        double r194265 = r194263 * r194264;
        double r194266 = 16.0;
        double r194267 = r194265 / r194266;
        double r194268 = r194262 + r194267;
        double r194269 = a;
        double r194270 = b;
        double r194271 = r194269 * r194270;
        double r194272 = 4.0;
        double r194273 = r194271 / r194272;
        double r194274 = r194268 - r194273;
        double r194275 = c;
        double r194276 = r194274 + r194275;
        return r194276;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r194277 = z;
        double r194278 = t;
        double r194279 = 16.0;
        double r194280 = r194278 / r194279;
        double r194281 = y;
        double r194282 = x;
        double r194283 = a;
        double r194284 = 4.0;
        double r194285 = r194283 / r194284;
        double r194286 = -r194285;
        double r194287 = b;
        double r194288 = c;
        double r194289 = fma(r194286, r194287, r194288);
        double r194290 = fma(r194281, r194282, r194289);
        double r194291 = fma(r194277, r194280, r194290);
        return r194291;
}

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 0.2

    \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]

  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 2020033 +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))