Average Error: 0.1 → 0.0
Time: 1.8s
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 r196318 = x;
        double r196319 = y;
        double r196320 = r196318 * r196319;
        double r196321 = z;
        double r196322 = t;
        double r196323 = r196321 * r196322;
        double r196324 = 16.0;
        double r196325 = r196323 / r196324;
        double r196326 = r196320 + r196325;
        double r196327 = a;
        double r196328 = b;
        double r196329 = r196327 * r196328;
        double r196330 = 4.0;
        double r196331 = r196329 / r196330;
        double r196332 = r196326 - r196331;
        double r196333 = c;
        double r196334 = r196332 + r196333;
        return r196334;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r196335 = z;
        double r196336 = t;
        double r196337 = 16.0;
        double r196338 = r196336 / r196337;
        double r196339 = y;
        double r196340 = x;
        double r196341 = a;
        double r196342 = 4.0;
        double r196343 = r196341 / r196342;
        double r196344 = -r196343;
        double r196345 = b;
        double r196346 = c;
        double r196347 = fma(r196344, r196345, r196346);
        double r196348 = fma(r196339, r196340, r196347);
        double r196349 = fma(r196335, r196338, r196348);
        return r196349;
}

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 2019353 +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))