Average Error: 0.2 → 0.0
Time: 1.3s
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 r191580 = x;
        double r191581 = y;
        double r191582 = r191580 * r191581;
        double r191583 = z;
        double r191584 = t;
        double r191585 = r191583 * r191584;
        double r191586 = 16.0;
        double r191587 = r191585 / r191586;
        double r191588 = r191582 + r191587;
        double r191589 = a;
        double r191590 = b;
        double r191591 = r191589 * r191590;
        double r191592 = 4.0;
        double r191593 = r191591 / r191592;
        double r191594 = r191588 - r191593;
        double r191595 = c;
        double r191596 = r191594 + r191595;
        return r191596;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r191597 = z;
        double r191598 = t;
        double r191599 = 16.0;
        double r191600 = r191598 / r191599;
        double r191601 = y;
        double r191602 = x;
        double r191603 = a;
        double r191604 = 4.0;
        double r191605 = r191603 / r191604;
        double r191606 = -r191605;
        double r191607 = b;
        double r191608 = c;
        double r191609 = fma(r191606, r191607, r191608);
        double r191610 = fma(r191601, r191602, r191609);
        double r191611 = fma(r191597, r191600, r191610);
        return r191611;
}

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 2020036 +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))