Average Error: 0.1 → 0.0
Time: 2.1s
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 r187646 = x;
        double r187647 = y;
        double r187648 = r187646 * r187647;
        double r187649 = z;
        double r187650 = t;
        double r187651 = r187649 * r187650;
        double r187652 = 16.0;
        double r187653 = r187651 / r187652;
        double r187654 = r187648 + r187653;
        double r187655 = a;
        double r187656 = b;
        double r187657 = r187655 * r187656;
        double r187658 = 4.0;
        double r187659 = r187657 / r187658;
        double r187660 = r187654 - r187659;
        double r187661 = c;
        double r187662 = r187660 + r187661;
        return r187662;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r187663 = z;
        double r187664 = t;
        double r187665 = 16.0;
        double r187666 = r187664 / r187665;
        double r187667 = y;
        double r187668 = x;
        double r187669 = a;
        double r187670 = 4.0;
        double r187671 = r187669 / r187670;
        double r187672 = -r187671;
        double r187673 = b;
        double r187674 = c;
        double r187675 = fma(r187672, r187673, r187674);
        double r187676 = fma(r187667, r187668, r187675);
        double r187677 = fma(r187663, r187666, r187676);
        return r187677;
}

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 2020089 +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))