Average Error: 0.2 → 0.0
Time: 2.9s
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 r202556 = x;
        double r202557 = y;
        double r202558 = r202556 * r202557;
        double r202559 = z;
        double r202560 = t;
        double r202561 = r202559 * r202560;
        double r202562 = 16.0;
        double r202563 = r202561 / r202562;
        double r202564 = r202558 + r202563;
        double r202565 = a;
        double r202566 = b;
        double r202567 = r202565 * r202566;
        double r202568 = 4.0;
        double r202569 = r202567 / r202568;
        double r202570 = r202564 - r202569;
        double r202571 = c;
        double r202572 = r202570 + r202571;
        return r202572;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r202573 = z;
        double r202574 = t;
        double r202575 = 16.0;
        double r202576 = r202574 / r202575;
        double r202577 = y;
        double r202578 = x;
        double r202579 = a;
        double r202580 = 4.0;
        double r202581 = r202579 / r202580;
        double r202582 = -r202581;
        double r202583 = b;
        double r202584 = c;
        double r202585 = fma(r202582, r202583, r202584);
        double r202586 = fma(r202577, r202578, r202585);
        double r202587 = fma(r202573, r202576, r202586);
        return r202587;
}

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 2020039 +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))