Average Error: 0.1 → 0.0
Time: 5.8s
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 r170575 = x;
        double r170576 = y;
        double r170577 = r170575 * r170576;
        double r170578 = z;
        double r170579 = t;
        double r170580 = r170578 * r170579;
        double r170581 = 16.0;
        double r170582 = r170580 / r170581;
        double r170583 = r170577 + r170582;
        double r170584 = a;
        double r170585 = b;
        double r170586 = r170584 * r170585;
        double r170587 = 4.0;
        double r170588 = r170586 / r170587;
        double r170589 = r170583 - r170588;
        double r170590 = c;
        double r170591 = r170589 + r170590;
        return r170591;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r170592 = z;
        double r170593 = t;
        double r170594 = 16.0;
        double r170595 = r170593 / r170594;
        double r170596 = y;
        double r170597 = x;
        double r170598 = a;
        double r170599 = 4.0;
        double r170600 = r170598 / r170599;
        double r170601 = -r170600;
        double r170602 = b;
        double r170603 = c;
        double r170604 = fma(r170601, r170602, r170603);
        double r170605 = fma(r170596, r170597, r170604);
        double r170606 = fma(r170592, r170595, r170605);
        return r170606;
}

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 2020056 +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))