Average Error: 0.1 → 0.0
Time: 1.5s
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 r259501 = x;
        double r259502 = y;
        double r259503 = r259501 * r259502;
        double r259504 = z;
        double r259505 = t;
        double r259506 = r259504 * r259505;
        double r259507 = 16.0;
        double r259508 = r259506 / r259507;
        double r259509 = r259503 + r259508;
        double r259510 = a;
        double r259511 = b;
        double r259512 = r259510 * r259511;
        double r259513 = 4.0;
        double r259514 = r259512 / r259513;
        double r259515 = r259509 - r259514;
        double r259516 = c;
        double r259517 = r259515 + r259516;
        return r259517;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r259518 = z;
        double r259519 = t;
        double r259520 = 16.0;
        double r259521 = r259519 / r259520;
        double r259522 = y;
        double r259523 = x;
        double r259524 = a;
        double r259525 = 4.0;
        double r259526 = r259524 / r259525;
        double r259527 = -r259526;
        double r259528 = b;
        double r259529 = c;
        double r259530 = fma(r259527, r259528, r259529);
        double r259531 = fma(r259522, r259523, r259530);
        double r259532 = fma(r259518, r259521, r259531);
        return r259532;
}

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 2020062 +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))