Average Error: 0.1 → 0.0
Time: 4.4s
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 r219643 = x;
        double r219644 = y;
        double r219645 = r219643 * r219644;
        double r219646 = z;
        double r219647 = t;
        double r219648 = r219646 * r219647;
        double r219649 = 16.0;
        double r219650 = r219648 / r219649;
        double r219651 = r219645 + r219650;
        double r219652 = a;
        double r219653 = b;
        double r219654 = r219652 * r219653;
        double r219655 = 4.0;
        double r219656 = r219654 / r219655;
        double r219657 = r219651 - r219656;
        double r219658 = c;
        double r219659 = r219657 + r219658;
        return r219659;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r219660 = z;
        double r219661 = t;
        double r219662 = 16.0;
        double r219663 = r219661 / r219662;
        double r219664 = y;
        double r219665 = x;
        double r219666 = a;
        double r219667 = 4.0;
        double r219668 = r219666 / r219667;
        double r219669 = -r219668;
        double r219670 = b;
        double r219671 = c;
        double r219672 = fma(r219669, r219670, r219671);
        double r219673 = fma(r219664, r219665, r219672);
        double r219674 = fma(r219660, r219663, r219673);
        return r219674;
}

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 2020021 +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))