Average Error: 0.2 → 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 r160177 = x;
        double r160178 = y;
        double r160179 = r160177 * r160178;
        double r160180 = z;
        double r160181 = t;
        double r160182 = r160180 * r160181;
        double r160183 = 16.0;
        double r160184 = r160182 / r160183;
        double r160185 = r160179 + r160184;
        double r160186 = a;
        double r160187 = b;
        double r160188 = r160186 * r160187;
        double r160189 = 4.0;
        double r160190 = r160188 / r160189;
        double r160191 = r160185 - r160190;
        double r160192 = c;
        double r160193 = r160191 + r160192;
        return r160193;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r160194 = z;
        double r160195 = t;
        double r160196 = 16.0;
        double r160197 = r160195 / r160196;
        double r160198 = y;
        double r160199 = x;
        double r160200 = a;
        double r160201 = 4.0;
        double r160202 = r160200 / r160201;
        double r160203 = -r160202;
        double r160204 = b;
        double r160205 = c;
        double r160206 = fma(r160203, r160204, r160205);
        double r160207 = fma(r160198, r160199, r160206);
        double r160208 = fma(r160194, r160197, r160207);
        return r160208;
}

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 2020083 +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))