Average Error: 0.1 → 0.0
Time: 1.4s
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 r197924 = x;
        double r197925 = y;
        double r197926 = r197924 * r197925;
        double r197927 = z;
        double r197928 = t;
        double r197929 = r197927 * r197928;
        double r197930 = 16.0;
        double r197931 = r197929 / r197930;
        double r197932 = r197926 + r197931;
        double r197933 = a;
        double r197934 = b;
        double r197935 = r197933 * r197934;
        double r197936 = 4.0;
        double r197937 = r197935 / r197936;
        double r197938 = r197932 - r197937;
        double r197939 = c;
        double r197940 = r197938 + r197939;
        return r197940;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r197941 = z;
        double r197942 = t;
        double r197943 = 16.0;
        double r197944 = r197942 / r197943;
        double r197945 = y;
        double r197946 = x;
        double r197947 = a;
        double r197948 = 4.0;
        double r197949 = r197947 / r197948;
        double r197950 = -r197949;
        double r197951 = b;
        double r197952 = c;
        double r197953 = fma(r197950, r197951, r197952);
        double r197954 = fma(r197945, r197946, r197953);
        double r197955 = fma(r197941, r197944, r197954);
        return r197955;
}

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 2020060 +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))