Average Error: 0.1 → 0.1
Time: 21.3s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{t}{16}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b \cdot a}{4}\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(\frac{t}{16}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b \cdot a}{4}\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r8905929 = x;
        double r8905930 = y;
        double r8905931 = r8905929 * r8905930;
        double r8905932 = z;
        double r8905933 = t;
        double r8905934 = r8905932 * r8905933;
        double r8905935 = 16.0;
        double r8905936 = r8905934 / r8905935;
        double r8905937 = r8905931 + r8905936;
        double r8905938 = a;
        double r8905939 = b;
        double r8905940 = r8905938 * r8905939;
        double r8905941 = 4.0;
        double r8905942 = r8905940 / r8905941;
        double r8905943 = r8905937 - r8905942;
        double r8905944 = c;
        double r8905945 = r8905943 + r8905944;
        return r8905945;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r8905946 = t;
        double r8905947 = 16.0;
        double r8905948 = r8905946 / r8905947;
        double r8905949 = z;
        double r8905950 = y;
        double r8905951 = x;
        double r8905952 = c;
        double r8905953 = fma(r8905950, r8905951, r8905952);
        double r8905954 = b;
        double r8905955 = a;
        double r8905956 = r8905954 * r8905955;
        double r8905957 = 4.0;
        double r8905958 = r8905956 / r8905957;
        double r8905959 = r8905953 - r8905958;
        double r8905960 = fma(r8905948, r8905949, r8905959);
        return r8905960;
}

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. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, \mathsf{fma}\left(y, x, c\right) - \frac{a \cdot b}{4}\right)}\]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\frac{t}{16}, z, \mathsf{fma}\left(y, x, c\right) - \frac{b \cdot a}{4}\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))