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 r189056 = x;
        double r189057 = y;
        double r189058 = r189056 * r189057;
        double r189059 = z;
        double r189060 = t;
        double r189061 = r189059 * r189060;
        double r189062 = 16.0;
        double r189063 = r189061 / r189062;
        double r189064 = r189058 + r189063;
        double r189065 = a;
        double r189066 = b;
        double r189067 = r189065 * r189066;
        double r189068 = 4.0;
        double r189069 = r189067 / r189068;
        double r189070 = r189064 - r189069;
        double r189071 = c;
        double r189072 = r189070 + r189071;
        return r189072;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r189073 = z;
        double r189074 = t;
        double r189075 = 16.0;
        double r189076 = r189074 / r189075;
        double r189077 = y;
        double r189078 = x;
        double r189079 = a;
        double r189080 = 4.0;
        double r189081 = r189079 / r189080;
        double r189082 = -r189081;
        double r189083 = b;
        double r189084 = c;
        double r189085 = fma(r189082, r189083, r189084);
        double r189086 = fma(r189077, r189078, r189085);
        double r189087 = fma(r189073, r189076, r189086);
        return r189087;
}

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 2020036 +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))