Average Error: 0.1 → 0.0
Time: 1.9s
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 r217059 = x;
        double r217060 = y;
        double r217061 = r217059 * r217060;
        double r217062 = z;
        double r217063 = t;
        double r217064 = r217062 * r217063;
        double r217065 = 16.0;
        double r217066 = r217064 / r217065;
        double r217067 = r217061 + r217066;
        double r217068 = a;
        double r217069 = b;
        double r217070 = r217068 * r217069;
        double r217071 = 4.0;
        double r217072 = r217070 / r217071;
        double r217073 = r217067 - r217072;
        double r217074 = c;
        double r217075 = r217073 + r217074;
        return r217075;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r217076 = z;
        double r217077 = t;
        double r217078 = 16.0;
        double r217079 = r217077 / r217078;
        double r217080 = y;
        double r217081 = x;
        double r217082 = a;
        double r217083 = 4.0;
        double r217084 = r217082 / r217083;
        double r217085 = -r217084;
        double r217086 = b;
        double r217087 = c;
        double r217088 = fma(r217085, r217086, r217087);
        double r217089 = fma(r217080, r217081, r217088);
        double r217090 = fma(r217076, r217079, r217089);
        return r217090;
}

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 2020027 +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))