Average Error: 0.1 → 0.0
Time: 4.3s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(\frac{a}{4}, -b, c\right)\right)\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(\frac{z}{16}, t, \mathsf{fma}\left(x, y, \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 r40107 = x;
        double r40108 = y;
        double r40109 = r40107 * r40108;
        double r40110 = z;
        double r40111 = t;
        double r40112 = r40110 * r40111;
        double r40113 = 16.0;
        double r40114 = r40112 / r40113;
        double r40115 = r40109 + r40114;
        double r40116 = a;
        double r40117 = b;
        double r40118 = r40116 * r40117;
        double r40119 = 4.0;
        double r40120 = r40118 / r40119;
        double r40121 = r40115 - r40120;
        double r40122 = c;
        double r40123 = r40121 + r40122;
        return r40123;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r40124 = z;
        double r40125 = 16.0;
        double r40126 = r40124 / r40125;
        double r40127 = t;
        double r40128 = x;
        double r40129 = y;
        double r40130 = a;
        double r40131 = 4.0;
        double r40132 = r40130 / r40131;
        double r40133 = b;
        double r40134 = -r40133;
        double r40135 = c;
        double r40136 = fma(r40132, r40134, r40135);
        double r40137 = fma(r40128, r40129, r40136);
        double r40138 = fma(r40126, r40127, r40137);
        return r40138;
}

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

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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))