Average Error: 0.1 → 0.0
Time: 2.8s
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 r1028 = x;
        double r1029 = y;
        double r1030 = r1028 * r1029;
        double r1031 = z;
        double r1032 = t;
        double r1033 = r1031 * r1032;
        double r1034 = 16.0;
        double r1035 = r1033 / r1034;
        double r1036 = r1030 + r1035;
        double r1037 = a;
        double r1038 = b;
        double r1039 = r1037 * r1038;
        double r1040 = 4.0;
        double r1041 = r1039 / r1040;
        double r1042 = r1036 - r1041;
        double r1043 = c;
        double r1044 = r1042 + r1043;
        return r1044;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r1045 = z;
        double r1046 = t;
        double r1047 = 16.0;
        double r1048 = r1046 / r1047;
        double r1049 = y;
        double r1050 = x;
        double r1051 = a;
        double r1052 = 4.0;
        double r1053 = r1051 / r1052;
        double r1054 = -r1053;
        double r1055 = b;
        double r1056 = c;
        double r1057 = fma(r1054, r1055, r1056);
        double r1058 = fma(r1049, r1050, r1057);
        double r1059 = fma(r1045, r1048, r1058);
        return r1059;
}

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 2020025 +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))