Average Error: 0.1 → 0.0
Time: 1.8s
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 r192836 = x;
        double r192837 = y;
        double r192838 = r192836 * r192837;
        double r192839 = z;
        double r192840 = t;
        double r192841 = r192839 * r192840;
        double r192842 = 16.0;
        double r192843 = r192841 / r192842;
        double r192844 = r192838 + r192843;
        double r192845 = a;
        double r192846 = b;
        double r192847 = r192845 * r192846;
        double r192848 = 4.0;
        double r192849 = r192847 / r192848;
        double r192850 = r192844 - r192849;
        double r192851 = c;
        double r192852 = r192850 + r192851;
        return r192852;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r192853 = z;
        double r192854 = t;
        double r192855 = 16.0;
        double r192856 = r192854 / r192855;
        double r192857 = y;
        double r192858 = x;
        double r192859 = a;
        double r192860 = 4.0;
        double r192861 = r192859 / r192860;
        double r192862 = -r192861;
        double r192863 = b;
        double r192864 = c;
        double r192865 = fma(r192862, r192863, r192864);
        double r192866 = fma(r192857, r192858, r192865);
        double r192867 = fma(r192853, r192856, r192866);
        return r192867;
}

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 2019353 +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))