Average Error: 0.1 → 0.0
Time: 13.5s
Precision: 64
\[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c\]
↓
\[\mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c\right) - \left(a \cdot 0.25\right) \cdot b\right)\]
\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c
↓
\mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c\right) - \left(a \cdot 0.25\right) \cdot b\right)
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r8060163 = x;
        double r8060164 = y;
        double r8060165 = r8060163 * r8060164;
        double r8060166 = z;
        double r8060167 = t;
        double r8060168 = r8060166 * r8060167;
        double r8060169 = 16.0;
        double r8060170 = r8060168 / r8060169;
        double r8060171 = r8060165 + r8060170;
        double r8060172 = a;
        double r8060173 = b;
        double r8060174 = r8060172 * r8060173;
        double r8060175 = 4.0;
        double r8060176 = r8060174 / r8060175;
        double r8060177 = r8060171 - r8060176;
        double r8060178 = c;
        double r8060179 = r8060177 + r8060178;
        return r8060179;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c) {
        double r8060180 = t;
        double r8060181 = z;
        double r8060182 = 16.0;
        double r8060183 = r8060181 / r8060182;
        double r8060184 = x;
        double r8060185 = y;
        double r8060186 = c;
        double r8060187 = fma(r8060184, r8060185, r8060186);
        double r8060188 = a;
        double r8060189 = 0.25;
        double r8060190 = r8060188 * r8060189;
        double r8060191 = b;
        double r8060192 = r8060190 * r8060191;
        double r8060193 = r8060187 - r8060192;
        double r8060194 = fma(r8060180, r8060183, r8060193);
        return r8060194;
}

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. Simplified0.1

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

  3. Taylor expanded around inf 0.0

    \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{\left(c + x \cdot y\right) - 0.25 \cdot \left(a \cdot b\right)}\right)\]

  4. Simplified0.0

    \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \color{blue}{\mathsf{fma}\left(x, y, c\right) - \left(0.25 \cdot a\right) \cdot b}\right)\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(t, \frac{z}{16}, \mathsf{fma}\left(x, y, c\right) - \left(a \cdot 0.25\right) \cdot b\right)\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))