Linear.V4:$cdot from linear-1.19.1.3

Average Error: 0.0 → 0.0

Time: 31.3s

Precision: 64

Math

\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]

↓

\[\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, \mathsf{fma}\left(y, x, c \cdot i\right)\right)\right)\]

C

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r4195265 = x;
        double r4195266 = y;
        double r4195267 = r4195265 * r4195266;
        double r4195268 = z;
        double r4195269 = t;
        double r4195270 = r4195268 * r4195269;
        double r4195271 = r4195267 + r4195270;
        double r4195272 = a;
        double r4195273 = b;
        double r4195274 = r4195272 * r4195273;
        double r4195275 = r4195271 + r4195274;
        double r4195276 = c;
        double r4195277 = i;
        double r4195278 = r4195276 * r4195277;
        double r4195279 = r4195275 + r4195278;
        return r4195279;
}

↓

double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r4195280 = t;
        double r4195281 = z;
        double r4195282 = a;
        double r4195283 = b;
        double r4195284 = y;
        double r4195285 = x;
        double r4195286 = c;
        double r4195287 = i;
        double r4195288 = r4195286 * r4195287;
        double r4195289 = fma(r4195284, r4195285, r4195288);
        double r4195290 = fma(r4195282, r4195283, r4195289);
        double r4195291 = fma(r4195280, r4195281, r4195290);
        return r4195291;
}

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

Bits error versus i

Derivation

1. Initial program 0.0

   \[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\]
3. Using strategy rm

4. Applied add-sqr-sqrt 32.2

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}}\]
5. Using strategy rm

6. Applied *-un-lft-identity 32.2

   \[\leadsto \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \cdot \sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}}\]

7. Applied sqrt-prod 32.2

   \[\leadsto \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \cdot \color{blue}{\left(\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)}\]

8. Applied *-un-lft-identity 32.2

   \[\leadsto \sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}} \cdot \left(\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)\]

9. Applied sqrt-prod 32.2

   \[\leadsto \color{blue}{\left(\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)} \cdot \left(\sqrt{1} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)\]

10. Applied swap-sqr 32.2

    \[\leadsto \color{blue}{\left(\sqrt{1} \cdot \sqrt{1}\right) \cdot \left(\sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)}\]

11. Simplified 32.2

    \[\leadsto \color{blue}{1} \cdot \left(\sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)} \cdot \sqrt{\mathsf{fma}\left(z, t, \mathsf{fma}\left(x, y, \mathsf{fma}\left(c, i, a \cdot b\right)\right)\right)}\right)\]

12. Simplified 0.0

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, \mathsf{fma}\left(y, x, c \cdot i\right)\right)\right)}\]

13. Final simplification 0.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Linear.V4:$cdot from linear-1.19.1.3"
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))