Average Error: 0.0 → 0.0
Time: 33.1s
Precision: 64
\[\left(\left(x \cdot y + z \cdot t\right) + a \cdot b\right) + c \cdot i\]
↓
\[\mathsf{fma}\left(t, z, \mathsf{fma}\left(a, b, \mathsf{fma}\left(y, x, c \cdot i\right)\right)\right)\]
\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)
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r6610105 = x;
        double r6610106 = y;
        double r6610107 = r6610105 * r6610106;
        double r6610108 = z;
        double r6610109 = t;
        double r6610110 = r6610108 * r6610109;
        double r6610111 = r6610107 + r6610110;
        double r6610112 = a;
        double r6610113 = b;
        double r6610114 = r6610112 * r6610113;
        double r6610115 = r6610111 + r6610114;
        double r6610116 = c;
        double r6610117 = i;
        double r6610118 = r6610116 * r6610117;
        double r6610119 = r6610115 + r6610118;
        return r6610119;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r6610120 = t;
        double r6610121 = z;
        double r6610122 = a;
        double r6610123 = b;
        double r6610124 = y;
        double r6610125 = x;
        double r6610126 = c;
        double r6610127 = i;
        double r6610128 = r6610126 * r6610127;
        double r6610129 = fma(r6610124, r6610125, r6610128);
        double r6610130 = fma(r6610122, r6610123, r6610129);
        double r6610131 = fma(r6610120, r6610121, r6610130);
        return r6610131;
}

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. Simplified0.0

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

  4. Applied add-sqr-sqrt32.2

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

  6. Applied *-un-lft-identity32.2

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

  7. Applied sqrt-prod32.2

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

  8. Applied *-un-lft-identity32.2

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

  9. Applied sqrt-prod32.2

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

  10. Applied swap-sqr32.2

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

  11. Simplified32.2

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

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, C"
  (+ (+ (+ (* x y) (* z t)) (* a b)) (* c i)))