Average Error: 0.2 → 0.2
Time: 13.1s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}
double f(double x, double y, double z) {
        double r139118 = 1.0;
        double r139119 = 2.0;
        double r139120 = r139118 / r139119;
        double r139121 = x;
        double r139122 = y;
        double r139123 = z;
        double r139124 = sqrt(r139123);
        double r139125 = r139122 * r139124;
        double r139126 = r139121 + r139125;
        double r139127 = r139120 * r139126;
        return r139127;
}
↓
double f(double x, double y, double z) {
        double r139128 = z;
        double r139129 = sqrt(r139128);
        double r139130 = y;
        double r139131 = x;
        double r139132 = fma(r139129, r139130, r139131);
        double r139133 = 1.0;
        double r139134 = 2.0;
        double r139135 = r139133 / r139134;
        double r139136 = r139132 * r139135;
        return r139136;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.2

    \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{2}\right)} \cdot \left(x + y \cdot \sqrt{z}\right)\]

  4. Applied associate-*l*0.2

    \[\leadsto \color{blue}{1 \cdot \left(\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\right)}\]

  5. Simplified0.2

    \[\leadsto 1 \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}\right)}\]

  6. Final simplification0.2

    \[\leadsto \mathsf{fma}\left(\sqrt{z}, y, x\right) \cdot \frac{1}{2}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1 2) (+ x (* y (sqrt z)))))