Average Error: 0.2 → 0.1
Time: 18.3s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{1}{2} \cdot \mathsf{fma}\left(\sqrt{z}, y, x\right)
double f(double x, double y, double z) {
        double r198101 = 1.0;
        double r198102 = 2.0;
        double r198103 = r198101 / r198102;
        double r198104 = x;
        double r198105 = y;
        double r198106 = z;
        double r198107 = sqrt(r198106);
        double r198108 = r198105 * r198107;
        double r198109 = r198104 + r198108;
        double r198110 = r198103 * r198109;
        return r198110;
}
↓
double f(double x, double y, double z) {
        double r198111 = 1.0;
        double r198112 = 2.0;
        double r198113 = r198111 / r198112;
        double r198114 = z;
        double r198115 = sqrt(r198114);
        double r198116 = y;
        double r198117 = x;
        double r198118 = fma(r198115, r198116, r198117);
        double r198119 = r198113 * r198118;
        return r198119;
}

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. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019347 +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)))))