Average Error: 0.1 → 0.1
Time: 15.8s
Precision: 64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)\]
↓
\[\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}\]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
\frac{\mathsf{fma}\left(y, \sqrt{z}, x\right) \cdot 1}{2}
double f(double x, double y, double z) {
        double r7545086 = 1.0;
        double r7545087 = 2.0;
        double r7545088 = r7545086 / r7545087;
        double r7545089 = x;
        double r7545090 = y;
        double r7545091 = z;
        double r7545092 = sqrt(r7545091);
        double r7545093 = r7545090 * r7545092;
        double r7545094 = r7545089 + r7545093;
        double r7545095 = r7545088 * r7545094;
        return r7545095;
}
↓
double f(double x, double y, double z) {
        double r7545096 = y;
        double r7545097 = z;
        double r7545098 = sqrt(r7545097);
        double r7545099 = x;
        double r7545100 = fma(r7545096, r7545098, r7545099);
        double r7545101 = 1.0;
        double r7545102 = r7545100 * r7545101;
        double r7545103 = 2.0;
        double r7545104 = r7545102 / r7545103;
        return r7545104;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))