Average Error: 0.1 → 0.1
Time: 6.6s
Precision: binary64
\[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
↓
\[0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]
\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right)
↓
0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))
↓
(FPCore (x y z) :precision binary64 (* 0.5 (fma y (sqrt z) x)))
double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}
↓
double code(double x, double y, double z) {
	return 0.5 * fma(y, sqrt(z), x);
}

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

    \[\leadsto \color{blue}{0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right)} \]

  3. Final simplification0.1

    \[\leadsto 0.5 \cdot \mathsf{fma}\left(y, \sqrt{z}, x\right) \]

Reproduce

herbie shell --seed 2022061 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))