Average Error: 1.9 → 1.9
Time: 12.3s
Precision: binary64
\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
↓
\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \frac{\frac{a \cdot {k}^{m}}{t_0}}{t_0} \end{array} \]
(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
↓
(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (sqrt (fma k (+ k 10.0) 1.0)))) (/ (/ (* a (pow k m)) t_0) t_0)))
double code(double a, double k, double m) {
	return (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}
↓
double code(double a, double k, double m) {
	double t_0 = sqrt(fma(k, (k + 10.0), 1.0));
	return ((a * pow(k, m)) / t_0) / t_0;
}

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end
↓
function code(a, k, m)
	t_0 = sqrt(fma(k, Float64(k + 10.0), 1.0))
	return Float64(Float64(Float64(a * (k ^ m)) / t_0) / t_0)
end

code[a_, k_, m_] := N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}
↓
\begin{array}{l}
t_0 := \sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}\\
\frac{\frac{a \cdot {k}^{m}}{t_0}}{t_0}
\end{array}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

  1. Initial program 1.9

    \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

  2. Simplified1.8

    \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]

  3. Applied add-sqr-sqrt_binary641.9

    \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot \sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}} \]

  4. Applied associate-/r*_binary641.9

    \[\leadsto \color{blue}{\frac{\frac{a \cdot {k}^{m}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}} \]

  5. Final simplification1.9

    \[\leadsto \frac{\frac{a \cdot {k}^{m}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}} \]

Reproduce

herbie shell --seed 2022130 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))