Falkner and Boettcher, Appendix A

Average Error: 2.0 → 2.0

Time: 23.7s

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(k, k + 10, 1\right)\\ \mathbf{if}\;{k}^{m} \ne 0:\\ \;\;\;\;\frac{a}{\frac{t_0}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{t_0}\\ \end{array} \]

FPCore

(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 (fma k (+ k 10.0) 1.0)))
   (if (!= (pow k m) 0.0) (/ a (/ t_0 (pow k m))) (/ (* a (pow k m)) t_0))))

C

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 = fma(k, (k + 10.0), 1.0);
	double tmp;
	if (pow(k, m) != 0.0) {
		tmp = a / (t_0 / pow(k, m));
	} else {
		tmp = (a * pow(k, m)) / t_0;
	}
	return tmp;
}

Julia

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 = fma(k, Float64(k + 10.0), 1.0)
	tmp = 0.0
	if ((k ^ m) != 0.0)
		tmp = Float64(a / Float64(t_0 / (k ^ m)));
	else
		tmp = Float64(Float64(a * (k ^ m)) / t_0);
	end
	return tmp
end

Wolfram

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[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]}, If[Unequal[N[Power[k, m], $MachinePrecision], 0.0], N[(a / N[(t$95$0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Initial program 2.0

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

2. Simplified 2.0

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

3. Applied egg-rr 2.0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{k}^{m} \ne 0:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, 10 + k, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10 + k, 1\right)}\\ } \end{array}} \]

4. Simplified 2.0

   \[\leadsto \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;{k}^{m} \ne 0:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ } \end{array}} \]
   Proof

Alternatives

Alternative 1
Error	2.5
Cost	7368

\[\begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + 10 \cdot k}\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{a \cdot m + a}{k \cdot \left(10 + k\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-\frac{{k}^{m}}{-1 - 10 \cdot k}\right)\\ \end{array} \]

Alternative 2
Error	2.5
Cost	7304

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1 + 10 \cdot k}\\ \mathbf{if}\;m \leq -3.3 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{a \cdot m + a}{k \cdot \left(10 + k\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.0
Cost	7168

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

Alternative 4
Error	2.7
Cost	7048

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -1500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 0.0023:\\ \;\;\;\;\frac{a \cdot m + a}{k \cdot \left(10 + k\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]

Alternative 6
Error	35.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{1 + 10 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]

Alternative 7
Error	43.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.27:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8
Error	43.0
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 1.8:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]

Alternative 9
Error	46.8
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023010 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))