Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.1

Time: 13.0s

Precision: binary64

Cost: 45828

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;k \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{t_0} \cdot \frac{a}{t_0}\\ \end{array} \]

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 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= k 5.2e+16)
     (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
     (* (/ (pow k m) t_0) (/ a 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 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (k <= 5.2e+16) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (pow(k, m) / t_0) * (a / 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 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (k <= 5.2e+16)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64((k ^ m) / t_0) * Float64(a / 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[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, 5.2e+16], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.2e16

   1. Initial program 0.1

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

   if 5.2e16 < k

   1. Initial program 5.4

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

   2. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

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

Alternative 2
Error	0.1
Cost	7300

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

Alternative 3
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 4
Error	0.7
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5
Error	2.6
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -4.2 \cdot 10^{-11} \lor \neg \left(m \leq 9 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \]

Alternative 6
Error	21.0
Cost	1736

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;k \leq -2 \cdot 10^{-96}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+101}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(t_0 \cdot \left(-10 - k\right)\right)}{1 - t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 7
Error	21.5
Cost	976

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{if}\;k \leq -2 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.0135:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{a}{\frac{k}{\frac{1}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 8
Error	19.1
Cost	972

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{if}\;m \leq -5.8 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{+295}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9
Error	19.0
Cost	972

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{if}\;m \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;m \leq 3 \cdot 10^{+301}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10
Error	23.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 11
Error	23.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 12
Error	23.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -10:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 13
Error	24.3
Cost	585

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

Alternative 14
Error	23.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 15
Error	23.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{+166}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 16
Error	40.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 17
Error	46.9
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022343 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))