Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 14.4s

Precision: binary64

Cost: 45956

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;k \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{a}{\frac{1}{-{k}^{m}} \cdot \left(-1 + k \cdot \left(-10 - k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{{k}^{m}}{\frac{t_0}{a}}\\ \end{array} \]

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 5e+151)
     (/ a (* (/ 1.0 (- (pow k m))) (+ -1.0 (* k (- -10.0 k)))))
     (* (/ 1.0 t_0) (/ (pow k m) (/ t_0 a))))))

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 <= 5e+151) {
		tmp = a / ((1.0 / -pow(k, m)) * (-1.0 + (k * (-10.0 - k))));
	} else {
		tmp = (1.0 / t_0) * (pow(k, m) / (t_0 / a));
	}
	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 <= 5e+151)
		tmp = Float64(a / Float64(Float64(1.0 / Float64(-(k ^ m))) * Float64(-1.0 + Float64(k * Float64(-10.0 - k)))));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64((k ^ m) / Float64(t_0 / a)));
	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, 5e+151], N[(a / N[(N[(1.0 / (-N[Power[k, m], $MachinePrecision])), $MachinePrecision] * N[(-1.0 + N[(k * N[(-10.0 - k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.0000000000000002e151

   1. Initial program 0.1

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

   2. Simplified 0.1

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

      [Start] 0.1	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 0.1	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 0.1	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 0.1

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

   if 5.0000000000000002e151 < k

   1. Initial program 9.6

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

   2. Applied egg-rr 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\frac{a}{\frac{1}{-{k}^{m}} \cdot \left(-1 + k \cdot \left(-10 - k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{{k}^{m}}{\frac{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7492

\[\begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{\frac{1}{-{k}^{m}} \cdot \left(-1 + k \cdot \left(-10 - k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+70}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 3
Error	0.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7044

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

Alternative 5
Error	2.4
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -4.4 \cdot 10^{-7} \lor \neg \left(m \leq 0.00062\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	2.4
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 0.116:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 7
Error	19.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+21} \lor \neg \left(m \leq 1.08 \cdot 10^{+16}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 8
Error	18.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{+21} \lor \neg \left(m \leq 4.9 \cdot 10^{+17}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9
Error	22.8
Cost	712

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

Alternative 10
Error	22.7
Cost	712

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

Alternative 11
Error	23.8
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 12
Error	22.9
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 13
Error	22.9
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{+71}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 14
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023237 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))