Falkner and Boettcher, Appendix A

Average Error: 97.1% → 99.9%

Time: 23.4s

Precision: binary64

Cost: 26436.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{k}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

↓

(FPCore (a k m)
 :precision binary64
 (if (<= k 2e+123)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (/ (pow k m) (hypot k (sqrt (fma k 10.0 1.0)))) (/ a k))))

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 tmp;
	if (k <= 2e+123) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (pow(k, m) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * (a / k);
	}
	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)
	tmp = 0.0
	if (k <= 2e+123)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64((k ^ m) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * Float64(a / k));
	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_] := If[LessEqual[k, 2e+123], 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] / N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.99999999999999996e123

   1. Initial program 99.9

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

   if 1.99999999999999996e123 < k

   1. Initial program 87.5

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

   2. Applied egg-rr 99.7

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

   3. Simplified 99.8

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

      [Start] 99.7	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{{k}^{m}}{\frac{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}{a}} \]
      associate-*l/ [=>] 99.7	\[ \color{blue}{\frac{1 \cdot \frac{{k}^{m}}{\frac{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}{a}}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
      *-lft-identity [=>] 99.7	\[ \frac{\color{blue}{\frac{{k}^{m}}{\frac{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}{a}}}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]
      associate-/r/ [=>] 99.7	\[ \frac{\color{blue}{\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot a}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]
      associate-*r/ [<=] 99.8	\[ \color{blue}{\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]

   4. Taylor expanded in k around inf 99.8

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

4. Final simplification 99.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.6%
Cost	19972.00

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+91}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\frac{a}{k}}}{e^{m \cdot \log k}}\right)}^{-1}\\ \end{array} \]

Alternative 2
Error	97.1%
Cost	7296.00

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

Alternative 3
Error	96.0%
Cost	6921.00

\[\begin{array}{l} \mathbf{if}\;m \leq -0.019 \lor \neg \left(m \leq 4.3 \cdot 10^{-19}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4
Error	74.8%
Cost	2504.00

\[\begin{array}{l} t_0 := \frac{-1}{a} + \left(k + 10\right) \cdot \frac{k}{a}\\ \mathbf{if}\;m \leq -9.5 \cdot 10^{+32}:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{-1}{t_0 \cdot \frac{-1 + k \cdot \left(-10 - k\right)}{a}}\\ \end{array} \]

Alternative 5
Error	74.6%
Cost	1224.00

\[\begin{array}{l} \mathbf{if}\;m \leq -9.5 \cdot 10^{+32}:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{elif}\;m \leq 140:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left(k \cdot a\right) \cdot \left(-10 - k\right) - a} \cdot \left(a \cdot a\right)\\ \end{array} \]

Alternative 6
Error	70.1%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+38} \lor \neg \left(m \leq 1.35 \cdot 10^{+15}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 7
Error	71.4%
Cost	841.00

\[\begin{array}{l} \mathbf{if}\;m \leq -9.5 \cdot 10^{+32} \lor \neg \left(m \leq 1.35 \cdot 10^{+15}\right):\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8
Error	64.4%
Cost	712.00

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

Alternative 9
Error	63.3%
Cost	585.00

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

Alternative 10
Error	64.1%
Cost	584.00

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

Alternative 11
Error	63.3%
Cost	448.00

\[\frac{a}{1 + k \cdot k} \]

Alternative 12
Error	27.4%
Cost	64.00

\[a \]

Reproduce

herbie shell --seed 2023093 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))