Falkner and Boettcher, Appendix A

Average Accuracy: 96.9% → 99.9%

Time: 19.8s

Precision: binary64

Cost: 45956

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{{k}^{m}}{\frac{t_0}{a}}\\ \end{array} \]

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 2e+56)
     (/ (/ a (fma k (+ k 10.0) 1.0)) (pow k (- m)))
     (* (/ 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 <= 2e+56) {
		tmp = (a / fma(k, (k + 10.0), 1.0)) / pow(k, -m);
	} 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 <= 2e+56)
		tmp = Float64(Float64(a / fma(k, Float64(k + 10.0), 1.0)) / (k ^ Float64(-m)));
	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, 2e+56], N[(N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, (-m)], $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 < 2.00000000000000018e56

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-/l* [=>] 99.8	\[ \color{blue}{\frac{a}{\frac{\left(1 + 10 \cdot k\right) + k \cdot k}{{k}^{m}}}} \]
      associate-+l+ [=>] 99.8	\[ \frac{a}{\frac{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}{{k}^{m}}} \]
      *-commutative [=>] 99.8	\[ \frac{a}{\frac{1 + \left(\color{blue}{k \cdot 10} + k \cdot k\right)}{{k}^{m}}} \]

   3. Applied egg-rr 99.9%

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

      [Start] 99.8	\[ \frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}} \]
      *-un-lft-identity [=>] 99.8	\[ \frac{\color{blue}{1 \cdot a}}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}} \]
      div-inv [=>] 99.8	\[ \frac{1 \cdot a}{\color{blue}{\left(1 + \left(k \cdot 10 + k \cdot k\right)\right) \cdot \frac{1}{{k}^{m}}}} \]
      times-frac [=>] 99.8	\[ \color{blue}{\frac{1}{1 + \left(k \cdot 10 + k \cdot k\right)} \cdot \frac{a}{\frac{1}{{k}^{m}}}} \]
      +-commutative [=>] 99.8	\[ \frac{1}{\color{blue}{\left(k \cdot 10 + k \cdot k\right) + 1}} \cdot \frac{a}{\frac{1}{{k}^{m}}} \]
      distribute-lft-out [=>] 99.9	\[ \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \cdot \frac{a}{\frac{1}{{k}^{m}}} \]
      fma-def [=>] 99.9	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \cdot \frac{a}{\frac{1}{{k}^{m}}} \]
      +-commutative [=>] 99.9	\[ \frac{1}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \cdot \frac{a}{\frac{1}{{k}^{m}}} \]
      pow-flip [=>] 99.9	\[ \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot \frac{a}{\color{blue}{{k}^{\left(-m\right)}}} \]

   4. Simplified 99.9%

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

      [Start] 99.9	\[ \frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot \frac{a}{{k}^{\left(-m\right)}} \]
      associate-*r/ [=>] 99.9	\[ \color{blue}{\frac{\frac{1}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot a}{{k}^{\left(-m\right)}}} \]
      associate-*l/ [=>] 99.9	\[ \frac{\color{blue}{\frac{1 \cdot a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]
      *-commutative [<=] 99.9	\[ \frac{\frac{\color{blue}{a \cdot 1}}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}} \]
      associate-*l/ [<=] 99.9	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)} \cdot 1}}{{k}^{\left(-m\right)}} \]
      *-rgt-identity [=>] 99.9	\[ \frac{\color{blue}{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}}{{k}^{\left(-m\right)}} \]

   if 2.00000000000000018e56 < k

   1. Initial program 90.5%

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

   2. Applied egg-rr 99.8%

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

      [Start] 90.5	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      *-un-lft-identity [=>] 90.5	\[ \frac{\color{blue}{1 \cdot \left(a \cdot {k}^{m}\right)}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      add-sqr-sqrt [=>] 90.5	\[ \frac{1 \cdot \left(a \cdot {k}^{m}\right)}{\color{blue}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]
      times-frac [=>] 90.5	\[ \color{blue}{\frac{1}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]
      +-commutative [=>] 90.5	\[ \frac{1}{\sqrt{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}}} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      add-sqr-sqrt [=>] 90.5	\[ \frac{1}{\sqrt{k \cdot k + \color{blue}{\sqrt{1 + 10 \cdot k} \cdot \sqrt{1 + 10 \cdot k}}}} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      hypot-def [=>] 90.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(k, \sqrt{1 + 10 \cdot k}\right)}} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      +-commutative [=>] 90.5	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\color{blue}{10 \cdot k + 1}}\right)} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      *-commutative [=>] 90.5	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\color{blue}{k \cdot 10} + 1}\right)} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      fma-def [=>] 90.5	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}}\right)} \cdot \frac{a \cdot {k}^{m}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      *-commutative [=>] 90.5	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{\color{blue}{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-/l* [=>] 90.5	\[ \frac{1}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \color{blue}{\frac{{k}^{m}}{\frac{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}{a}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{{k}^{m}}{\frac{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	13636

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+47}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+31}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 3
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 2.2 \cdot 10^{+32}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot k + k \cdot 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 4
Accuracy	99.8%
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 5
Accuracy	99.8%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{+16}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 - k \cdot \left(-10 - k\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 6
Accuracy	98.9%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-6}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 7
Accuracy	99.2%
Cost	7172

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

Alternative 8
Accuracy	96.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-18}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 - k \cdot \left(-10 - k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 9
Accuracy	96.0%
Cost	7048

\[\begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 - k \cdot \left(-10 - k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 10
Accuracy	98.8%
Cost	7044

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

Alternative 11
Accuracy	96.1%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-18} \lor \neg \left(m \leq 0.145\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 - k \cdot \left(-10 - k\right)}\\ \end{array} \]

Alternative 12
Accuracy	74.0%
Cost	1736

\[\begin{array}{l} t_0 := a \cdot \left(k \cdot 10\right)\\ \mathbf{if}\;m \leq -0.34:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 50000000000:\\ \;\;\;\;\frac{a}{1 - k \cdot \left(-10 - k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot a + \left(a \cdot \left(k \cdot -10\right)\right) \cdot t_0}{a + t_0}\\ \end{array} \]

Alternative 13
Accuracy	65.1%
Cost	844

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.47:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Accuracy	66.4%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.47:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 15
Accuracy	66.5%
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.47:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 10.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 16
Accuracy	74.7%
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.145:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{1 - k \cdot \left(-10 - k\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 17
Accuracy	64.8%
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.47:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Accuracy	73.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -1.9:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.8:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 19
Accuracy	67.9%
Cost	580

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

Alternative 20
Accuracy	32.7%
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.49:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 21
Accuracy	26.8%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023139 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))