Falkner and Boettcher, Appendix A

Average Error: 2.2 → 0.1

Time: 11.8s

Precision: binary64

Cost: 13572

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k + 10}{a}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (if (<= k 2e+22)
   (/ (* a (pow k m)) (fma k (+ k 10.0) 1.0))
   (/ (/ (pow k m) k) (/ (+ k 10.0) 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 tmp;
	if (k <= 2e+22) {
		tmp = (a * pow(k, m)) / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = (pow(k, m) / k) / ((k + 10.0) / 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)
	tmp = 0.0
	if (k <= 2e+22)
		tmp = Float64(Float64(a * (k ^ m)) / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(Float64((k ^ m) / k) / Float64(Float64(k + 10.0) / 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_] := If[LessEqual[k, 2e+22], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e22

   1. Initial program 0.1

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      Proof
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (+.f64 k 10) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (fma.f64 k (Rewrite<= +-commutative_binary64 (+.f64 10 k)) 1)): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 k (+.f64 10 k)) 1))): 1 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 10 k) (*.f64 k k))) 1)): 2 points increase in error, 1 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= +-commutative_binary64 (+.f64 1 (+.f64 (*.f64 10 k) (*.f64 k k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 a (pow.f64 k m)) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))): 0 points increase in error, 0 points decrease in error

   if 2e22 < k

   1. Initial program 6.1

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

   2. Taylor expanded in k around inf 6.1

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{10 \cdot k} + k \cdot k} \]

   3. Simplified 6.1

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot 10} + k \cdot k} \]
      Proof
      (*.f64 k 10): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 10 k)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}} \]

   5. Applied egg-rr 0.1

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m}}{k}}{\frac{k + 10}{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+22}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k + 10}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k + 10}{a}}\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m}}{k}}{\frac{k + 10}{a}}\\ \end{array} \]

Alternative 4
Error	1.8
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+156}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 5
Error	0.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 1.3 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 6
Error	2.7
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.2 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -10:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-207}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k + 10}}{k}\\ \end{array} \]

Alternative 8
Error	17.1
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -52000000000000:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \end{array} \]

Alternative 9
Error	17.1
Cost	840

\[\begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -52000000000000:\\ \;\;\;\;1 + \left(\frac{a}{t_0} + -1\right)\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \end{array} \]

Alternative 10
Error	23.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	23.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 12
Error	23.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.076:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 13
Error	23.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -0.43:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.078:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k + 10}}{k}\\ \end{array} \]

Alternative 14
Error	20.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -6 \cdot 10^{-207}:\\ \;\;\;\;\left(1 + a \cdot \frac{0.1}{k}\right) + -1\\ \mathbf{elif}\;k \leq 0.076:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k + 10}}{k}\\ \end{array} \]

Alternative 15
Error	38.8
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot 10}\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	24.4
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	23.3
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 18
Error	46.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022330 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))