Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.1

Time: 11.4s

Precision: binary64

Cost: 13572

Math

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

↓

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

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 1e+19)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (/ (* a (/ (- (pow k m)) k)) (- 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 <= 1e+19) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = (a * (-pow(k, m) / k)) / -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 <= 1e+19)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(Float64(a * Float64(Float64(-(k ^ m)) / k)) / Float64(-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, 1e+19], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[((-N[Power[k, m], $MachinePrecision]) / k), $MachinePrecision]), $MachinePrecision] / (-k)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1e19

   1. Initial program 0.1

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      Proof
      (*.f64 a (/.f64 (pow.f64 k m) (fma.f64 k (+.f64 k 10) 1))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (/.f64 (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 a (/.f64 (pow.f64 k m) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 k (+.f64 10 k)) 1)))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (/.f64 (pow.f64 k m) (+.f64 (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 10 k) (*.f64 k k))) 1))): 1 points increase in error, 0 points decrease in error
      (*.f64 a (/.f64 (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 a (/.f64 (pow.f64 k m) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))))): 1 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))): 3 points increase in error, 10 points decrease in error

   if 1e19 < k

   1. Initial program 5.7

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

   2. Simplified 5.7

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

   3. Taylor expanded in k around inf 5.7

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]

   4. Simplified 0.2

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{k}^{m}}{k}} \]
      Proof
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 k m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 k) m))) k)): 50 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (exp.f64 (*.f64 (log.f64 k) m)) (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 k))))): 57 points increase in error, 21 points decrease in error
      (*.f64 (/.f64 a k) (Rewrite=> div-exp_binary64 (exp.f64 (-.f64 (*.f64 (log.f64 k) m) (log.f64 k))))): 11 points increase in error, 14 points decrease in error
      (*.f64 (/.f64 a k) (exp.f64 (-.f64 (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 k)))) m) (log.f64 k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (exp.f64 (-.f64 (*.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 k)))) m) (log.f64 k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (exp.f64 (-.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (log.f64 (/.f64 1 k)) m))) (log.f64 k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (exp.f64 (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) (log.f64 k)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (Rewrite<= div-exp_binary64 (/.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) (exp.f64 (log.f64 k))))): 14 points increase in error, 11 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) (Rewrite=> rem-exp-log_binary64 k))): 21 points increase in error, 57 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) (*.f64 k k))): 45 points increase in error, 14 points decrease in error
      (/.f64 (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 0.2

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m}}{k} \cdot \left(-a\right)}{-k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+19}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{-{k}^{m}}{k}}{-k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	21768

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-296}:\\ \;\;\;\;\frac{a}{\frac{1 + \left(k \cdot 10 + k \cdot k\right)}{{k}^{m}}}\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{a \cdot \frac{-{k}^{m}}{k}}{-k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	7172

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

Alternative 3
Error	1.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+151}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 4
Error	0.9
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 5
Error	0.9
Cost	7044

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

Alternative 6
Error	4.8
Cost	6852

\[\begin{array}{l} \mathbf{if}\;k \leq 21000:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 7
Error	4.8
Cost	6788

\[\begin{array}{l} \mathbf{if}\;k \leq 21000:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 8
Error	20.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{-144}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 9
Error	23.0
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 10
Error	22.8
Cost	712

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

Alternative 11
Error	22.9
Cost	712

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

Alternative 12
Error	22.8
Cost	712

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

Alternative 13
Error	20.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -2.3 \cdot 10^{-216}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 14
Error	20.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1.8 \cdot 10^{-134}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 15
Error	38.1
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	24.0
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	23.0
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	23.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 19
Error	46.1
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022329 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))