Falkner and Boettcher, Appendix A

Average Error: 2.0 → 0.2

Time: 12.8s

Precision: binary64

Cost: 40388

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}\\ t_1 := a \cdot {k}^{m}\\ t_2 := \frac{t_1}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;\frac{\frac{t_1}{t_0}}{t_0}\\ \mathbf{elif}\;t_2 \leq 0:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (sqrt (fma k (+ k 10.0) 1.0)))
        (t_1 (* a (pow k m)))
        (t_2 (/ t_1 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_2 -1e-320)
     (/ (/ t_1 t_0) t_0)
     (if (<= t_2 0.0) (/ 1.0 (* k (/ k t_1))) t_2))))

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 = sqrt(fma(k, (k + 10.0), 1.0));
	double t_1 = a * pow(k, m);
	double t_2 = t_1 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_2 <= -1e-320) {
		tmp = (t_1 / t_0) / t_0;
	} else if (t_2 <= 0.0) {
		tmp = 1.0 / (k * (k / t_1));
	} else {
		tmp = t_2;
	}
	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 = sqrt(fma(k, Float64(k + 10.0), 1.0))
	t_1 = Float64(a * (k ^ m))
	t_2 = Float64(t_1 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_2 <= -1e-320)
		tmp = Float64(Float64(t_1 / t_0) / t_0);
	elseif (t_2 <= 0.0)
		tmp = Float64(1.0 / Float64(k * Float64(k / t_1)));
	else
		tmp = t_2;
	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[N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-320], N[(N[(t$95$1 / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(1.0 / N[(k * N[(k / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < -9.99989e-321

   1. Initial program 0.1

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      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, 1 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))))): 0 points increase in error, 1 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, 7 points decrease in error

   3. Applied egg-rr 0.1

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

   if -9.99989e-321 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 0.0

   1. Initial program 2.9

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

   2. Simplified 2.9

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      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, 1 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))))): 0 points increase in error, 1 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, 7 points decrease in error

   3. Applied egg-rr 2.9

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

   4. Taylor expanded in k around inf 33.1

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

   5. Simplified 0.1

      \[\leadsto \frac{1}{\color{blue}{\frac{k}{{k}^{m} \cdot a} \cdot k}} \]
      Proof
      (*.f64 (/.f64 k (*.f64 (pow.f64 k m) a)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (*.f64 (pow.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 k))) m) a)) k): 0 points increase in error, 1 points decrease in error
      (*.f64 (/.f64 k (*.f64 (pow.f64 (exp.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 k))))) m) a)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (*.f64 (pow.f64 (exp.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 k))))) m) a)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (*.f64 (pow.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 k))))) m) a)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (*.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 1 k))) m))) a)) k): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (*.f64 (exp.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) a)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 k (Rewrite=> *-commutative_binary64 (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))))) k): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 k (/.f64 (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) k))): 7 points increase in error, 12 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 k k) (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))))): 21 points increase in error, 15 points decrease in error
      (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (pow.f64 k 2) (Rewrite<= *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) a))): 0 points increase in error, 0 points decrease in error

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 0.5

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq -1 \cdot 10^{-320}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a \cdot {k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	22024

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-320}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13828

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+115}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{\frac{{\left(-{\left(\frac{-1}{k}\right)}^{-1}\right)}^{m}}{\frac{k}{a}}}}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	7172

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

Alternative 4
Error	0.9
Cost	7172

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

Alternative 5
Error	0.8
Cost	7172

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 100000:\\ \;\;\;\;\frac{t_0}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{t_0}}\\ \end{array} \]

Alternative 6
Error	2.3
Cost	7048

\[\begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+183}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+224}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 7
Error	2.5
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.0058:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 5.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	20.2
Cost	840

\[\begin{array}{l} t_0 := -1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{if}\;m \leq -5.5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 195000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	19.4
Cost	840

\[\begin{array}{l} t_0 := -1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \mathbf{if}\;m \leq -5.5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 195000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	17.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{+20}:\\ \;\;\;\;a \cdot \left(1 + \left(-1 + \frac{-1}{k \cdot k}\right)\right)\\ \mathbf{elif}\;m \leq 195000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \frac{a}{k \cdot k}\right)\\ \end{array} \]

Alternative 11
Error	24.9
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq -13500000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	23.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -13500000000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-7}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	24.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{+184}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+224}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 14
Error	47.1
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022297 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))