Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 15.5s

Precision: binary64

Cost: 33156

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{if}\;k \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

FPCore

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

↓

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (sqrt (fma k (+ k 10.0) 1.0))))
   (if (<= k 5e+16)
     (/ (/ (* a (pow k m)) t_0) t_0)
     (* (pow k m) (/ (/ a 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 t_0 = sqrt(fma(k, (k + 10.0), 1.0));
	double tmp;
	if (k <= 5e+16) {
		tmp = ((a * pow(k, m)) / t_0) / t_0;
	} else {
		tmp = pow(k, m) * ((a / 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)
	t_0 = sqrt(fma(k, Float64(k + 10.0), 1.0))
	tmp = 0.0
	if (k <= 5e+16)
		tmp = Float64(Float64(Float64(a * (k ^ m)) / t_0) / t_0);
	else
		tmp = Float64((k ^ m) * Float64(Float64(a / k) / 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_] := Block[{t$95$0 = N[Sqrt[N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 5e+16], N[(N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5e16

   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))): 0 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, 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)))): 6 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 5e16 < k

   1. Initial program 5.4

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

   2. Simplified 5.4

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

   3. Taylor expanded in k around inf 5.4

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

   4. Simplified 5.4

      \[\leadsto \frac{a}{\frac{1 + \color{blue}{k \cdot k}}{{k}^{m}}} \]
      Proof
      (*.f64 k k): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= remove-double-div_binary64 (/.f64 1 (/.f64 1 k))) k): 9 points increase in error, 13 points decrease in error
      (*.f64 (Rewrite<= unpow-1_binary64 (pow.f64 (/.f64 1 k) -1)) k): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (/.f64 1 k) -1) (Rewrite<= remove-double-div_binary64 (/.f64 1 (/.f64 1 k)))): 8 points increase in error, 10 points decrease in error
      (*.f64 (pow.f64 (/.f64 1 k) -1) (Rewrite<= unpow-1_binary64 (pow.f64 (/.f64 1 k) -1))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> pow-sqr_binary64 (pow.f64 (/.f64 1 k) (*.f64 2 -1))): 33 points increase in error, 26 points decrease in error
      (pow.f64 (/.f64 1 k) (Rewrite=> metadata-eval -2)): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in k around inf 5.1

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

   6. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k} \cdot {k}^{m}} \]
      Proof
      (*.f64 (/.f64 (/.f64 a k) k) (pow.f64 k m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 a 1)) k) k) (pow.f64 k m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 a (/.f64 1 k))) k) (pow.f64 k m)): 17 points increase in error, 6 points decrease in error
      (*.f64 (Rewrite<= associate-*r/_binary64 (*.f64 a (/.f64 (/.f64 1 k) k))) (pow.f64 k m)): 36 points increase in error, 8 points decrease in error
      (*.f64 (*.f64 a (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 1 k))) k)) (pow.f64 k m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 k) (/.f64 1 k)))) (pow.f64 k m)): 10 points increase in error, 2 points decrease in error
      (*.f64 (*.f64 a (Rewrite<= unpow2_binary64 (pow.f64 (/.f64 1 k) 2))) (pow.f64 k m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (pow.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 k))) m)): 42 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (pow.f64 (exp.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 k))))) m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (pow.f64 (exp.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 k))))) m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (pow.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 k))))) m)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 1 k))) m)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 a (pow.f64 (/.f64 1 k) 2)) (exp.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) (*.f64 a (pow.f64 (/.f64 1 k) 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}}{\sqrt{\mathsf{fma}\left(k, k + 10, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	13572

\[\begin{array}{l} \mathbf{if}\;k \leq 3.8 \cdot 10^{+21}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 2
Error	0.1
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7172

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

Alternative 4
Error	0.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 5
Error	2.4
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 6 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	17.5
Cost	1476

\[\begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{1 + \frac{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k + -100\right)}{k}}{k + -10}}\\ \mathbf{elif}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 7
Error	17.5
Cost	1348

\[\begin{array}{l} \mathbf{if}\;m \leq -50:\\ \;\;\;\;\frac{a}{1 + \frac{\frac{\left(k \cdot k\right) \cdot \left(k \cdot k + -100\right)}{k}}{k + -10}}\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 8
Error	21.4
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.62 \cdot 10^{-307}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	21.3
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10
Error	37.2
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot 10}\\ \mathbf{if}\;k \leq -0.1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.3
Cost	716

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	21.6
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{-307}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	19.6
Cost	708

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

Alternative 14
Error	20.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 15
Error	42.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 0.39:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 16
Error	42.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 1.2:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 17
Error	46.5
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022325 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))