Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.1

Time: 15.1s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+150}:\\ \;\;\;\;a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \left(\frac{{\left(\sqrt[3]{a}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a}}{k}\right)\\ \end{array} \]

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+150)
   (* a (expm1 (log1p (/ (pow k m) (fma k (+ k 10.0) 1.0)))))
   (* (pow k m) (* (/ (pow (cbrt a) 2.0) k) (/ (cbrt a) 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+150) {
		tmp = a * expm1(log1p((pow(k, m) / fma(k, (k + 10.0), 1.0))));
	} else {
		tmp = pow(k, m) * ((pow(cbrt(a), 2.0) / k) * (cbrt(a) / 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+150)
		tmp = Float64(a * expm1(log1p(Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)))));
	else
		tmp = Float64((k ^ m) * Float64(Float64((cbrt(a) ^ 2.0) / k) * Float64(cbrt(a) / 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+150], N[(a * N[(Exp[N[Log[1 + N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * N[(N[(N[Power[N[Power[a, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / k), $MachinePrecision] * N[(N[Power[a, 1/3], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999981e149

   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, 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)))): 2 points increase in error, 3 points decrease in error

   3. Applied egg-rr 0.1

      \[\leadsto a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\right)} \]

   if 9.99999999999999981e149 < k

   1. Initial program 10.5

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

   2. Simplified 10.5

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      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)))): 2 points increase in error, 3 points decrease in error

   3. Taylor expanded in k around inf 10.5

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

   4. Simplified 10.5

      \[\leadsto \color{blue}{{k}^{m} \cdot \frac{a}{k \cdot k}} \]
      Proof
      (*.f64 (pow.f64 k m) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 k))) m) (/.f64 a (*.f64 k k))): 50 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (exp.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 k))))) m) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (exp.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 k))))) m) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 k))))) m) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 1 k))) m))) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) (/.f64 a (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (*.f64 (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))) (/.f64 a (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 a (pow.f64 k 2)) (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 a (exp.f64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) (pow.f64 k 2))): 1 points increase in error, 6 points decrease in error

   5. Applied egg-rr 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{+150}:\\ \;\;\;\;a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot \left(\frac{{\left(\sqrt[3]{a}\right)}^{2}}{k} \cdot \frac{\sqrt[3]{a}}{k}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	26372

\[\begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+127}:\\ \;\;\;\;a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 2
Error	0.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{+127}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternative 3
Error	1.0
Cost	7044

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

Alternative 4
Error	2.8
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 2200000:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	2.8
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 2200000:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 6
Error	13.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.12:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 7
Error	13.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.12:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 8
Error	11.4
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -15600000000000:\\ \;\;\;\;a \cdot \left(1 + \left(-1 - \frac{1}{k \cdot k}\right)\right)\\ \mathbf{elif}\;m \leq 0.12:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 9
Error	18.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{+116}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0:\\ \;\;\;\;\left(a + 1\right) + -1\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10
Error	13.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.12:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 11
Error	19.9
Cost	716

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq -1.25 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0:\\ \;\;\;\;\left(a + 1\right) + -1\\ \mathbf{elif}\;k \leq 2900:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	18.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{+116}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0:\\ \;\;\;\;\left(a + 1\right) + -1\\ \mathbf{elif}\;k \leq 2900:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 13
Error	13.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -1.1 \cdot 10^{+16}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 0.12:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 14
Error	30.5
Cost	584

\[\begin{array}{l} t_0 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq 0:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-38}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	17.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 0.12:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 16
Error	42.7
Cost	452

\[\begin{array}{l} \mathbf{if}\;m \leq 2200000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 17
Error	46.3
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022311 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))