Falkner and Boettcher, Appendix A

Average Error: 2.2 → 0.3

Time: 12.3s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 20000000000000:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

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 20000000000000.0)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (/ (pow k m) (/ k (/ 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 <= 20000000000000.0) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = pow(k, m) / (k / (a / k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

↓

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 20000000000000.0d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (k ** m) / (k / (a / k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	return (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
}

↓

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 20000000000000.0) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = Math.pow(k, m) / (k / (a / k));
	}
	return tmp;
}

Python

def code(a, k, m):
	return (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))

↓

def code(a, k, m):
	tmp = 0
	if k <= 20000000000000.0:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = math.pow(k, m) / (k / (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 <= 20000000000000.0)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) / Float64(k / Float64(a / k)));
	end
	return tmp
end

MATLAB

function tmp = code(a, k, m)
	tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
end

↓

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 20000000000000.0)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (k ^ m) / (k / (a / k));
	end
	tmp_2 = 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, 20000000000000.0], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] / N[(k / N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2e13

   1. Initial program 0.1

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

   if 2e13 < k

   1. Initial program 5.8

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

   2. Simplified 5.9

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

   3. Taylor expanded in k around inf 5.9

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

   4. Simplified 5.9

      \[\leadsto \color{blue}{{k}^{m} \cdot \frac{a}{k \cdot k}} \]
      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))): 27 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, 3 points decrease in error

   5. Applied egg-rr 0.7

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 20000000000000:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{k}{\frac{a}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -5700:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1368

\[\begin{array}{l} t_0 := \left(a + 1\right) + -1\\ t_1 := \left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{if}\;k \leq -7 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 600000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.4
Cost	1108

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	21.2
Cost	1108

\[\begin{array}{l} t_0 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5
Error	21.3
Cost	1108

\[\begin{array}{l} t_0 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq -5 \cdot 10^{+62}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 600000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6
Error	11.4
Cost	1096

\[\begin{array}{l} \mathbf{if}\;m \leq -5700:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 7
Error	21.6
Cost	980

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 600000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.2
Cost	848

\[\begin{array}{l} t_0 := \left(a + 1\right) + -1\\ \mathbf{if}\;k \leq -1 \cdot 10^{-277}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-267}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 600000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 9
Error	13.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{+61}:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a + 1\right) + -1\\ \end{array} \]

Alternative 10
Error	24.5
Cost	584

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq -7 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 600000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	46.7
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022300 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))