Falkner and Boettcher, Appendix A

Average Error: 1.9 → 0.1

Time: 12.7s

Precision: binary64

Cost: 7620

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

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 1.1e+17)
   (/ (* a (pow (/ 1.0 k) (- m))) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (/ a k) (/ (pow k m) 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 <= 1.1e+17) {
		tmp = (a * pow((1.0 / k), -m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * (pow(k, m) / 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 <= 1.1d+17) then
        tmp = (a * ((1.0d0 / k) ** -m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (a / k) * ((k ** m) / 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 <= 1.1e+17) {
		tmp = (a * Math.pow((1.0 / k), -m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * (Math.pow(k, m) / 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 <= 1.1e+17:
		tmp = (a * math.pow((1.0 / k), -m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (a / k) * (math.pow(k, m) / 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 <= 1.1e+17)
		tmp = Float64(Float64(a * (Float64(1.0 / k) ^ Float64(-m))) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / 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 <= 1.1e+17)
		tmp = (a * ((1.0 / k) ^ -m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (a / k) * ((k ^ m) / 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, 1.1e+17], N[(N[(a * N[Power[N[(1.0 / k), $MachinePrecision], (-m)], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.1e17

   1. Initial program 0.1

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

   2. Taylor expanded in k around inf 23.8

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. Simplified 0.1

      \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      Proof
      (*.f64 a (pow.f64 (/.f64 1 k) (neg.f64 m))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 1 k)) (neg.f64 m))))): 57 points increase in error, 0 points decrease in error
      (*.f64 a (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (log.f64 (/.f64 1 k)) m))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (exp.f64 (Rewrite<= mul-1-neg_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))) a)): 0 points increase in error, 0 points decrease in error

   if 1.1e17 < k

   1. Initial program 5.1

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

   2. Taylor expanded in k around inf 5.1

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. Simplified 5.1

      \[\leadsto \frac{\color{blue}{a \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      Proof
      (*.f64 a (pow.f64 (/.f64 1 k) (neg.f64 m))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 1 k)) (neg.f64 m))))): 57 points increase in error, 0 points decrease in error
      (*.f64 a (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (log.f64 (/.f64 1 k)) m))))): 0 points increase in error, 0 points decrease in error
      (*.f64 a (exp.f64 (Rewrite<= mul-1-neg_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))) a)): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in k around inf 5.1

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

   5. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{k}^{m}}{k}} \]
      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 (pow.f64 (Rewrite<= rem-exp-log_binary64 (exp.f64 (log.f64 k))) m) k)): 51 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (exp.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (log.f64 k))))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (exp.f64 (neg.f64 (Rewrite<= log-rec_binary64 (log.f64 (/.f64 1 k))))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (log.f64 (/.f64 1 k))))) m) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (*.f64 -1 (log.f64 (/.f64 1 k))) m))) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (exp.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1 (*.f64 (log.f64 (/.f64 1 k)) m)))) k)): 0 points increase in error, 0 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))): 38 points increase in error, 17 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
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.1
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 2
Error	0.7
Cost	7172

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

Alternative 3
Error	0.7
Cost	7172

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

Alternative 4
Error	2.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;m \leq -0.068:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 1.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {\left(\frac{1}{k}\right)}^{\left(-m\right)}\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7044

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

Alternative 6
Error	2.4
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.0045:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -0.68:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{1}{k \cdot a}}\\ \end{array} \]

Alternative 8
Error	15.8
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -0.205:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{1}{k \cdot a}}\\ \end{array} \]

Alternative 9
Error	21.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-289}:\\ \;\;\;\;\frac{1}{\frac{k}{k \cdot a}}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 10
Error	21.7
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-291}:\\ \;\;\;\;\frac{1}{\frac{k}{k \cdot a}}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 11
Error	21.6
Cost	844

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{-290}:\\ \;\;\;\;\frac{1}{\frac{k}{k \cdot a}}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 12
Error	16.9
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.54:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 7 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 13
Error	16.5
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -1.82:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 7 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{1}{k \cdot a}}\\ \end{array} \]

Alternative 14
Error	15.8
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -0.52:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{1}{k \cdot a}}\\ \end{array} \]

Alternative 15
Error	23.3
Cost	712

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

Alternative 16
Error	23.1
Cost	712

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

Alternative 17
Error	20.9
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq 7 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 18
Error	38.9
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 19
Error	24.1
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 20
Error	23.3
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 21
Error	20.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;m \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{k \cdot a}}\\ \end{array} \]

Alternative 22
Error	46.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022337 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))