Falkner and Boettcher, Appendix A

Average Error: 2.1 → 0.1

Time: 9.8s

Precision: binary64

Cost: 7428

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot {k}^{\left(m + -1\right)}\\ \end{array} \]

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 1.4e+154)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (* (/ a k) (pow k (+ m -1.0)))))

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 <= 1.4e+154) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * pow(k, (m + -1.0));
	}
	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 <= 1.4d+154) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (a / k) * (k ** (m + (-1.0d0)))
    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 <= 1.4e+154) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a / k) * Math.pow(k, (m + -1.0));
	}
	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 <= 1.4e+154:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (a / k) * math.pow(k, (m + -1.0))
	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 <= 1.4e+154)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(a / k) * (k ^ Float64(m + -1.0)));
	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 <= 1.4e+154)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (a / k) * (k ^ (m + -1.0));
	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, 1.4e+154], 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[(a / k), $MachinePrecision] * N[Power[k, N[(m + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e154

   1. Initial program 0.1

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

   if 1.4e154 < k

   1. Initial program 10.3

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

   2. Taylor expanded in k around inf 10.3

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{{\left(\frac{1}{k}\right)}^{\left(-m\right)}}{k}} \]
      Proof
      (*.f64 (/.f64 a k) (/.f64 (pow.f64 (/.f64 1 k) (neg.f64 m)) k)): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 1 k)) (neg.f64 m)))) k)): 55 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 a k) (/.f64 (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (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<= mul-1-neg_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))): 30 points increase in error, 14 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

   4. Applied egg-rr 0.1

      \[\leadsto \frac{a}{k} \cdot \color{blue}{{k}^{\left(m - 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot {k}^{\left(m + -1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	7300

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 0.0072:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{t_0}{k + 10}\\ \end{array} \]

Alternative 2
Error	0.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 0.0072:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot {k}^{\left(m + -1\right)}\\ \end{array} \]

Alternative 3
Error	2.8
Cost	6920

\[\begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4
Error	2.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -2.7:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.46:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	2.7
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -1.25:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.7:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	13.9
Cost	848

\[\begin{array}{l} \mathbf{if}\;k \leq 8.1 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{+59}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	848

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-303}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	2.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -1.95:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.24:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Error	15.2
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq 8.1 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-15}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+59}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 10
Error	14.3
Cost	716

\[\begin{array}{l} \mathbf{if}\;k \leq 8.1 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-13}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{+60}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	3.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;m \leq -0.235:\\ \;\;\;\;0\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	16.3
Cost	328

\[\begin{array}{l} \mathbf{if}\;k \leq 8.1 \cdot 10^{-305}:\\ \;\;\;\;0\\ \mathbf{elif}\;k \leq 4.6 \cdot 10^{-14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 13
Error	46.6
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022339 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))