Falkner and Boettcher, Appendix A

Average Error: 2.1 → 1.4

Time: 15.6s

Precision: binary64

Cost: 13512

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{\sqrt{k}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

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 4.1e+169)
   (/ (* a (pow k m)) (+ (+ 1.0 (* k 10.0)) (* k k)))
   (if (<= k 2.9e+259) (/ (/ (/ a k) (sqrt k)) (sqrt k)) (/ 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 tmp;
	if (k <= 4.1e+169) {
		tmp = (a * pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else if (k <= 2.9e+259) {
		tmp = ((a / k) / sqrt(k)) / sqrt(k);
	} else {
		tmp = a / (k * 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 <= 4.1d+169) then
        tmp = (a * (k ** m)) / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else if (k <= 2.9d+259) then
        tmp = ((a / k) / sqrt(k)) / sqrt(k)
    else
        tmp = a / (k * 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 <= 4.1e+169) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k));
	} else if (k <= 2.9e+259) {
		tmp = ((a / k) / Math.sqrt(k)) / Math.sqrt(k);
	} else {
		tmp = a / (k * 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 <= 4.1e+169:
		tmp = (a * math.pow(k, m)) / ((1.0 + (k * 10.0)) + (k * k))
	elif k <= 2.9e+259:
		tmp = ((a / k) / math.sqrt(k)) / math.sqrt(k)
	else:
		tmp = 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)
	tmp = 0.0
	if (k <= 4.1e+169)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	elseif (k <= 2.9e+259)
		tmp = Float64(Float64(Float64(a / k) / sqrt(k)) / sqrt(k));
	else
		tmp = Float64(a / Float64(k * 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 <= 4.1e+169)
		tmp = (a * (k ^ m)) / ((1.0 + (k * 10.0)) + (k * k));
	elseif (k <= 2.9e+259)
		tmp = ((a / k) / sqrt(k)) / sqrt(k);
	else
		tmp = a / (k * 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, 4.1e+169], 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], If[LessEqual[k, 2.9e+259], N[(N[(N[(a / k), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.1000000000000003e169

   1. Initial program 0.5

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

   if 4.1000000000000003e169 < k < 2.8999999999999999e259

   1. Initial program 12.9

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

   2. Simplified 12.9

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      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))))): 1 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, 5 points decrease in error

   3. Taylor expanded in m around 0 12.9

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   4. Taylor expanded in k around inf 12.9

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]

   5. Simplified 6.4

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
      Proof
      (/.f64 (/.f64 a k) k): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 a (*.f64 k k))): 34 points increase in error, 28 points decrease in error
      (/.f64 a (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 6.4

      \[\leadsto \color{blue}{\frac{a}{k} \cdot \frac{1}{k}} \]

   7. Applied egg-rr 6.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{a}{k}}{\sqrt{k}}}{\sqrt{k}}} \]

   if 2.8999999999999999e259 < k

   1. Initial program 3.4

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

   2. Simplified 3.4

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      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))))): 1 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, 5 points decrease in error

   3. Taylor expanded in m around 0 3.4

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]

   4. Taylor expanded in k around inf 3.4

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]

   5. Simplified 1.7

      \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
      Proof
      (/.f64 (/.f64 a k) k): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r*_binary64 (/.f64 a (*.f64 k k))): 34 points increase in error, 28 points decrease in error
      (/.f64 a (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in a around 0 3.4

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]

   7. Simplified 3.4

      \[\leadsto \color{blue}{\frac{a}{k \cdot k}} \]
      Proof
      (/.f64 a (*.f64 k k)): 0 points increase in error, 0 points decrease in error
      (/.f64 a (Rewrite<= unpow2_binary64 (pow.f64 k 2))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{\frac{a}{k}}{\sqrt{k}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	7428

\[\begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 2
Error	2.6
Cost	7172

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -1.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{t_0}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 185000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	2.7
Cost	6920

\[\begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.09:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 185000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	20.1
Cost	840

\[\begin{array}{l} t_0 := \left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \mathbf{if}\;m \leq -3800:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	18.0
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -3800:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 6
Error	17.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;m \leq -3800:\\ \;\;\;\;\left(1 + \frac{a}{k \cdot k}\right) + -1\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{a}{k} \cdot 0.1\right) + -1\\ \end{array} \]

Alternative 7
Error	23.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;k \leq -1:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8
Error	39.3
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot 10}\\ \mathbf{if}\;k \leq -0.00038:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	24.9
Cost	584

\[\begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-9}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11
Error	24.1
Cost	584

\[\begin{array}{l} \mathbf{if}\;k \leq 4.1 \cdot 10^{+169}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+259}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 12
Error	46.7
Cost	64

\[a \]

Reproduce

herbie shell --seed 2022308 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))