Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 97.3%

Time: 9.1s

Precision: binary64

Cost: 7172

• 23.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -1.9999999999999999e-154
  2. k = 0.0
  3. m = -2.00000000000000007e154

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.00056:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k + 10} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

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 0.00056) (* a (pow k m)) (* (/ (pow k m) (+ k 10.0)) (/ 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 <= 0.00056) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) / (k + 10.0)) * (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 <= 0.00056d0) then
        tmp = a * (k ** m)
    else
        tmp = ((k ** m) / (k + 10.0d0)) * (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 <= 0.00056) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) / (k + 10.0)) * (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 <= 0.00056:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) / (k + 10.0)) * (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 <= 0.00056)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) / Float64(k + 10.0)) * 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 <= 0.00056)
		tmp = a * (k ^ m);
	else
		tmp = ((k ^ m) / (k + 10.0)) * (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, 0.00056], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5999999999999995e-4

   1. Initial program 97.9%

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

   2. Simplified 97.9%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}} \]
      Step-by-step derivation

      [Start] 97.9%	\[ \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
      associate-*r/ [<=] 97.9%	\[ \color{blue}{a \cdot \frac{{k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}} \]
      associate-+l+ [=>] 97.9%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}} \]
      +-commutative [=>] 97.9%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}} \]
      distribute-rgt-out [=>] 97.9%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{k \cdot \left(10 + k\right)} + 1} \]
      fma-def [=>] 97.9%	\[ a \cdot \frac{{k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]
      +-commutative [=>] 97.9%	\[ a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, \color{blue}{k + 10}, 1\right)} \]

   3. Taylor expanded in k around 0 54.5%

      \[\leadsto a \cdot \color{blue}{e^{\log k \cdot m}} \]

   4. Simplified 99.8%

      \[\leadsto a \cdot \color{blue}{{k}^{m}} \]
      Step-by-step derivation

      [Start] 54.5%	\[ a \cdot e^{\log k \cdot m} \]
      exp-to-pow [=>] 99.8%	\[ a \cdot \color{blue}{{k}^{m}} \]

   if 5.5999999999999995e-4 < k

   1. Initial program 83.8%

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

   2. Taylor expanded in k around inf 82.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{k}^{2} + 10 \cdot k}} \]

   3. Simplified 82.5%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot \left(k + 10\right)}} \]
      Step-by-step derivation

      [Start] 82.5%	\[ \frac{a \cdot {k}^{m}}{{k}^{2} + 10 \cdot k} \]
      unpow2 [=>] 82.5%	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k} + 10 \cdot k} \]
      distribute-rgt-in [<=] 82.5%	\[ \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot \left(k + 10\right)}} \]

   4. Applied egg-rr 94.9%

      \[\leadsto \color{blue}{\frac{{k}^{m}}{k + 10} \cdot \frac{a}{k}} \]
      Step-by-step derivation

      [Start] 82.5%	\[ \frac{a \cdot {k}^{m}}{k \cdot \left(k + 10\right)} \]
      *-commutative [=>] 82.5%	\[ \frac{\color{blue}{{k}^{m} \cdot a}}{k \cdot \left(k + 10\right)} \]
      *-commutative [=>] 82.5%	\[ \frac{{k}^{m} \cdot a}{\color{blue}{\left(k + 10\right) \cdot k}} \]
      times-frac [=>] 94.9%	\[ \color{blue}{\frac{{k}^{m}}{k + 10} \cdot \frac{a}{k}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.00056:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k + 10} \cdot \frac{a}{k}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.3%
Cost	7172

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00056:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k + 10} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 2
Accuracy	97.0%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00056:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 3
Accuracy	97.1%
Cost	6921

\[\begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-7} \lor \neg \left(m \leq 0.205\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4
Accuracy	97.1%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;k \leq 0.00056:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 5
Accuracy	46.0%
Cost	976

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq 8.2 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.00056:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	45.9%
Cost	976

\[\begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-301}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-195}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.00056:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{k}}{\frac{k}{a}}\\ \end{array} \]

Alternative 7
Accuracy	46.2%
Cost	976

\[\begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-302}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.00056:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 8
Accuracy	54.7%
Cost	968

\[\begin{array}{l} \mathbf{if}\;m \leq -0.46:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1 + 0.001 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9
Accuracy	46.3%
Cost	850

\[\begin{array}{l} \mathbf{if}\;k \leq 7.7 \cdot 10^{-302} \lor \neg \left(k \leq 3.7 \cdot 10^{-218}\right) \land \left(k \leq 1.75 \cdot 10^{-194} \lor \neg \left(k \leq 0.00056\right)\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10
Accuracy	45.8%
Cost	848

\[\begin{array}{l} t_0 := \frac{\frac{a}{k}}{k}\\ \mathbf{if}\;k \leq 2.15 \cdot 10^{-302}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-197}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.00056:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	53.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;m \leq -0.58:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 12
Accuracy	27.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-302} \lor \neg \left(k \leq 0.00056\right):\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 13
Accuracy	20.0%
Cost	64

\[a \]

Reproduce

herbie shell --seed 2023167 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))