Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 99.2%

Time: 10.4s

Alternatives: 15

Speedup: 1.1×

• 22.9% 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

Specification

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

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

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

Java

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

Python

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

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

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

Initial Program: 90.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))

C

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

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

Java

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

Python

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

Julia

function code(a, k, m)
	return Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
end

MATLAB

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (sqrt (pow k m))))
   (if (<= k 1.0) (* a (pow k m)) (/ (* t_0 (* a (/ t_0 k))) k))))

C

double code(double a, double k, double m) {
	double t_0 = sqrt(pow(k, m));
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = (t_0 * (a * (t_0 / 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((k ** m))
    if (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = (t_0 * (a * (t_0 / k))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.sqrt(Math.pow(k, m));
	double tmp;
	if (k <= 1.0) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (t_0 * (a * (t_0 / k))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.sqrt(math.pow(k, m))
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = (t_0 * (a * (t_0 / k))) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = sqrt((k ^ m))
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(t_0 * Float64(a * Float64(t_0 / k))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = sqrt((k ^ m));
	tmp = 0.0;
	if (k <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = (t_0 * (a * (t_0 / k))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[N[Power[k, m], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(a * N[(t$95$0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 95.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 95.6%

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

      2. associate-+l+ 95.6%

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

      3. +-commutative 95.6%

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

      4. distribute-rgt-out 95.6%

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

      5. fma-def 95.6%

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

      6. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 57.4%

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

      1. exp-to-pow 99.2%

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

   6. Simplified 99.2%

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

   if 1 < k

   1. Initial program 78.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 78.3%

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

      2. associate-+l+ 78.3%

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

      3. +-commutative 78.3%

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

      4. distribute-rgt-out 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in k around inf 77.2%

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. associate-/l* 77.2%

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

      4. associate-*r* 77.2%

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

      5. exp-prod 77.2%

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

      6. mul-1-neg 77.2%

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

      7. log-rec 77.2%

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

      8. associate-/l* 87.9%

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

   6. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{-\left(-\log k\right)}\right)}^{m}}{\frac{k}{\frac{a}{k}}}} \]
   7. Step-by-step derivation

      1. pow-exp 87.9%

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

      2. remove-double-neg 87.9%

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

      3. pow-to-exp 87.9%

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

      4. add-sqr-sqrt 87.9%

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

      5. associate-/r/ 87.9%

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

      6. times-frac 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.6%

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

      2. associate-/r/ 98.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right)} \cdot \sqrt{{k}^{m}}}{k} \]

   10. Applied egg-rr 98.7%

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

4. Final simplification 99.0%

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

Alternative 2: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.05:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.05)
   (* (pow k m) (/ a (fma k k (fma k 10.0 1.0))))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.05) {
		tmp = pow(k, m) * (a / fma(k, k, fma(k, 10.0, 1.0)));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.05)
		tmp = Float64((k ^ m) * Float64(a / fma(k, k, fma(k, 10.0, 1.0))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.05], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(k * k + N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.050000000000000003

   1. Initial program 95.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*l/ 95.8%

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

      2. +-commutative 95.8%

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

      3. fma-def 95.8%

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

      4. +-commutative 95.8%

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

      5. *-commutative 95.8%

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

      6. fma-def 95.8%

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot {k}^{m} \]

   3. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)} \cdot {k}^{m}} \]

   if 0.050000000000000003 < m

   1. Initial program 75.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. associate-+l+ 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-rgt-out 75.3%

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

      5. fma-def 75.3%

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

      6. +-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in k around 0 56.4%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.05:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(k, 10, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 0.05:\\ \;\;\;\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 0.05) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 0.05) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 0.05d0) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 0.05) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 0.05:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 0.05)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 0.05)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 0.05], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 0.050000000000000003

   1. Initial program 95.8%

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

   if 0.050000000000000003 < m

   1. Initial program 75.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. associate-+l+ 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-rgt-out 75.3%

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

      5. fma-def 75.3%

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

      6. +-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in k around 0 56.4%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.05:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \mathbf{elif}\;m \leq 0.012:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.05e-21)
   (* (/ a k) (/ (pow k m) k))
   (if (<= m 0.012) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.05e-21) {
		tmp = (a / k) * (pow(k, m) / k);
	} else if (m <= 0.012) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * pow(k, m);
	}
	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
    real(8) :: tmp
    if (m <= (-2.05d-21)) then
        tmp = (a / k) * ((k ** m) / k)
    else if (m <= 0.012d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.05e-21) {
		tmp = (a / k) * (Math.pow(k, m) / k);
	} else if (m <= 0.012) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.05e-21:
		tmp = (a / k) * (math.pow(k, m) / k)
	elif m <= 0.012:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.05e-21)
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / k));
	elseif (m <= 0.012)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.05e-21)
		tmp = (a / k) * ((k ^ m) / k);
	elseif (m <= 0.012)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.05e-21], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.012], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.04999999999999997e-21

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 97.8%

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

      2. associate-+l+ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-rgt-out 97.8%

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

      5. fma-def 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in k around inf 62.5%

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 62.5%

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

      2. unpow2 62.5%

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

      3. associate-/l* 62.5%

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

      4. associate-*r* 62.5%

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

      5. exp-prod 62.5%

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

      6. mul-1-neg 62.5%

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

      7. log-rec 62.5%

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

      8. associate-/l* 64.7%

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

   6. Simplified 64.7%

      \[\leadsto \color{blue}{\frac{{\left(e^{-\left(-\log k\right)}\right)}^{m}}{\frac{k}{\frac{a}{k}}}} \]
   7. Step-by-step derivation

      1. pow-exp 64.7%

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

      2. remove-double-neg 64.7%

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

      3. pow-to-exp 98.8%

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

      4. add-sqr-sqrt 98.8%

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

      5. associate-/r/ 98.8%

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

      6. times-frac 98.8%

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

   8. Applied egg-rr 98.8%

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

   9. Taylor expanded in k around 0 62.5%

      \[\leadsto \color{blue}{\frac{e^{\log k \cdot m} \cdot a}{{k}^{2}}} \]
   10. Step-by-step derivation

       1. exp-to-pow 96.6%

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

       2. *-commutative 96.6%

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

       3. unpow2 96.6%

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

       4. times-frac 98.8%

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

   11. Simplified 98.8%

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

   if -2.04999999999999997e-21 < m < 0.012

   1. Initial program 93.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 93.7%

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

      2. associate-+l+ 93.7%

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

      3. +-commutative 93.7%

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

      4. distribute-rgt-out 93.7%

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

      5. fma-def 93.7%

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

      6. +-commutative 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in m around 0 93.1%

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

   if 0.012 < m

   1. Initial program 75.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. associate-+l+ 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-rgt-out 75.3%

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

      5. fma-def 75.3%

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

      6. +-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in k around 0 56.4%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-21}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \mathbf{elif}\;m \leq 0.012:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0058 \lor \neg \left(m \leq 0.0116\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.0058) (not (<= m 0.0116)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.0058) || !(m <= 0.0116)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	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
    real(8) :: tmp
    if ((m <= (-0.0058d0)) .or. (.not. (m <= 0.0116d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.0058) || !(m <= 0.0116)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.0058) or not (m <= 0.0116):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.0058) || !(m <= 0.0116))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.0058) || ~((m <= 0.0116)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.0058], N[Not[LessEqual[m, 0.0116]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0058 or 0.0116 < m

   1. Initial program 87.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 87.3%

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

      2. associate-+l+ 87.3%

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

      3. +-commutative 87.3%

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

      4. distribute-rgt-out 87.3%

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

      5. fma-def 87.3%

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

      6. +-commutative 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in k around 0 60.2%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if -0.0058 < m < 0.0116

   1. Initial program 91.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 91.9%

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

      2. associate-+l+ 91.9%

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

      3. +-commutative 91.9%

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

      4. distribute-rgt-out 91.9%

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

      5. fma-def 91.9%

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

      6. +-commutative 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in m around 0 90.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0058 \lor \neg \left(m \leq 0.0116\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6: 62.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.05:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.3e+19)
   (/ a (* k k))
   (if (<= m 0.05)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (+ (* k (* k 100.0)) (* k -10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.05) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	}
	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
    real(8) :: tmp
    if (m <= (-3.3d+19)) then
        tmp = a / (k * k)
    else if (m <= 0.05d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * ((k * (k * 100.0d0)) + (k * (-10.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.05) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.3e+19:
		tmp = a / (k * k)
	elif m <= 0.05:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.3e+19)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.05)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(Float64(k * Float64(k * 100.0)) + Float64(k * -10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.3e+19)
		tmp = a / (k * k);
	elseif (m <= 0.05)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * ((k * (k * 100.0)) + (k * -10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.3e+19], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.05], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(N[(k * N[(k * 100.0), $MachinePrecision]), $MachinePrecision] + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.3e19

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 42.7%

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

   5. Taylor expanded in k around inf 66.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 66.4%

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

   7. Simplified 66.4%

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

   if -3.3e19 < m < 0.050000000000000003

   1. Initial program 92.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 92.2%

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

      2. associate-+l+ 92.2%

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

      3. +-commutative 92.2%

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

      4. distribute-rgt-out 92.2%

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

      5. fma-def 92.2%

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

      6. +-commutative 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in m around 0 90.2%

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

   if 0.050000000000000003 < m

   1. Initial program 75.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.3%

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

      2. associate-+l+ 75.3%

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

      3. +-commutative 75.3%

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

      4. distribute-rgt-out 75.3%

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

      5. fma-def 75.3%

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

      6. +-commutative 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 3.0%

      \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 3.0%

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   7. Simplified 3.0%

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   8. Taylor expanded in k around 0 23.8%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 23.8%

         \[\leadsto a + \left(\color{blue}{\left(-10 \cdot k\right) \cdot a} + 100 \cdot \left({k}^{2} \cdot a\right)\right) \]

      2. metadata-eval 23.8%

         \[\leadsto a + \left(\left(\color{blue}{\left(-10\right)} \cdot k\right) \cdot a + 100 \cdot \left({k}^{2} \cdot a\right)\right) \]

      3. distribute-lft-neg-in 23.8%

         \[\leadsto a + \left(\color{blue}{\left(-10 \cdot k\right)} \cdot a + 100 \cdot \left({k}^{2} \cdot a\right)\right) \]

      4. *-commutative 23.8%

         \[\leadsto a + \left(\left(-\color{blue}{k \cdot 10}\right) \cdot a + 100 \cdot \left({k}^{2} \cdot a\right)\right) \]

      5. unpow2 23.8%

         \[\leadsto a + \left(\left(-k \cdot 10\right) \cdot a + 100 \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot a\right)\right) \]

      6. associate-*r* 23.8%

         \[\leadsto a + \left(\left(-k \cdot 10\right) \cdot a + \color{blue}{\left(100 \cdot \left(k \cdot k\right)\right) \cdot a}\right) \]

      7. associate-*l* 23.8%

         \[\leadsto a + \left(\left(-k \cdot 10\right) \cdot a + \color{blue}{\left(\left(100 \cdot k\right) \cdot k\right)} \cdot a\right) \]

      8. distribute-rgt-out 33.2%

         \[\leadsto a + \color{blue}{a \cdot \left(\left(-k \cdot 10\right) + \left(100 \cdot k\right) \cdot k\right)} \]

      9. +-commutative 33.2%

         \[\leadsto a + a \cdot \color{blue}{\left(\left(100 \cdot k\right) \cdot k + \left(-k \cdot 10\right)\right)} \]

      10. *-commutative 33.2%

          \[\leadsto a + a \cdot \left(\color{blue}{k \cdot \left(100 \cdot k\right)} + \left(-k \cdot 10\right)\right) \]

      11. *-commutative 33.2%

          \[\leadsto a + a \cdot \left(k \cdot \color{blue}{\left(k \cdot 100\right)} + \left(-k \cdot 10\right)\right) \]

      12. distribute-rgt-neg-in 33.2%

          \[\leadsto a + a \cdot \left(k \cdot \left(k \cdot 100\right) + \color{blue}{k \cdot \left(-10\right)}\right) \]

      13. metadata-eval 33.2%

          \[\leadsto a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot \color{blue}{-10}\right) \]

   10. Simplified 33.2%

       \[\leadsto \color{blue}{a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.05:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \]

Alternative 7: 47.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.75 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-288}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 1.75e-298)
     t_0
     (if (<= k 9.5e-288)
       a
       (if (<= k 2.35e-253)
         t_0
         (if (<= k 0.1) (+ a (* -10.0 (* k a))) (/ (/ a k) k)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.75e-298) {
		tmp = t_0;
	} else if (k <= 9.5e-288) {
		tmp = a;
	} else if (k <= 2.35e-253) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (k * a));
	} 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.75d-298) then
        tmp = t_0
    else if (k <= 9.5d-288) then
        tmp = a
    else if (k <= 2.35d-253) then
        tmp = t_0
    else if (k <= 0.1d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.75e-298) {
		tmp = t_0;
	} else if (k <= 9.5e-288) {
		tmp = a;
	} else if (k <= 2.35e-253) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.75e-298:
		tmp = t_0
	elif k <= 9.5e-288:
		tmp = a
	elif k <= 2.35e-253:
		tmp = t_0
	elif k <= 0.1:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.75e-298)
		tmp = t_0;
	elseif (k <= 9.5e-288)
		tmp = a;
	elseif (k <= 2.35e-253)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 1.75e-298)
		tmp = t_0;
	elseif (k <= 9.5e-288)
		tmp = a;
	elseif (k <= 2.35e-253)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.75e-298], t$95$0, If[LessEqual[k, 9.5e-288], a, If[LessEqual[k, 2.35e-253], t$95$0, If[LessEqual[k, 0.1], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.7499999999999999e-298 or 9.49999999999999955e-288 < k < 2.34999999999999991e-253

   1. Initial program 90.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 90.9%

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

      2. associate-+l+ 90.9%

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

      3. +-commutative 90.9%

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

      4. distribute-rgt-out 90.9%

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

      5. fma-def 90.9%

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

      6. +-commutative 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in m around 0 24.6%

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

   5. Taylor expanded in k around inf 37.6%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 37.6%

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

   7. Simplified 37.6%

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

   if 1.7499999999999999e-298 < k < 9.49999999999999955e-288

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 90.5%

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

   5. Taylor expanded in k around 0 90.5%

      \[\leadsto \color{blue}{a} \]

   if 2.34999999999999991e-253 < k < 0.10000000000000001

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 52.5%

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

   5. Taylor expanded in k around 0 51.5%

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

   if 0.10000000000000001 < k

   1. Initial program 78.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 78.3%

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

      2. associate-+l+ 78.3%

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

      3. +-commutative 78.3%

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

      4. distribute-rgt-out 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in k around inf 77.2%

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. associate-/l* 77.2%

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

      4. associate-*r* 77.2%

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

      5. exp-prod 77.2%

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

      6. mul-1-neg 77.2%

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

      7. log-rec 77.2%

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

      8. associate-/l* 87.9%

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

   6. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{-\left(-\log k\right)}\right)}^{m}}{\frac{k}{\frac{a}{k}}}} \]
   7. Step-by-step derivation

      1. pow-exp 87.9%

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

      2. remove-double-neg 87.9%

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

      3. pow-to-exp 87.9%

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

      4. add-sqr-sqrt 87.9%

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

      5. associate-/r/ 87.9%

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

      6. times-frac 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.6%

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

      2. associate-/r/ 98.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right)} \cdot \sqrt{{k}^{m}}}{k} \]

   10. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right) \cdot \sqrt{{k}^{m}}}{k}} \]

   11. Taylor expanded in m around 0 56.7%

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 56.7%

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

       2. associate-/r* 59.8%

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

   13. Simplified 59.8%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-298}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-288}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{-253}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 8: 48.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;m \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{-200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.85:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (+ 1.0 (* k 10.0)))))
   (if (<= m -1.05e-61)
     (/ a (* k k))
     (if (<= m 6e-200)
       t_0
       (if (<= m 7.5e-157)
         (/ (/ a k) k)
         (if (<= m 0.85) t_0 (* 0.1 (* k a))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -1.05e-61) {
		tmp = a / (k * k);
	} else if (m <= 6e-200) {
		tmp = t_0;
	} else if (m <= 7.5e-157) {
		tmp = (a / k) / k;
	} else if (m <= 0.85) {
		tmp = t_0;
	} else {
		tmp = 0.1 * (k * a);
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (1.0d0 + (k * 10.0d0))
    if (m <= (-1.05d-61)) then
        tmp = a / (k * k)
    else if (m <= 6d-200) then
        tmp = t_0
    else if (m <= 7.5d-157) then
        tmp = (a / k) / k
    else if (m <= 0.85d0) then
        tmp = t_0
    else
        tmp = 0.1d0 * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -1.05e-61) {
		tmp = a / (k * k);
	} else if (m <= 6e-200) {
		tmp = t_0;
	} else if (m <= 7.5e-157) {
		tmp = (a / k) / k;
	} else if (m <= 0.85) {
		tmp = t_0;
	} else {
		tmp = 0.1 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (1.0 + (k * 10.0))
	tmp = 0
	if m <= -1.05e-61:
		tmp = a / (k * k)
	elif m <= 6e-200:
		tmp = t_0
	elif m <= 7.5e-157:
		tmp = (a / k) / k
	elif m <= 0.85:
		tmp = t_0
	else:
		tmp = 0.1 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(1.0 + Float64(k * 10.0)))
	tmp = 0.0
	if (m <= -1.05e-61)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 6e-200)
		tmp = t_0;
	elseif (m <= 7.5e-157)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 0.85)
		tmp = t_0;
	else
		tmp = Float64(0.1 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (1.0 + (k * 10.0));
	tmp = 0.0;
	if (m <= -1.05e-61)
		tmp = a / (k * k);
	elseif (m <= 6e-200)
		tmp = t_0;
	elseif (m <= 7.5e-157)
		tmp = (a / k) / k;
	elseif (m <= 0.85)
		tmp = t_0;
	else
		tmp = 0.1 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.05e-61], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6e-200], t$95$0, If[LessEqual[m, 7.5e-157], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.85], t$95$0, N[(0.1 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -1.05e-61

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 97.8%

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

      2. associate-+l+ 97.8%

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

      3. +-commutative 97.8%

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

      4. distribute-rgt-out 97.8%

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

      5. fma-def 97.8%

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

      6. +-commutative 97.8%

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

   3. Simplified 97.8%

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

   4. Taylor expanded in m around 0 44.8%

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

   5. Taylor expanded in k around inf 65.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 65.4%

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

   7. Simplified 65.4%

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

   if -1.05e-61 < m < 5.99999999999999989e-200 or 7.500000000000001e-157 < m < 0.849999999999999978

   1. Initial program 95.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 95.1%

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

      2. associate-+l+ 95.0%

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

      3. +-commutative 95.0%

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

      4. distribute-rgt-out 95.1%

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

      5. fma-def 95.1%

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

      6. +-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in m around 0 93.3%

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

   5. Taylor expanded in k around 0 70.5%

      \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 70.5%

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   7. Simplified 70.5%

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   if 5.99999999999999989e-200 < m < 7.500000000000001e-157

   1. Initial program 78.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 78.5%

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

      2. associate-+l+ 78.5%

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

      3. +-commutative 78.5%

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

      4. distribute-rgt-out 78.5%

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

      5. fma-def 78.5%

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

      6. +-commutative 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in k around inf 65.4%

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. associate-/l* 65.4%

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

      4. associate-*r* 65.4%

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

      5. exp-prod 65.4%

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

      6. mul-1-neg 65.4%

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

      7. log-rec 65.4%

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

      8. associate-/l* 78.9%

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

   6. Simplified 78.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{-\left(-\log k\right)}\right)}^{m}}{\frac{k}{\frac{a}{k}}}} \]
   7. Step-by-step derivation

      1. pow-exp 78.9%

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

      2. remove-double-neg 78.9%

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

      3. pow-to-exp 78.9%

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

      4. add-sqr-sqrt 78.9%

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

      5. associate-/r/ 78.9%

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

      6. times-frac 86.4%

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

   8. Applied egg-rr 86.4%

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

      1. associate-*r/ 86.4%

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

      2. associate-/r/ 86.4%

         \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right)} \cdot \sqrt{{k}^{m}}}{k} \]

   10. Applied egg-rr 86.4%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right) \cdot \sqrt{{k}^{m}}}{k}} \]

   11. Taylor expanded in m around 0 65.4%

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 65.4%

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

       2. associate-/r* 86.6%

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

   13. Simplified 86.6%

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

   if 0.849999999999999978 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.0%

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

      2. associate-+l+ 75.0%

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

      3. +-commutative 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. fma-def 75.0%

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

      6. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 2.8%

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

      2. add-exp-log 1.5%

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

      3. remove-double-neg 1.5%

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

      4. exp-neg 1.5%

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

      5. neg-log 1.5%

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

      6. add-exp-log 2.8%

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

      7. un-div-inv 2.8%

         \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{\frac{1}{k}}}} \]

      8. add-exp-log 1.5%

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

      9. neg-log 1.5%

         \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{-\log k}}}} \]

      10. add-sqr-sqrt 0.9%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{-\log k} \cdot \sqrt{-\log k}}}}} \]

      11. sqrt-unprod 2.1%

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

      12. sqr-neg 2.1%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\sqrt{\color{blue}{\log k \cdot \log k}}}}} \]

      13. sqrt-unprod 1.2%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{\log k} \cdot \sqrt{\log k}}}}} \]

      14. add-sqr-sqrt 5.7%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\log k}}}} \]

      15. add-exp-log 13.3%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{\color{blue}{k}}} \]

   6. Applied egg-rr 13.3%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{k}}} \]

   7. Taylor expanded in k around 0 22.0%

      \[\leadsto \color{blue}{0.1 \cdot \left(k \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{-61}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6 \cdot 10^{-200}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9: 46.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-298} \lor \neg \left(k \leq 9.4 \cdot 10^{-289} \lor \neg \left(k \leq 2.55 \cdot 10^{-253}\right) \land k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.8e-298)
         (not (or (<= k 9.4e-289) (and (not (<= k 2.55e-253)) (<= k 1.0)))))
   (/ a (* k k))
   a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.8e-298) || !((k <= 9.4e-289) || (!(k <= 2.55e-253) && (k <= 1.0)))) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	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
    real(8) :: tmp
    if ((k <= 1.8d-298) .or. (.not. (k <= 9.4d-289) .or. (.not. (k <= 2.55d-253)) .and. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.8e-298) || !((k <= 9.4e-289) || (!(k <= 2.55e-253) && (k <= 1.0)))) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.8e-298) or not ((k <= 9.4e-289) or (not (k <= 2.55e-253) and (k <= 1.0))):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.8e-298) || !((k <= 9.4e-289) || (!(k <= 2.55e-253) && (k <= 1.0))))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 1.8e-298) || ~(((k <= 9.4e-289) || (~((k <= 2.55e-253)) && (k <= 1.0)))))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.8e-298], N[Not[Or[LessEqual[k, 9.4e-289], And[N[Not[LessEqual[k, 2.55e-253]], $MachinePrecision], LessEqual[k, 1.0]]]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.80000000000000001e-298 or 9.39999999999999934e-289 < k < 2.55000000000000004e-253 or 1 < k

   1. Initial program 83.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 83.8%

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

      2. associate-+l+ 83.8%

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

      3. +-commutative 83.8%

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

      4. distribute-rgt-out 83.8%

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

      5. fma-def 83.8%

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

      6. +-commutative 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in m around 0 43.2%

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

   5. Taylor expanded in k around inf 48.3%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 48.3%

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

   7. Simplified 48.3%

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

   if 1.80000000000000001e-298 < k < 9.39999999999999934e-289 or 2.55000000000000004e-253 < k < 1

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.8%

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

   5. Taylor expanded in k around 0 53.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-298} \lor \neg \left(k \leq 9.4 \cdot 10^{-289} \lor \neg \left(k \leq 2.55 \cdot 10^{-253}\right) \land k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 10: 47.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-287}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 9.5e-299)
     t_0
     (if (<= k 5.8e-287)
       a
       (if (<= k 1.2e-253) t_0 (if (<= k 1.0) a (/ (/ a k) k)))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 9.5e-299) {
		tmp = t_0;
	} else if (k <= 5.8e-287) {
		tmp = a;
	} else if (k <= 1.2e-253) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} 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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 9.5d-299) then
        tmp = t_0
    else if (k <= 5.8d-287) then
        tmp = a
    else if (k <= 1.2d-253) then
        tmp = t_0
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 9.5e-299) {
		tmp = t_0;
	} else if (k <= 5.8e-287) {
		tmp = a;
	} else if (k <= 1.2e-253) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 9.5e-299:
		tmp = t_0
	elif k <= 5.8e-287:
		tmp = a
	elif k <= 1.2e-253:
		tmp = t_0
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 9.5e-299)
		tmp = t_0;
	elseif (k <= 5.8e-287)
		tmp = a;
	elseif (k <= 1.2e-253)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 9.5e-299)
		tmp = t_0;
	elseif (k <= 5.8e-287)
		tmp = a;
	elseif (k <= 1.2e-253)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 9.5e-299], t$95$0, If[LessEqual[k, 5.8e-287], a, If[LessEqual[k, 1.2e-253], t$95$0, If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.5000000000000001e-299 or 5.7999999999999996e-287 < k < 1.20000000000000005e-253

   1. Initial program 90.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 90.9%

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

      2. associate-+l+ 90.9%

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

      3. +-commutative 90.9%

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

      4. distribute-rgt-out 90.9%

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

      5. fma-def 90.9%

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

      6. +-commutative 90.9%

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

   3. Simplified 90.9%

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

   4. Taylor expanded in m around 0 24.6%

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

   5. Taylor expanded in k around inf 37.6%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 37.6%

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

   7. Simplified 37.6%

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

   if 9.5000000000000001e-299 < k < 5.7999999999999996e-287 or 1.20000000000000005e-253 < k < 1

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 54.8%

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

   5. Taylor expanded in k around 0 53.2%

      \[\leadsto \color{blue}{a} \]

   if 1 < k

   1. Initial program 78.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 78.3%

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

      2. associate-+l+ 78.3%

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

      3. +-commutative 78.3%

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

      4. distribute-rgt-out 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in k around inf 77.2%

      \[\leadsto \color{blue}{\frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{{k}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 77.2%

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

      2. unpow2 77.2%

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

      3. associate-/l* 77.2%

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

      4. associate-*r* 77.2%

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

      5. exp-prod 77.2%

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

      6. mul-1-neg 77.2%

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

      7. log-rec 77.2%

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

      8. associate-/l* 87.9%

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

   6. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{{\left(e^{-\left(-\log k\right)}\right)}^{m}}{\frac{k}{\frac{a}{k}}}} \]
   7. Step-by-step derivation

      1. pow-exp 87.9%

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

      2. remove-double-neg 87.9%

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

      3. pow-to-exp 87.9%

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

      4. add-sqr-sqrt 87.9%

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

      5. associate-/r/ 87.9%

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

      6. times-frac 93.6%

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

   8. Applied egg-rr 93.6%

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

      1. associate-*r/ 93.6%

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

      2. associate-/r/ 98.7%

         \[\leadsto \frac{\color{blue}{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right)} \cdot \sqrt{{k}^{m}}}{k} \]

   10. Applied egg-rr 98.7%

       \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt{{k}^{m}}}{k} \cdot a\right) \cdot \sqrt{{k}^{m}}}{k}} \]

   11. Taylor expanded in m around 0 56.7%

       \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   12. Step-by-step derivation

       1. unpow2 56.7%

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

       2. associate-/r* 59.8%

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

   13. Simplified 59.8%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-287}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-253}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 11: 59.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.78:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.3e+19)
   (/ a (* k k))
   (if (<= m 0.78) (/ a (+ 1.0 (* k (+ k 10.0)))) (* 0.1 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.78) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = 0.1 * (k * a);
	}
	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
    real(8) :: tmp
    if (m <= (-3.3d+19)) then
        tmp = a / (k * k)
    else if (m <= 0.78d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = 0.1d0 * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.78) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = 0.1 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.3e+19:
		tmp = a / (k * k)
	elif m <= 0.78:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = 0.1 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.3e+19)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.78)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(0.1 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.3e+19)
		tmp = a / (k * k);
	elseif (m <= 0.78)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = 0.1 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.3e+19], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.78], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.3e19

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 42.7%

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

   5. Taylor expanded in k around inf 66.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 66.4%

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

   7. Simplified 66.4%

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

   if -3.3e19 < m < 0.78000000000000003

   1. Initial program 92.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 92.3%

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

      2. associate-+l+ 92.3%

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

      3. +-commutative 92.3%

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

      4. distribute-rgt-out 92.3%

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

      5. fma-def 92.3%

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

      6. +-commutative 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in m around 0 89.3%

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

   if 0.78000000000000003 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.0%

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

      2. associate-+l+ 75.0%

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

      3. +-commutative 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. fma-def 75.0%

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

      6. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 2.8%

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

      2. add-exp-log 1.5%

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

      3. remove-double-neg 1.5%

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

      4. exp-neg 1.5%

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

      5. neg-log 1.5%

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

      6. add-exp-log 2.8%

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

      7. un-div-inv 2.8%

         \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{\frac{1}{k}}}} \]

      8. add-exp-log 1.5%

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

      9. neg-log 1.5%

         \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{-\log k}}}} \]

      10. add-sqr-sqrt 0.9%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{-\log k} \cdot \sqrt{-\log k}}}}} \]

      11. sqrt-unprod 2.1%

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

      12. sqr-neg 2.1%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\sqrt{\color{blue}{\log k \cdot \log k}}}}} \]

      13. sqrt-unprod 1.2%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{\log k} \cdot \sqrt{\log k}}}}} \]

      14. add-sqr-sqrt 5.7%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\log k}}}} \]

      15. add-exp-log 13.3%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{\color{blue}{k}}} \]

   6. Applied egg-rr 13.3%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{k}}} \]

   7. Taylor expanded in k around 0 22.0%

      \[\leadsto \color{blue}{0.1 \cdot \left(k \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.78:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 12: 59.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.78:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.3e+19)
   (/ a (* k k))
   (if (<= m 0.78) (/ a (+ 1.0 (* k k))) (* 0.1 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.78) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = 0.1 * (k * a);
	}
	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
    real(8) :: tmp
    if (m <= (-3.3d+19)) then
        tmp = a / (k * k)
    else if (m <= 0.78d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = 0.1d0 * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.3e+19) {
		tmp = a / (k * k);
	} else if (m <= 0.78) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = 0.1 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.3e+19:
		tmp = a / (k * k)
	elif m <= 0.78:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = 0.1 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.3e+19)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.78)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(0.1 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.3e+19)
		tmp = a / (k * k);
	elseif (m <= 0.78)
		tmp = a / (1.0 + (k * k));
	else
		tmp = 0.1 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.3e+19], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.78], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.3e19

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 42.7%

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

   5. Taylor expanded in k around inf 66.4%

      \[\leadsto \color{blue}{\frac{a}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 66.4%

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

   7. Simplified 66.4%

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

   if -3.3e19 < m < 0.78000000000000003

   1. Initial program 92.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 92.3%

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

      2. associate-+l+ 92.3%

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

      3. +-commutative 92.3%

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

      4. distribute-rgt-out 92.3%

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

      5. fma-def 92.3%

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

      6. +-commutative 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in m around 0 89.3%

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

   5. Taylor expanded in k around inf 86.8%

      \[\leadsto \frac{a}{1 + \color{blue}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 86.8%

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

   7. Simplified 86.8%

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

   if 0.78000000000000003 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.0%

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

      2. associate-+l+ 75.0%

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

      3. +-commutative 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. fma-def 75.0%

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

      6. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 2.8%

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

      2. add-exp-log 1.5%

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

      3. remove-double-neg 1.5%

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

      4. exp-neg 1.5%

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

      5. neg-log 1.5%

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

      6. add-exp-log 2.8%

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

      7. un-div-inv 2.8%

         \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{\frac{1}{k}}}} \]

      8. add-exp-log 1.5%

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

      9. neg-log 1.5%

         \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{-\log k}}}} \]

      10. add-sqr-sqrt 0.9%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{-\log k} \cdot \sqrt{-\log k}}}}} \]

      11. sqrt-unprod 2.1%

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

      12. sqr-neg 2.1%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\sqrt{\color{blue}{\log k \cdot \log k}}}}} \]

      13. sqrt-unprod 1.2%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{\log k} \cdot \sqrt{\log k}}}}} \]

      14. add-sqr-sqrt 5.7%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\log k}}}} \]

      15. add-exp-log 13.3%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{\color{blue}{k}}} \]

   6. Applied egg-rr 13.3%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{k}}} \]

   7. Taylor expanded in k around 0 22.0%

      \[\leadsto \color{blue}{0.1 \cdot \left(k \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.3 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.78:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13: 32.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.58:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8e-8) (/ a (* k 10.0)) (if (<= m 0.58) a (* 0.1 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8e-8) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = 0.1 * (k * a);
	}
	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
    real(8) :: tmp
    if (m <= (-8d-8)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 0.58d0) then
        tmp = a
    else
        tmp = 0.1d0 * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -8e-8) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = 0.1 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8e-8:
		tmp = a / (k * 10.0)
	elif m <= 0.58:
		tmp = a
	else:
		tmp = 0.1 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8e-8)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 0.58)
		tmp = a;
	else
		tmp = Float64(0.1 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8e-8)
		tmp = a / (k * 10.0);
	elseif (m <= 0.58)
		tmp = a;
	else
		tmp = 0.1 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8e-8], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.58], a, N[(0.1 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.0000000000000002e-8

   1. Initial program 98.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 98.8%

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

      2. associate-+l+ 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-rgt-out 98.8%

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

      5. fma-def 98.8%

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

      6. +-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in m around 0 44.5%

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

   5. Taylor expanded in k around 0 25.2%

      \[\leadsto \frac{a}{1 + \color{blue}{10 \cdot k}} \]
   6. Step-by-step derivation

      1. *-commutative 25.2%

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   7. Simplified 25.2%

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot 10}} \]

   8. Taylor expanded in k around inf 33.1%

      \[\leadsto \frac{a}{\color{blue}{10 \cdot k}} \]
   9. Step-by-step derivation

      1. *-commutative 33.1%

         \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

   10. Simplified 33.1%

       \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]

   if -8.0000000000000002e-8 < m < 0.57999999999999996

   1. Initial program 92.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 92.9%

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

      2. associate-+l+ 92.9%

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

      3. +-commutative 92.9%

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

      4. distribute-rgt-out 92.9%

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

      5. fma-def 92.9%

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

      6. +-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in m around 0 90.8%

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

   5. Taylor expanded in k around 0 50.7%

      \[\leadsto \color{blue}{a} \]

   if 0.57999999999999996 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.0%

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

      2. associate-+l+ 75.0%

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

      3. +-commutative 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. fma-def 75.0%

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

      6. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 2.8%

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

      2. add-exp-log 1.5%

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

      3. remove-double-neg 1.5%

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

      4. exp-neg 1.5%

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

      5. neg-log 1.5%

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

      6. add-exp-log 2.8%

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

      7. un-div-inv 2.8%

         \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{\frac{1}{k}}}} \]

      8. add-exp-log 1.5%

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

      9. neg-log 1.5%

         \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{-\log k}}}} \]

      10. add-sqr-sqrt 0.9%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{-\log k} \cdot \sqrt{-\log k}}}}} \]

      11. sqrt-unprod 2.1%

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

      12. sqr-neg 2.1%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\sqrt{\color{blue}{\log k \cdot \log k}}}}} \]

      13. sqrt-unprod 1.2%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{\log k} \cdot \sqrt{\log k}}}}} \]

      14. add-sqr-sqrt 5.7%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\log k}}}} \]

      15. add-exp-log 13.3%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{\color{blue}{k}}} \]

   6. Applied egg-rr 13.3%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{k}}} \]

   7. Taylor expanded in k around 0 22.0%

      \[\leadsto \color{blue}{0.1 \cdot \left(k \cdot a\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.58:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 14: 26.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.34:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.34) a (* 0.1 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.34) {
		tmp = a;
	} else {
		tmp = 0.1 * (k * a);
	}
	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
    real(8) :: tmp
    if (m <= 0.34d0) then
        tmp = a
    else
        tmp = 0.1d0 * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.34) {
		tmp = a;
	} else {
		tmp = 0.1 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.34:
		tmp = a
	else:
		tmp = 0.1 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.34)
		tmp = a;
	else
		tmp = Float64(0.1 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.34)
		tmp = a;
	else
		tmp = 0.1 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.34], a, N[(0.1 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.340000000000000024

   1. Initial program 95.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 95.8%

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

      2. associate-+l+ 95.7%

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

      3. +-commutative 95.7%

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

      4. distribute-rgt-out 95.8%

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

      5. fma-def 95.8%

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

      6. +-commutative 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in m around 0 68.4%

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

   5. Taylor expanded in k around 0 28.3%

      \[\leadsto \color{blue}{a} \]

   if 0.340000000000000024 < m

   1. Initial program 75.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Step-by-step derivation

      1. associate-*r/ 75.0%

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

      2. associate-+l+ 75.0%

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

      3. +-commutative 75.0%

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

      4. distribute-rgt-out 75.0%

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

      5. fma-def 75.0%

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

      6. +-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in m around 0 2.8%

      \[\leadsto \color{blue}{\frac{a}{1 + k \cdot \left(k + 10\right)}} \]
   5. Step-by-step derivation

      1. *-commutative 2.8%

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

      2. add-exp-log 1.5%

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

      3. remove-double-neg 1.5%

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

      4. exp-neg 1.5%

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

      5. neg-log 1.5%

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

      6. add-exp-log 2.8%

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

      7. un-div-inv 2.8%

         \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{\frac{1}{k}}}} \]

      8. add-exp-log 1.5%

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

      9. neg-log 1.5%

         \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{-\log k}}}} \]

      10. add-sqr-sqrt 0.9%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{-\log k} \cdot \sqrt{-\log k}}}}} \]

      11. sqrt-unprod 2.1%

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

      12. sqr-neg 2.1%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\sqrt{\color{blue}{\log k \cdot \log k}}}}} \]

      13. sqrt-unprod 1.2%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\sqrt{\log k} \cdot \sqrt{\log k}}}}} \]

      14. add-sqr-sqrt 5.7%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{e^{\color{blue}{\log k}}}} \]

      15. add-exp-log 13.3%

          \[\leadsto \frac{a}{1 + \frac{k + 10}{\color{blue}{k}}} \]

   6. Applied egg-rr 13.3%

      \[\leadsto \frac{a}{1 + \color{blue}{\frac{k + 10}{k}}} \]

   7. Taylor expanded in k around 0 22.0%

      \[\leadsto \color{blue}{0.1 \cdot \left(k \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.34:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 15: 20.4% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

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
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.0%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Step-by-step derivation

   1. associate-*r/ 88.9%

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

   2. associate-+l+ 88.9%

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

   3. +-commutative 88.9%

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

   4. distribute-rgt-out 88.9%

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

   5. fma-def 88.9%

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

   6. +-commutative 88.9%

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

3. Simplified 88.9%

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

4. Taylor expanded in m around 0 46.9%

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

5. Taylor expanded in k around 0 20.3%

   \[\leadsto \color{blue}{a} \]

6. Final simplification 20.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023182 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))