Falkner and Boettcher, Appendix A

Percentage Accurate: 89.7% → 97.6%

Time: 9.8s

Alternatives: 17

Speedup: 1.1×

• 24.0% 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: 89.7% 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: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.86:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.86)
   (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
   (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.86) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.86)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.86], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.859999999999999987

   1. Initial program 97.6%

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

      1. associate-*r/ 97.6%

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

      2. associate-+l+ 97.6%

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

      3. +-commutative 97.6%

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

      4. distribute-rgt-out 97.7%

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

      5. fma-def 97.7%

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

      6. +-commutative 97.7%

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

   3. Simplified 97.7%

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

   if 0.859999999999999987 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. associate-+l+ 82.5%

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

      3. +-commutative 82.5%

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

      4. distribute-rgt-out 82.5%

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

      5. fma-def 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in k around inf 28.4%

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

      1. mul-1-neg 28.4%

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

      2. exp-neg 28.4%

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

      3. log-rec 28.4%

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

      4. distribute-lft-neg-in 28.4%

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

      5. rec-exp 28.4%

         \[\leadsto a \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{{k}^{2}} \]

      6. exp-to-pow 55.4%

         \[\leadsto a \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{{k}^{2}} \]

      7. unpow2 55.4%

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

   6. Simplified 55.4%

      \[\leadsto a \cdot \color{blue}{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k \cdot k}} \]
   7. Step-by-step derivation

      1. remove-double-div 55.4%

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

      2. pow2 55.4%

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

      3. pow-div 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.86:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 2: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.85)
   (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
   (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.85) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	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 <= 0.85d0) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = a * (k ** (m - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.85) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.85:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.85)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.85)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.85], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.849999999999999978

   1. Initial program 97.6%

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

      1. associate-/l* 97.6%

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

      2. associate-+l+ 97.6%

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

      3. *-commutative 97.6%

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

   3. Simplified 97.6%

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

      1. distribute-lft-out 97.6%

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

      2. +-commutative 97.6%

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

      3. *-commutative 97.6%

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

   5. Applied egg-rr 97.6%

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

   if 0.849999999999999978 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. associate-+l+ 82.5%

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

      3. +-commutative 82.5%

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

      4. distribute-rgt-out 82.5%

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

      5. fma-def 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in k around inf 28.4%

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

      1. mul-1-neg 28.4%

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

      2. exp-neg 28.4%

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

      3. log-rec 28.4%

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

      4. distribute-lft-neg-in 28.4%

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

      5. rec-exp 28.4%

         \[\leadsto a \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{{k}^{2}} \]

      6. exp-to-pow 55.4%

         \[\leadsto a \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{{k}^{2}} \]

      7. unpow2 55.4%

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

   6. Simplified 55.4%

      \[\leadsto a \cdot \color{blue}{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k \cdot k}} \]
   7. Step-by-step derivation

      1. remove-double-div 55.4%

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

      2. pow2 55.4%

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

      3. pow-div 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.86:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.86) (/ (* a (pow k m)) (+ 1.0 (* k k))) (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.86) {
		tmp = (a * pow(k, m)) / (1.0 + (k * k));
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	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 <= 0.86d0) then
        tmp = (a * (k ** m)) / (1.0d0 + (k * k))
    else
        tmp = a * (k ** (m - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.86) {
		tmp = (a * Math.pow(k, m)) / (1.0 + (k * k));
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.86:
		tmp = (a * math.pow(k, m)) / (1.0 + (k * k))
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.86)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.86)
		tmp = (a * (k ^ m)) / (1.0 + (k * k));
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.86], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.859999999999999987

   1. Initial program 97.6%

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

   2. Taylor expanded in k around 0 96.5%

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

   if 0.859999999999999987 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. associate-+l+ 82.5%

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

      3. +-commutative 82.5%

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

      4. distribute-rgt-out 82.5%

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

      5. fma-def 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in k around inf 28.4%

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

      1. mul-1-neg 28.4%

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

      2. exp-neg 28.4%

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

      3. log-rec 28.4%

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

      4. distribute-lft-neg-in 28.4%

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

      5. rec-exp 28.4%

         \[\leadsto a \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{{k}^{2}} \]

      6. exp-to-pow 55.4%

         \[\leadsto a \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{{k}^{2}} \]

      7. unpow2 55.4%

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

   6. Simplified 55.4%

      \[\leadsto a \cdot \color{blue}{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k \cdot k}} \]
   7. Step-by-step derivation

      1. remove-double-div 55.4%

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

      2. pow2 55.4%

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

      3. pow-div 99.4%

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

   8. Applied egg-rr 99.4%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.86:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-10} \lor \neg \left(m \leq 0.9\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -8.6e-10) (not (<= m 0.9)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -8.6e-10) || !(m <= 0.9)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-8.6d-10)) .or. (.not. (m <= 0.9d0))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -8.6e-10) || !(m <= 0.9)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -8.6e-10) or not (m <= 0.9):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -8.6e-10) || !(m <= 0.9))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -8.6e-10) || ~((m <= 0.9)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -8.6e-10], N[Not[LessEqual[m, 0.9]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -8.60000000000000029e-10 or 0.900000000000000022 < m

   1. Initial program 91.8%

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

      1. associate-*r/ 91.8%

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

      2. associate-+l+ 91.8%

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

      3. +-commutative 91.8%

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

      4. distribute-rgt-out 91.8%

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

      5. fma-def 91.8%

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

      6. +-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in k around 0 52.6%

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

      1. exp-to-pow 99.4%

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

      2. *-commutative 99.4%

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

   6. Simplified 99.4%

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

   if -8.60000000000000029e-10 < m < 0.900000000000000022

   1. Initial program 94.2%

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

      1. associate-*r/ 94.2%

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

      2. associate-+l+ 94.2%

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

      3. +-commutative 94.2%

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

      4. distribute-rgt-out 94.2%

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

      5. fma-def 94.2%

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

      6. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 92.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.6 \cdot 10^{-10} \lor \neg \left(m \leq 0.9\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* a (pow k m)) (* a (pow k (- m 2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m - 2.0));
	}
	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 (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m - 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m - 2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m - 2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m - 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m - 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m - 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.2%

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

      1. associate-*r/ 98.2%

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

      2. associate-+l+ 98.2%

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

      3. +-commutative 98.2%

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

      4. distribute-rgt-out 98.2%

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

      5. fma-def 98.2%

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

      6. +-commutative 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in k around 0 50.7%

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

      1. exp-to-pow 98.9%

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

      2. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   if 1 < k

   1. Initial program 82.3%

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

      1. associate-*r/ 82.3%

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

      2. associate-+l+ 82.3%

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

      3. +-commutative 82.3%

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

      4. distribute-rgt-out 82.3%

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

      5. fma-def 82.3%

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

      6. +-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in k around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. exp-neg 82.2%

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

      3. log-rec 82.2%

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

      4. distribute-lft-neg-in 82.2%

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

      5. rec-exp 82.2%

         \[\leadsto a \cdot \frac{\frac{1}{\color{blue}{\frac{1}{e^{\log k \cdot m}}}}}{{k}^{2}} \]

      6. exp-to-pow 82.2%

         \[\leadsto a \cdot \frac{\frac{1}{\frac{1}{\color{blue}{{k}^{m}}}}}{{k}^{2}} \]

      7. unpow2 82.2%

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

   6. Simplified 82.2%

      \[\leadsto a \cdot \color{blue}{\frac{\frac{1}{\frac{1}{{k}^{m}}}}{k \cdot k}} \]
   7. Step-by-step derivation

      1. remove-double-div 82.2%

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

      2. pow2 82.2%

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

      3. pow-div 94.8%

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

   8. Applied egg-rr 94.8%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m - 2\right)}\\ \end{array} \]

Alternative 6: 62.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.4:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.115)
   (/ a (* k k))
   (if (<= m 2.4)
     (* a (/ 1.0 (+ 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 <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 2.4) {
		tmp = a * (1.0 / (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 <= (-0.115d0)) then
        tmp = a / (k * k)
    else if (m <= 2.4d0) then
        tmp = a * (1.0d0 / (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 <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 2.4) {
		tmp = a * (1.0 / (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 <= -0.115:
		tmp = a / (k * k)
	elif m <= 2.4:
		tmp = a * (1.0 / (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 <= -0.115)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.4)
		tmp = Float64(a * Float64(1.0 / 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 <= -0.115)
		tmp = a / (k * k);
	elseif (m <= 2.4)
		tmp = a * (1.0 / (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, -0.115], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $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 < -0.115000000000000005

   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 36.2%

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

   5. Taylor expanded in k around inf 62.7%

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

      1. unpow2 62.7%

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

   7. Simplified 62.7%

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

   if -0.115000000000000005 < m < 2.39999999999999991

   1. Initial program 94.2%

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

      1. associate-*r/ 94.2%

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

      2. associate-+l+ 94.2%

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

      3. +-commutative 94.2%

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

      4. distribute-rgt-out 94.2%

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

      5. fma-def 94.2%

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

      6. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 92.1%

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

   if 2.39999999999999991 < m

   1. Initial program 83.3%

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

   2. Taylor expanded in k around 0 79.8%

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

   3. Taylor expanded in m around 0 2.8%

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

   4. Taylor expanded in k around 0 16.2%

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

      1. +-commutative 16.2%

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

      2. unpow2 16.2%

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

      3. associate-*r* 16.2%

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

      4. associate-*r* 16.2%

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

      5. *-commutative 16.2%

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

      6. distribute-rgt-out 24.5%

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

      7. *-commutative 24.5%

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

      8. associate-*l* 24.5%

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

      9. *-commutative 24.5%

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

   6. Simplified 24.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.4:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 100\right) + k \cdot -10\right)\\ \end{array} \]

Alternative 7: 44.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.8 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-216}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-179} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 1.8e-265)
     t_0
     (if (<= k 6.6e-216)
       a
       (if (or (<= k 1.05e-179) (not (<= k 0.1)))
         t_0
         (* a (+ 1.0 (* k -10.0))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.8e-265) {
		tmp = t_0;
	} else if (k <= 6.6e-216) {
		tmp = a;
	} else if ((k <= 1.05e-179) || !(k <= 0.1)) {
		tmp = t_0;
	} else {
		tmp = a * (1.0 + (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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.8d-265) then
        tmp = t_0
    else if (k <= 6.6d-216) then
        tmp = a
    else if ((k <= 1.05d-179) .or. (.not. (k <= 0.1d0))) then
        tmp = t_0
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    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 <= 1.8e-265) {
		tmp = t_0;
	} else if (k <= 6.6e-216) {
		tmp = a;
	} else if ((k <= 1.05e-179) || !(k <= 0.1)) {
		tmp = t_0;
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.8e-265:
		tmp = t_0
	elif k <= 6.6e-216:
		tmp = a
	elif (k <= 1.05e-179) or not (k <= 0.1):
		tmp = t_0
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.8e-265)
		tmp = t_0;
	elseif (k <= 6.6e-216)
		tmp = a;
	elseif ((k <= 1.05e-179) || !(k <= 0.1))
		tmp = t_0;
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 1.8e-265)
		tmp = t_0;
	elseif (k <= 6.6e-216)
		tmp = a;
	elseif ((k <= 1.05e-179) || ~((k <= 0.1)))
		tmp = t_0;
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.8e-265], t$95$0, If[LessEqual[k, 6.6e-216], a, If[Or[LessEqual[k, 1.05e-179], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], t$95$0, N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.8000000000000001e-265 or 6.59999999999999937e-216 < k < 1.0499999999999999e-179 or 0.10000000000000001 < k

   1. Initial program 90.2%

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

      1. associate-*r/ 90.2%

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

      2. associate-+l+ 90.2%

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

      3. +-commutative 90.2%

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

      4. distribute-rgt-out 90.2%

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

      5. fma-def 90.2%

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

      6. +-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in m around 0 38.9%

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

   5. Taylor expanded in k around inf 47.4%

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

      1. unpow2 47.4%

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

   7. Simplified 47.4%

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

   if 1.8000000000000001e-265 < k < 6.59999999999999937e-216

   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 74.5%

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

   5. Taylor expanded in k around 0 74.5%

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

   if 1.0499999999999999e-179 < k < 0.10000000000000001

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 55.6%

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

   5. Taylor expanded in k around 0 55.6%

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

   6. Taylor expanded in a around 0 55.6%

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

      1. *-commutative 55.6%

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

   8. Simplified 55.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-216}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-179} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 8: 45.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.9 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{-214}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 1.9e-265)
     t_0
     (if (<= k 3.3e-214)
       a
       (if (<= k 8.2e-180)
         t_0
         (if (<= k 0.1) (* a (+ 1.0 (* k -10.0))) (* (/ 1.0 k) (/ a k))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.9e-265) {
		tmp = t_0;
	} else if (k <= 3.3e-214) {
		tmp = a;
	} else if (k <= 8.2e-180) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.9d-265) then
        tmp = t_0
    else if (k <= 3.3d-214) then
        tmp = a
    else if (k <= 8.2d-180) then
        tmp = t_0
    else if (k <= 0.1d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (1.0d0 / k) * (a / 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 <= 1.9e-265) {
		tmp = t_0;
	} else if (k <= 3.3e-214) {
		tmp = a;
	} else if (k <= 8.2e-180) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.9e-265:
		tmp = t_0
	elif k <= 3.3e-214:
		tmp = a
	elif k <= 8.2e-180:
		tmp = t_0
	elif k <= 0.1:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (1.0 / k) * (a / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.9e-265)
		tmp = t_0;
	elseif (k <= 3.3e-214)
		tmp = a;
	elseif (k <= 8.2e-180)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 1.9e-265)
		tmp = t_0;
	elseif (k <= 3.3e-214)
		tmp = a;
	elseif (k <= 8.2e-180)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (1.0 / k) * (a / 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, 1.9e-265], t$95$0, If[LessEqual[k, 3.3e-214], a, If[LessEqual[k, 8.2e-180], t$95$0, If[LessEqual[k, 0.1], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.8999999999999999e-265 or 3.2999999999999998e-214 < k < 8.2e-180

   1. Initial program 97.1%

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

      1. associate-*r/ 97.1%

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

      2. associate-+l+ 97.1%

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

      3. +-commutative 97.1%

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

      4. distribute-rgt-out 97.1%

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

      5. fma-def 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in m around 0 20.1%

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

   5. Taylor expanded in k around inf 35.9%

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

      1. unpow2 35.9%

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

   7. Simplified 35.9%

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

   if 1.8999999999999999e-265 < k < 3.2999999999999998e-214

   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 74.5%

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

   5. Taylor expanded in k around 0 74.5%

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

   if 8.2e-180 < k < 0.10000000000000001

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 55.6%

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

   5. Taylor expanded in k around 0 55.6%

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

   6. Taylor expanded in a around 0 55.6%

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

      1. *-commutative 55.6%

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

   8. Simplified 55.6%

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

   if 0.10000000000000001 < k

   1. Initial program 82.3%

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

      1. associate-*r/ 82.3%

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

      2. associate-+l+ 82.3%

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

      3. +-commutative 82.3%

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

      4. distribute-rgt-out 82.3%

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

      5. fma-def 82.3%

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

      6. +-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in m around 0 60.5%

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

   5. Taylor expanded in k around inf 60.5%

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

      1. unpow2 60.5%

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

   7. Simplified 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 64.8%

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

   9. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.9 \cdot 10^{-265}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{-214}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 8.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 9: 45.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.75 \cdot 10^{-265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 1.75e-265)
     t_0
     (if (<= k 4e-218)
       a
       (if (<= k 1.25e-179)
         t_0
         (if (<= k 0.1) (+ a (* -10.0 (* a k))) (* (/ 1.0 k) (/ a k))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.75e-265) {
		tmp = t_0;
	} else if (k <= 4e-218) {
		tmp = a;
	} else if (k <= 1.25e-179) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.75d-265) then
        tmp = t_0
    else if (k <= 4d-218) then
        tmp = a
    else if (k <= 1.25d-179) then
        tmp = t_0
    else if (k <= 0.1d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = (1.0d0 / k) * (a / 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-265) {
		tmp = t_0;
	} else if (k <= 4e-218) {
		tmp = a;
	} else if (k <= 1.25e-179) {
		tmp = t_0;
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = (1.0 / k) * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.75e-265:
		tmp = t_0
	elif k <= 4e-218:
		tmp = a
	elif k <= 1.25e-179:
		tmp = t_0
	elif k <= 0.1:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = (1.0 / k) * (a / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.75e-265)
		tmp = t_0;
	elseif (k <= 4e-218)
		tmp = a;
	elseif (k <= 1.25e-179)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(a / 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-265)
		tmp = t_0;
	elseif (k <= 4e-218)
		tmp = a;
	elseif (k <= 1.25e-179)
		tmp = t_0;
	elseif (k <= 0.1)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = (1.0 / k) * (a / 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-265], t$95$0, If[LessEqual[k, 4e-218], a, If[LessEqual[k, 1.25e-179], t$95$0, If[LessEqual[k, 0.1], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 1.75000000000000008e-265 or 4.0000000000000001e-218 < k < 1.2499999999999999e-179

   1. Initial program 97.1%

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

      1. associate-*r/ 97.1%

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

      2. associate-+l+ 97.1%

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

      3. +-commutative 97.1%

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

      4. distribute-rgt-out 97.1%

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

      5. fma-def 97.1%

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

      6. +-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in m around 0 20.1%

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

   5. Taylor expanded in k around inf 35.9%

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

      1. unpow2 35.9%

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

   7. Simplified 35.9%

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

   if 1.75000000000000008e-265 < k < 4.0000000000000001e-218

   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 74.5%

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

   5. Taylor expanded in k around 0 74.5%

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

   if 1.2499999999999999e-179 < k < 0.10000000000000001

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 55.6%

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

   5. Taylor expanded in k around 0 55.6%

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

   if 0.10000000000000001 < k

   1. Initial program 82.3%

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

      1. associate-*r/ 82.3%

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

      2. associate-+l+ 82.3%

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

      3. +-commutative 82.3%

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

      4. distribute-rgt-out 82.3%

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

      5. fma-def 82.3%

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

      6. +-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in m around 0 60.5%

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

   5. Taylor expanded in k around inf 60.5%

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

      1. unpow2 60.5%

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

   7. Simplified 60.5%

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

      1. *-un-lft-identity 60.5%

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

      2. times-frac 64.8%

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

   9. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{-265}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-218}:\\ \;\;\;\;a\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-179}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 10: 58.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.115)
   (/ a (* k k))
   (if (<= m 1.15) (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0))))) (* k (* a 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 1.15) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = k * (a * 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 <= (-0.115d0)) then
        tmp = a / (k * k)
    else if (m <= 1.15d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = k * (a * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 1.15) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = k * (a * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.115:
		tmp = a / (k * k)
	elif m <= 1.15:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = k * (a * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.115)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.15)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(k * Float64(a * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.115)
		tmp = a / (k * k);
	elseif (m <= 1.15)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = k * (a * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.115], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.115000000000000005

   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 36.2%

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

   5. Taylor expanded in k around inf 62.7%

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

      1. unpow2 62.7%

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

   7. Simplified 62.7%

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

   if -0.115000000000000005 < m < 1.1499999999999999

   1. Initial program 94.2%

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

      1. associate-*r/ 94.2%

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

      2. associate-+l+ 94.2%

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

      3. +-commutative 94.2%

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

      4. distribute-rgt-out 94.2%

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

      5. fma-def 94.2%

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

      6. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 92.1%

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

   if 1.1499999999999999 < m

   1. Initial program 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. distribute-rgt-out 83.3%

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

      5. fma-def 83.3%

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

      6. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   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 6.2%

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

   6. Taylor expanded in a around 0 6.2%

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

      1. *-commutative 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in k around inf 18.1%

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

       1. *-commutative 18.1%

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

       2. *-commutative 18.1%

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

       3. associate-*r* 18.1%

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

       4. rem-square-sqrt 13.4%

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

       5. fabs-sqr 13.4%

          \[\leadsto a \cdot \color{blue}{\left|\sqrt{k \cdot -10} \cdot \sqrt{k \cdot -10}\right|} \]

       6. rem-square-sqrt 27.1%

          \[\leadsto a \cdot \left|\color{blue}{k \cdot -10}\right| \]

       7. *-lft-identity 27.1%

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

       8. fabs-mul 27.1%

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

       9. *-lft-identity 27.1%

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

       10. rem-square-sqrt 13.7%

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

       11. fabs-sqr 13.7%

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

       12. rem-square-sqrt 23.0%

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

       13. metadata-eval 23.0%

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

       14. associate-*l* 23.0%

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

       15. *-commutative 23.0%

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

       16. associate-*l* 23.0%

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

   11. Simplified 23.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.15:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \]

Alternative 11: 44.1% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-265} \lor \neg \left(k \leq 4.4 \cdot 10^{-214}\right) \land \left(k \leq 3.3 \cdot 10^{-180} \lor \neg \left(k \leq 32500000\right)\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 1.95e-265)
         (and (not (<= k 4.4e-214))
              (or (<= k 3.3e-180) (not (<= k 32500000.0)))))
   (/ a (* k k))
   a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 1.95e-265) || (!(k <= 4.4e-214) && ((k <= 3.3e-180) || !(k <= 32500000.0)))) {
		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 <= 1.95d-265) .or. (.not. (k <= 4.4d-214)) .and. (k <= 3.3d-180) .or. (.not. (k <= 32500000.0d0))) 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 <= 1.95e-265) || (!(k <= 4.4e-214) && ((k <= 3.3e-180) || !(k <= 32500000.0)))) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 1.95e-265) or (not (k <= 4.4e-214) and ((k <= 3.3e-180) or not (k <= 32500000.0))):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 1.95e-265) || (!(k <= 4.4e-214) && ((k <= 3.3e-180) || !(k <= 32500000.0))))
		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 <= 1.95e-265) || (~((k <= 4.4e-214)) && ((k <= 3.3e-180) || ~((k <= 32500000.0)))))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 1.95e-265], And[N[Not[LessEqual[k, 4.4e-214]], $MachinePrecision], Or[LessEqual[k, 3.3e-180], N[Not[LessEqual[k, 32500000.0]], $MachinePrecision]]]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 1.9499999999999999e-265 or 4.40000000000000003e-214 < k < 3.29999999999999998e-180 or 3.25e7 < k

   1. Initial program 90.1%

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

      1. associate-*r/ 90.1%

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

      2. associate-+l+ 90.1%

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

      3. +-commutative 90.1%

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

      4. distribute-rgt-out 90.1%

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

      5. fma-def 90.1%

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

      6. +-commutative 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in m around 0 39.3%

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

   5. Taylor expanded in k around inf 47.8%

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

      1. unpow2 47.8%

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

   7. Simplified 47.8%

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

   if 1.9499999999999999e-265 < k < 4.40000000000000003e-214 or 3.29999999999999998e-180 < k < 3.25e7

   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 57.1%

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

   5. Taylor expanded in k around 0 55.8%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-265} \lor \neg \left(k \leq 4.4 \cdot 10^{-214}\right) \land \left(k \leq 3.3 \cdot 10^{-180} \lor \neg \left(k \leq 32500000\right)\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12: 58.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.115)
   (/ a (* k k))
   (if (<= m 1.4) (/ a (+ 1.0 (* k (+ k 10.0)))) (* k (* a 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 1.4) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * 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 <= (-0.115d0)) then
        tmp = a / (k * k)
    else if (m <= 1.4d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = k * (a * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.115) {
		tmp = a / (k * k);
	} else if (m <= 1.4) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = k * (a * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.115:
		tmp = a / (k * k)
	elif m <= 1.4:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = k * (a * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.115)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.4)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(k * Float64(a * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.115)
		tmp = a / (k * k);
	elseif (m <= 1.4)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = k * (a * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.115], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.4], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.115000000000000005

   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 36.2%

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

   5. Taylor expanded in k around inf 62.7%

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

      1. unpow2 62.7%

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

   7. Simplified 62.7%

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

   if -0.115000000000000005 < m < 1.3999999999999999

   1. Initial program 94.2%

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

      1. associate-*r/ 94.2%

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

      2. associate-+l+ 94.2%

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

      3. +-commutative 94.2%

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

      4. distribute-rgt-out 94.2%

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

      5. fma-def 94.2%

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

      6. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in m around 0 92.1%

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

   if 1.3999999999999999 < m

   1. Initial program 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. distribute-rgt-out 83.3%

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

      5. fma-def 83.3%

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

      6. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   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 6.2%

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

   6. Taylor expanded in a around 0 6.2%

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

      1. *-commutative 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in k around inf 18.1%

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

       1. *-commutative 18.1%

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

       2. *-commutative 18.1%

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

       3. associate-*r* 18.1%

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

       4. rem-square-sqrt 13.4%

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

       5. fabs-sqr 13.4%

          \[\leadsto a \cdot \color{blue}{\left|\sqrt{k \cdot -10} \cdot \sqrt{k \cdot -10}\right|} \]

       6. rem-square-sqrt 27.1%

          \[\leadsto a \cdot \left|\color{blue}{k \cdot -10}\right| \]

       7. *-lft-identity 27.1%

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

       8. fabs-mul 27.1%

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

       9. *-lft-identity 27.1%

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

       10. rem-square-sqrt 13.7%

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

       11. fabs-sqr 13.7%

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

       12. rem-square-sqrt 23.0%

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

       13. metadata-eval 23.0%

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

       14. associate-*l* 23.0%

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

       15. *-commutative 23.0%

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

       16. associate-*l* 23.0%

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

   11. Simplified 23.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.115:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \]

Alternative 13: 28.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k} \cdot 0.1\\ \mathbf{if}\;k \leq -1.75 \cdot 10^{+183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-276}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 32500000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ a k) 0.1)))
   (if (<= k -1.75e+183)
     t_0
     (if (<= k 3e-276) (* -10.0 (* a k)) (if (<= k 32500000.0) a t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = (a / k) * 0.1;
	double tmp;
	if (k <= -1.75e+183) {
		tmp = t_0;
	} else if (k <= 3e-276) {
		tmp = -10.0 * (a * k);
	} else if (k <= 32500000.0) {
		tmp = a;
	} 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) * 0.1d0
    if (k <= (-1.75d+183)) then
        tmp = t_0
    else if (k <= 3d-276) then
        tmp = (-10.0d0) * (a * k)
    else if (k <= 32500000.0d0) then
        tmp = a
    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 / k) * 0.1;
	double tmp;
	if (k <= -1.75e+183) {
		tmp = t_0;
	} else if (k <= 3e-276) {
		tmp = -10.0 * (a * k);
	} else if (k <= 32500000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (a / k) * 0.1
	tmp = 0
	if k <= -1.75e+183:
		tmp = t_0
	elif k <= 3e-276:
		tmp = -10.0 * (a * k)
	elif k <= 32500000.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a / k) * 0.1)
	tmp = 0.0
	if (k <= -1.75e+183)
		tmp = t_0;
	elseif (k <= 3e-276)
		tmp = Float64(-10.0 * Float64(a * k));
	elseif (k <= 32500000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (a / k) * 0.1;
	tmp = 0.0;
	if (k <= -1.75e+183)
		tmp = t_0;
	elseif (k <= 3e-276)
		tmp = -10.0 * (a * k);
	elseif (k <= 32500000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision]}, If[LessEqual[k, -1.75e+183], t$95$0, If[LessEqual[k, 3e-276], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 32500000.0], a, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.74999999999999994e183 or 3.25e7 < k

   1. Initial program 82.1%

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

   2. Taylor expanded in k around 0 71.6%

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

   3. Taylor expanded in k around inf 57.5%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}} \]
   4. Step-by-step derivation

      1. *-commutative 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right) \cdot -1}}}{k} \]

      2. log-rec 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\left(\color{blue}{\left(-\log k\right)} \cdot m\right) \cdot -1}}{k} \]

      3. distribute-lft-neg-in 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(-\log k \cdot m\right)} \cdot -1}}{k} \]

      4. distribute-rgt-neg-out 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log k \cdot \left(-m\right)\right)} \cdot -1}}{k} \]

      5. *-commutative 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\left(-m\right) \cdot \log k\right)} \cdot -1}}{k} \]

      6. *-commutative 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log k \cdot \left(-m\right)\right)} \cdot -1}}{k} \]

      7. distribute-rgt-neg-out 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(-\log k \cdot m\right)} \cdot -1}}{k} \]

      8. distribute-lft-neg-out 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{-\left(\log k \cdot m\right) \cdot -1}}}{k} \]

      9. *-commutative 57.5%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{-1 \cdot \left(\log k \cdot m\right)}}}{k} \]

      10. mul-1-neg 57.5%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{\left(-\log k \cdot m\right)}}}{k} \]

      11. distribute-rgt-neg-out 57.5%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{\log k \cdot \left(-m\right)}}}{k} \]

      12. distribute-rgt-neg-out 57.5%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\log k \cdot \left(-\left(-m\right)\right)}}}{k} \]

      13. remove-double-neg 57.5%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{\log k \cdot \color{blue}{m}}}{k} \]

      14. exp-to-pow 71.6%

          \[\leadsto 0.1 \cdot \frac{a \cdot \color{blue}{{k}^{m}}}{k} \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a \cdot {k}^{m}}{k}} \]

   6. Taylor expanded in m around 0 32.6%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]

   if -1.74999999999999994e183 < k < 2.99999999999999988e-276

   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 3.7%

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

   5. Taylor expanded in k around 0 4.8%

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

   6. Taylor expanded in k around inf 15.9%

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

   if 2.99999999999999988e-276 < k < 3.25e7

   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 50.2%

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

   5. Taylor expanded in k around 0 49.2%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{+183}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-276}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 32500000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \end{array} \]

Alternative 14: 48.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8.2e-12)
   (/ a (* k k))
   (if (<= m 1.75) (/ a (+ 1.0 (* k 10.0))) (* k (* a 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.2e-12) {
		tmp = a / (k * k);
	} else if (m <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = k * (a * 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 <= (-8.2d-12)) then
        tmp = a / (k * k)
    else if (m <= 1.75d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = k * (a * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.2e-12) {
		tmp = a / (k * k);
	} else if (m <= 1.75) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = k * (a * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8.2e-12:
		tmp = a / (k * k)
	elif m <= 1.75:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = k * (a * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8.2e-12)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.75)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(k * Float64(a * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8.2e-12)
		tmp = a / (k * k);
	elseif (m <= 1.75)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = k * (a * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8.2e-12], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.75], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.19999999999999979e-12

   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 36.2%

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

   5. Taylor expanded in k around inf 62.7%

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

      1. unpow2 62.7%

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

   7. Simplified 62.7%

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

   if -8.19999999999999979e-12 < m < 1.75

   1. Initial program 94.2%

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

   2. Taylor expanded in k around 0 68.1%

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

   3. Taylor expanded in m around 0 67.8%

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

   if 1.75 < m

   1. Initial program 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. distribute-rgt-out 83.3%

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

      5. fma-def 83.3%

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

      6. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   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 6.2%

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

   6. Taylor expanded in a around 0 6.2%

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

      1. *-commutative 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in k around inf 18.1%

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

       1. *-commutative 18.1%

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

       2. *-commutative 18.1%

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

       3. associate-*r* 18.1%

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

       4. rem-square-sqrt 13.4%

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

       5. fabs-sqr 13.4%

          \[\leadsto a \cdot \color{blue}{\left|\sqrt{k \cdot -10} \cdot \sqrt{k \cdot -10}\right|} \]

       6. rem-square-sqrt 27.1%

          \[\leadsto a \cdot \left|\color{blue}{k \cdot -10}\right| \]

       7. *-lft-identity 27.1%

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

       8. fabs-mul 27.1%

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

       9. *-lft-identity 27.1%

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

       10. rem-square-sqrt 13.7%

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

       11. fabs-sqr 13.7%

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

       12. rem-square-sqrt 23.0%

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

       13. metadata-eval 23.0%

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

       14. associate-*l* 23.0%

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

       15. *-commutative 23.0%

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

       16. associate-*l* 23.0%

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

   11. Simplified 23.0%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.75:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \]

Alternative 15: 31.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8.2e-26) (* (/ a k) 0.1) (if (<= m 0.9) a (* k (* a 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.2e-26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.9) {
		tmp = a;
	} else {
		tmp = k * (a * 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 <= (-8.2d-26)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 0.9d0) then
        tmp = a
    else
        tmp = k * (a * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -8.2e-26) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.9) {
		tmp = a;
	} else {
		tmp = k * (a * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8.2e-26:
		tmp = (a / k) * 0.1
	elif m <= 0.9:
		tmp = a
	else:
		tmp = k * (a * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8.2e-26)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 0.9)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8.2e-26)
		tmp = (a / k) * 0.1;
	elseif (m <= 0.9)
		tmp = a;
	else
		tmp = k * (a * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8.2e-26], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 0.9], a, N[(k * N[(a * 10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.1999999999999997e-26

   1. Initial program 100.0%

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

   2. Taylor expanded in k around 0 98.9%

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

   3. Taylor expanded in k around inf 52.3%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}{k}} \]
   4. Step-by-step derivation

      1. *-commutative 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log \left(\frac{1}{k}\right) \cdot m\right) \cdot -1}}}{k} \]

      2. log-rec 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\left(\color{blue}{\left(-\log k\right)} \cdot m\right) \cdot -1}}{k} \]

      3. distribute-lft-neg-in 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(-\log k \cdot m\right)} \cdot -1}}{k} \]

      4. distribute-rgt-neg-out 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log k \cdot \left(-m\right)\right)} \cdot -1}}{k} \]

      5. *-commutative 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\left(-m\right) \cdot \log k\right)} \cdot -1}}{k} \]

      6. *-commutative 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(\log k \cdot \left(-m\right)\right)} \cdot -1}}{k} \]

      7. distribute-rgt-neg-out 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\left(-\log k \cdot m\right)} \cdot -1}}{k} \]

      8. distribute-lft-neg-out 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{-\left(\log k \cdot m\right) \cdot -1}}}{k} \]

      9. *-commutative 52.3%

         \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{-1 \cdot \left(\log k \cdot m\right)}}}{k} \]

      10. mul-1-neg 52.3%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{\left(-\log k \cdot m\right)}}}{k} \]

      11. distribute-rgt-neg-out 52.3%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{-\color{blue}{\log k \cdot \left(-m\right)}}}{k} \]

      12. distribute-rgt-neg-out 52.3%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{\color{blue}{\log k \cdot \left(-\left(-m\right)\right)}}}{k} \]

      13. remove-double-neg 52.3%

          \[\leadsto 0.1 \cdot \frac{a \cdot e^{\log k \cdot \color{blue}{m}}}{k} \]

      14. exp-to-pow 82.3%

          \[\leadsto 0.1 \cdot \frac{a \cdot \color{blue}{{k}^{m}}}{k} \]

   5. Simplified 82.3%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a \cdot {k}^{m}}{k}} \]

   6. Taylor expanded in m around 0 27.2%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]

   if -8.1999999999999997e-26 < m < 0.900000000000000022

   1. Initial program 93.9%

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

      1. associate-*r/ 94.0%

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

      2. associate-+l+ 94.0%

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

      3. +-commutative 94.0%

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

      4. distribute-rgt-out 94.0%

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

      5. fma-def 94.0%

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

      6. +-commutative 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in m around 0 91.8%

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

   5. Taylor expanded in k around 0 47.9%

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

   if 0.900000000000000022 < m

   1. Initial program 83.3%

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

      1. associate-*r/ 83.3%

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

      2. associate-+l+ 83.3%

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

      3. +-commutative 83.3%

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

      4. distribute-rgt-out 83.3%

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

      5. fma-def 83.3%

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

      6. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   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 6.2%

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

   6. Taylor expanded in a around 0 6.2%

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

      1. *-commutative 6.2%

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

   8. Simplified 6.2%

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

   9. Taylor expanded in k around inf 18.1%

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

       1. *-commutative 18.1%

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

       2. *-commutative 18.1%

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

       3. associate-*r* 18.1%

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

       4. rem-square-sqrt 13.4%

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

       5. fabs-sqr 13.4%

          \[\leadsto a \cdot \color{blue}{\left|\sqrt{k \cdot -10} \cdot \sqrt{k \cdot -10}\right|} \]

       6. rem-square-sqrt 27.1%

          \[\leadsto a \cdot \left|\color{blue}{k \cdot -10}\right| \]

       7. *-lft-identity 27.1%

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

       8. fabs-mul 27.1%

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

       9. *-lft-identity 27.1%

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

       10. rem-square-sqrt 13.7%

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

       11. fabs-sqr 13.7%

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

       12. rem-square-sqrt 23.0%

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

       13. metadata-eval 23.0%

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

       14. associate-*l* 23.0%

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

       15. *-commutative 23.0%

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

       16. associate-*l* 23.0%

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

   11. Simplified 23.0%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot 10\right)\\ \end{array} \]

Alternative 16: 25.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 5.6e+14) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+14) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.6d+14) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.6e+14) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.6e+14:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.6e+14)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.6e+14)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.6e+14], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.6e14

   1. Initial program 96.6%

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

      1. associate-*r/ 96.6%

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

      2. associate-+l+ 96.6%

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

      3. +-commutative 96.6%

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

      4. distribute-rgt-out 96.6%

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

      5. fma-def 96.6%

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

      6. +-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in m around 0 63.1%

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

   5. Taylor expanded in k around 0 25.2%

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

   if 5.6e14 < m

   1. Initial program 84.1%

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

      1. associate-*r/ 84.1%

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

      2. associate-+l+ 84.1%

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

      3. +-commutative 84.1%

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

      4. distribute-rgt-out 84.1%

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

      5. fma-def 84.1%

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

      6. +-commutative 84.1%

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

   3. Simplified 84.1%

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

   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 6.3%

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

   6. Taylor expanded in k around inf 18.6%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.6 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 17: 19.9% 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 92.6%

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

   1. associate-*r/ 92.6%

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

   2. associate-+l+ 92.6%

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

   3. +-commutative 92.6%

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

   4. distribute-rgt-out 92.6%

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

   5. fma-def 92.6%

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

   6. +-commutative 92.6%

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

3. Simplified 92.6%

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

4. Taylor expanded in m around 0 43.8%

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

5. Taylor expanded in k around 0 18.4%

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

6. Final simplification 18.4%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023192 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))