Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 98.0%

Time: 12.4s

Alternatives: 18

Speedup: 1.1×

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

• could not determine a ground truth

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

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: 89.9% 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: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.5 \cdot 10^{+162}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -2.5e+162)
   (* a (pow k m))
   (if (<= k 5e+140)
     (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
     (/ (* (pow k m) (/ a k)) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.5e+162) {
		tmp = a * pow(k, m);
	} else if (k <= 5e+140) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = (pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-2.5d+162)) then
        tmp = a * (k ** m)
    else if (k <= 5d+140) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = ((k ** m) * (a / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -2.5e+162) {
		tmp = a * Math.pow(k, m);
	} else if (k <= 5e+140) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = (Math.pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -2.5e+162:
		tmp = a * math.pow(k, m)
	elif k <= 5e+140:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = (math.pow(k, m) * (a / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -2.5e+162)
		tmp = Float64(a * (k ^ m));
	elseif (k <= 5e+140)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(Float64((k ^ m) * Float64(a / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -2.5e+162)
		tmp = a * (k ^ m);
	elseif (k <= 5e+140)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = ((k ^ m) * (a / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -2.5e+162], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5e+140], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.4999999999999998e162

   1. Initial program 56.3%

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

      1. associate-*r/ 56.3%

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

      2. associate-+l+ 56.3%

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

      3. +-commutative 56.3%

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

      4. distribute-rgt-out 62.5%

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

      5. fma-def 62.5%

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

      6. +-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in k around 0 0.0%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -2.4999999999999998e162 < k < 5.00000000000000008e140

   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-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   if 5.00000000000000008e140 < k

   1. Initial program 61.9%

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

      1. associate-/l* 61.9%

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

      2. associate-+l+ 61.9%

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

      3. *-commutative 61.9%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   6. Simplified 61.9%

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

      1. add-cube-cbrt 61.8%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1 + k \cdot k}{{k}^{m}}} \]

      2. add-sqr-sqrt 61.8%

         \[\leadsto \frac{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}{\color{blue}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}} \cdot \sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      3. times-frac 61.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      4. pow2 61.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      5. sqrt-div 61.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      6. hypot-1-def 61.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      7. sqrt-div 61.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \]

      8. hypot-1-def 99.4%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]

   9. Taylor expanded in k around inf 61.9%

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

       1. unpow2 61.9%

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

       2. times-frac 96.4%

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

       3. mul-1-neg 96.4%

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

       4. exp-neg 96.4%

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

       5. log-rec 96.4%

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

       6. distribute-lft-neg-in 96.4%

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

       7. distribute-rgt-neg-out 96.4%

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

       8. exp-to-pow 96.4%

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

   11. Simplified 96.4%

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

       1. associate-*r/ 96.5%

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

       2. pow-flip 96.5%

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

   13. Applied egg-rr 96.5%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.5 \cdot 10^{+162}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternative 2: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}\\ \frac{{\left(\sqrt[3]{a}\right)}^{2}}{t_0} \cdot \frac{\sqrt[3]{a}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (hypot 1.0 k) (sqrt (pow k m)))))
   (* (/ (pow (cbrt a) 2.0) t_0) (/ (cbrt a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(1.0, k) / sqrt(pow(k, m));
	return (pow(cbrt(a), 2.0) / t_0) * (cbrt(a) / t_0);
}

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.hypot(1.0, k) / Math.sqrt(Math.pow(k, m));
	return (Math.pow(Math.cbrt(a), 2.0) / t_0) * (Math.cbrt(a) / t_0);
}

Julia

function code(a, k, m)
	t_0 = Float64(hypot(1.0, k) / sqrt((k ^ m)))
	return Float64(Float64((cbrt(a) ^ 2.0) / t_0) * Float64(cbrt(a) / t_0))
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision] / N[Sqrt[N[Power[k, m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Power[N[Power[a, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[Power[a, 1/3], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 88.5%

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

   1. associate-/l* 88.5%

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

   2. associate-+l+ 88.5%

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

   3. *-commutative 88.5%

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

3. Simplified 88.5%

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

4. Taylor expanded in k around inf 87.9%

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

   1. unpow2 87.9%

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

6. Simplified 87.9%

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

   1. add-cube-cbrt 87.5%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1 + k \cdot k}{{k}^{m}}} \]

   2. add-sqr-sqrt 87.5%

      \[\leadsto \frac{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}{\color{blue}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}} \cdot \sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

   3. times-frac 87.5%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

   4. pow2 87.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

   5. sqrt-div 87.5%

      \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

   6. hypot-1-def 87.5%

      \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

   7. sqrt-div 87.5%

      \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \]

   8. hypot-1-def 98.5%

      \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \]

8. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]

9. Final simplification 98.5%

   \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \]

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{k}^{m}}\\ \mathbf{if}\;m \leq 6.8:\\ \;\;\;\;\frac{t_0 \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}}{\frac{\mathsf{hypot}\left(1, k\right)}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (sqrt (pow k m))))
   (if (<= m 6.8)
     (/ (* t_0 (/ a (hypot 1.0 k))) (/ (hypot 1.0 k) t_0))
     (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double t_0 = sqrt(pow(k, m));
	double tmp;
	if (m <= 6.8) {
		tmp = (t_0 * (a / hypot(1.0, k))) / (hypot(1.0, k) / t_0);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.sqrt(Math.pow(k, m));
	double tmp;
	if (m <= 6.8) {
		tmp = (t_0 * (a / Math.hypot(1.0, k))) / (Math.hypot(1.0, k) / t_0);
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.sqrt(math.pow(k, m))
	tmp = 0
	if m <= 6.8:
		tmp = (t_0 * (a / math.hypot(1.0, k))) / (math.hypot(1.0, k) / t_0)
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	t_0 = sqrt((k ^ m))
	tmp = 0.0
	if (m <= 6.8)
		tmp = Float64(Float64(t_0 * Float64(a / hypot(1.0, k))) / Float64(hypot(1.0, k) / t_0));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = sqrt((k ^ m));
	tmp = 0.0;
	if (m <= 6.8)
		tmp = (t_0 * (a / hypot(1.0, k))) / (hypot(1.0, k) / t_0);
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[N[Power[k, m], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, 6.8], N[(N[(t$95$0 * N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < 6.79999999999999982

   1. Initial program 90.5%

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

      1. associate-/l* 90.5%

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

      2. associate-+l+ 90.5%

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

      3. *-commutative 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in k around inf 89.8%

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

      1. unpow2 89.8%

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

   6. Simplified 89.8%

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

      1. *-un-lft-identity 89.8%

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

      2. add-sqr-sqrt 89.8%

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

      3. times-frac 89.7%

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

      4. sqrt-div 89.7%

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

      5. hypot-1-def 89.7%

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

      6. sqrt-div 89.7%

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

      7. hypot-1-def 98.6%

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

   8. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{a}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 98.6%

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

      2. *-lft-identity 98.6%

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

      3. associate-/r/ 98.6%

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

   10. Simplified 98.6%

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

   if 6.79999999999999982 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in k around 0 54.9%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 6.8:\\ \;\;\;\;\frac{\sqrt{{k}^{m}} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 6.6e-8) (* a (pow k m)) (* (/ a k) (/ (pow k m) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = a * pow(k, m);
	} else {
		tmp = (a / k) * (pow(k, m) / 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 (k <= 6.6d-8) then
        tmp = a * (k ** m)
    else
        tmp = (a / k) * ((k ** m) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (a / k) * (Math.pow(k, m) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 6.6e-8:
		tmp = a * math.pow(k, m)
	else:
		tmp = (a / k) * (math.pow(k, m) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 6.6e-8)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(a / k) * Float64((k ^ m) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 6.6e-8)
		tmp = a * (k ^ m);
	else
		tmp = (a / k) * ((k ^ m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 6.6e-8], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.59999999999999954e-8

   1. Initial program 95.5%

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

      1. associate-*r/ 95.5%

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

      2. associate-+l+ 95.6%

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

      3. +-commutative 95.6%

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

      4. distribute-rgt-out 96.2%

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

      5. fma-def 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]
   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 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-/l* 77.0%

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

      2. associate-+l+ 77.0%

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

      3. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in k around inf 75.5%

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

      1. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-cube-cbrt 75.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1 + k \cdot k}{{k}^{m}}} \]

      2. add-sqr-sqrt 75.2%

         \[\leadsto \frac{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}{\color{blue}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}} \cdot \sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      3. times-frac 75.2%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      4. pow2 75.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      5. sqrt-div 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      6. hypot-1-def 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      7. sqrt-div 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \]

      8. hypot-1-def 97.8%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \]

   8. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]

   9. Taylor expanded in k around inf 75.5%

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

       1. unpow2 75.5%

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

       2. times-frac 94.2%

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

       3. mul-1-neg 94.2%

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

       4. exp-neg 94.2%

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

       5. log-rec 94.2%

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

       6. distribute-lft-neg-in 94.2%

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

       7. distribute-rgt-neg-out 94.2%

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

       8. exp-to-pow 94.2%

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

   11. Simplified 94.2%

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

   12. Taylor expanded in k around 0 94.2%

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

       1. mul-1-neg 94.2%

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

       2. *-commutative 94.2%

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

       3. distribute-lft-neg-out 94.2%

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

       4. rec-exp 94.2%

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

       5. *-commutative 94.2%

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

       6. distribute-rgt-neg-out 94.2%

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

       7. exp-to-pow 94.2%

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

       8. remove-double-neg 94.2%

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

   14. Simplified 94.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 6.6e-8) (* a (pow k m)) (/ (* (pow k m) (/ a k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 6.6d-8) then
        tmp = a * (k ** m)
    else
        tmp = ((k ** m) * (a / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.6e-8) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 6.6e-8:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) * (a / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 6.6e-8)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) * Float64(a / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 6.6e-8)
		tmp = a * (k ^ m);
	else
		tmp = ((k ^ m) * (a / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 6.6e-8], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.59999999999999954e-8

   1. Initial program 95.5%

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

      1. associate-*r/ 95.5%

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

      2. associate-+l+ 95.6%

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

      3. +-commutative 95.6%

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

      4. distribute-rgt-out 96.2%

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

      5. fma-def 96.2%

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

      6. +-commutative 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in k around 0 53.8%

      \[\leadsto \color{blue}{e^{\log k \cdot m} \cdot a} \]
   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 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-/l* 77.0%

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

      2. associate-+l+ 77.0%

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

      3. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in k around inf 75.5%

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

      1. unpow2 75.5%

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

   6. Simplified 75.5%

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

      1. add-cube-cbrt 75.2%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1 + k \cdot k}{{k}^{m}}} \]

      2. add-sqr-sqrt 75.2%

         \[\leadsto \frac{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}{\color{blue}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}} \cdot \sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      3. times-frac 75.2%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      4. pow2 75.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      5. sqrt-div 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      6. hypot-1-def 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      7. sqrt-div 75.2%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \]

      8. hypot-1-def 97.8%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \]

   8. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]

   9. Taylor expanded in k around inf 75.5%

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

       1. unpow2 75.5%

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

       2. times-frac 94.2%

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

       3. mul-1-neg 94.2%

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

       4. exp-neg 94.2%

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

       5. log-rec 94.2%

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

       6. distribute-lft-neg-in 94.2%

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

       7. distribute-rgt-neg-out 94.2%

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

       8. exp-to-pow 94.2%

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

   11. Simplified 94.2%

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

       1. associate-*r/ 94.2%

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

       2. pow-flip 94.2%

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

   13. Applied egg-rr 94.2%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternative 6: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-6} \lor \neg \left(m \leq 3.6 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.35e-6) (not (<= m 3.6e-7)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.35e-6) || !(m <= 3.6e-7)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-1.35d-6)) .or. (.not. (m <= 3.6d-7))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.35e-6) || !(m <= 3.6e-7)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -1.35e-6) or not (m <= 3.6e-7):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.35e-6) || !(m <= 3.6e-7))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -1.35e-6) || ~((m <= 3.6e-7)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.35e-6], N[Not[LessEqual[m, 3.6e-7]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.34999999999999999e-6 or 3.59999999999999994e-7 < m

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. associate-+l+ 91.4%

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

      3. +-commutative 91.4%

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

      4. distribute-rgt-out 92.1%

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

      5. fma-def 92.1%

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

      6. +-commutative 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in k around 0 52.3%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   if -1.34999999999999999e-6 < m < 3.59999999999999994e-7

   1. Initial program 84.2%

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

      1. associate-*r/ 84.2%

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

      2. associate-+l+ 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-rgt-out 84.2%

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

      5. fma-def 84.2%

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

      6. +-commutative 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in m around 0 83.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.35 \cdot 10^{-6} \lor \neg \left(m \leq 3.6 \cdot 10^{-7}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7: 58.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.32)
   (/ a (* k k))
   (if (<= m 0.92) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.32) {
		tmp = a / (k * k);
	} else if (m <= 0.92) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} 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 <= (-0.32d0)) then
        tmp = a / (k * k)
    else if (m <= 0.92d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    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 <= -0.32) {
		tmp = a / (k * k);
	} else if (m <= 0.92) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.32:
		tmp = a / (k * k)
	elif m <= 0.92:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.32)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 <= -0.32)
		tmp = a / (k * k);
	elseif (m <= 0.92)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.32], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.320000000000000007

   1. Initial program 98.8%

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

      1. associate-*r/ 98.8%

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

      2. associate-+l+ 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-rgt-out 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 39.0%

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

   5. Taylor expanded in k around inf 64.3%

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

      1. unpow2 64.3%

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

   7. Simplified 64.3%

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

   if -0.320000000000000007 < m < 0.92000000000000004

   1. Initial program 84.2%

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

      1. associate-*r/ 84.2%

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

      2. associate-+l+ 84.2%

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

      3. +-commutative 84.2%

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

      4. distribute-rgt-out 84.2%

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

      5. fma-def 84.2%

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

      6. +-commutative 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in m around 0 83.8%

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

   if 0.92000000000000004 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 9.8%

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

      1. *-commutative 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in k around inf 26.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.32:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 46.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-310} \lor \neg \left(k \leq 6.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 2e-310) (not (<= k 6.6e-8)))
   (/ a (* k k))
   (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2e-310) || !(k <= 6.6e-8)) {
		tmp = a / (k * k);
	} 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) :: tmp
    if ((k <= 2d-310) .or. (.not. (k <= 6.6d-8))) then
        tmp = a / (k * k)
    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 tmp;
	if ((k <= 2e-310) || !(k <= 6.6e-8)) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 2e-310) or not (k <= 6.6e-8):
		tmp = a / (k * k)
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 2e-310) || !(k <= 6.6e-8))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 2e-310) || ~((k <= 6.6e-8)))
		tmp = a / (k * k);
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 2e-310], N[Not[LessEqual[k, 6.6e-8]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.999999999999994e-310 or 6.59999999999999954e-8 < k

   1. Initial program 82.6%

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

      1. associate-*r/ 82.6%

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

      2. associate-+l+ 82.6%

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

      3. +-commutative 82.6%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 45.2%

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

   5. Taylor expanded in k around inf 49.9%

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

      1. unpow2 49.9%

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

   7. Simplified 49.9%

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

   if 1.999999999999994e-310 < k < 6.59999999999999954e-8

   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 51.9%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. *-commutative 51.7%

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

   7. Simplified 51.7%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-310} \lor \neg \left(k \leq 6.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 9: 46.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{1}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 9e-309)
   (/ a (* k k))
   (if (<= k 6.6e-8) (* a (+ 1.0 (* k -10.0))) (* a (/ (/ 1.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 9e-309) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * ((1.0 / k) / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 9d-309) then
        tmp = a / (k * k)
    else if (k <= 6.6d-8) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = a * ((1.0d0 / k) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 9e-309) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = a * ((1.0 / k) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 9e-309:
		tmp = a / (k * k)
	elif k <= 6.6e-8:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = a * ((1.0 / k) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 9e-309)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 6.6e-8)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(a * Float64(Float64(1.0 / k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 9e-309)
		tmp = a / (k * k);
	elseif (k <= 6.6e-8)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = a * ((1.0 / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 9e-309], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e-8], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(1.0 / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 9.0000000000000021e-309

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

      3. +-commutative 90.3%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 22.4%

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

   5. Taylor expanded in k around inf 35.3%

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

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   if 9.0000000000000021e-309 < k < 6.59999999999999954e-8

   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 51.9%

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

   5. Taylor expanded in k around 0 51.7%

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

      1. *-commutative 51.7%

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

   7. Simplified 51.7%

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

   if 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-+l+ 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-out 77.0%

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

      5. fma-def 77.0%

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

      6. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in m around 0 61.8%

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

   5. Taylor expanded in k around inf 60.5%

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

      1. unpow2 60.5%

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

      2. associate-/r* 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{-309}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{1}{k}}{k}\\ \end{array} \]

Alternative 10: 46.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{1}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.6e-307)
   (/ a (* k k))
   (if (<= k 6.6e-8) (+ a (* -10.0 (* a k))) (* a (/ (/ 1.0 k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-307) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a * ((1.0 / k) / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.6d-307) then
        tmp = a / (k * k)
    else if (k <= 6.6d-8) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = a * ((1.0d0 / k) / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.6e-307) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a * ((1.0 / k) / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.6e-307:
		tmp = a / (k * k)
	elif k <= 6.6e-8:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = a * ((1.0 / k) / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.6e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 6.6e-8)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(a * Float64(Float64(1.0 / k) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.6e-307)
		tmp = a / (k * k);
	elseif (k <= 6.6e-8)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = a * ((1.0 / k) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.6e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e-8], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(1.0 / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.60000000000000005e-307

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

      3. +-commutative 90.3%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 22.4%

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

   5. Taylor expanded in k around inf 35.3%

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

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   if 1.60000000000000005e-307 < k < 6.59999999999999954e-8

   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 51.9%

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

   5. Taylor expanded in k around 0 51.8%

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

   if 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-+l+ 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-out 77.0%

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

      5. fma-def 77.0%

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

      6. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in m around 0 61.8%

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

   5. Taylor expanded in k around inf 60.5%

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

      1. unpow2 60.5%

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

      2. associate-/r* 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{1}{k}}{k}\\ \end{array} \]

Alternative 11: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -5e-310)
   (/ a (* k k))
   (if (<= k 6.6e-8) (+ a (* -10.0 (* a k))) (/ 1.0 (* k (/ k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5d-310)) then
        tmp = a / (k * k)
    else if (k <= 6.6d-8) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -5e-310:
		tmp = a / (k * k)
	elif k <= 6.6e-8:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -5e-310)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 6.6e-8)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = a / (k * k);
	elseif (k <= 6.6e-8)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -5e-310], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e-8], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

      3. +-commutative 90.3%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 22.4%

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

   5. Taylor expanded in k around inf 35.3%

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

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   if -4.999999999999985e-310 < k < 6.59999999999999954e-8

   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 51.9%

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

   5. Taylor expanded in k around 0 51.8%

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

   if 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-+l+ 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-out 77.0%

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

      5. fma-def 77.0%

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

      6. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in m around 0 61.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

      1. clear-num 60.6%

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

      2. inv-pow 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. unpow-1 60.6%

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

       2. *-lft-identity 60.6%

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

       3. times-frac 75.8%

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

       4. /-rgt-identity 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 12: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2e-310)
   (/ a (* k k))
   (if (<= k 6.6e-8) (/ a (+ 1.0 (* k 10.0))) (/ 1.0 (* k (/ k a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2e-310) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 2d-310) then
        tmp = a / (k * k)
    else if (k <= 6.6d-8) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2e-310) {
		tmp = a / (k * k);
	} else if (k <= 6.6e-8) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2e-310:
		tmp = a / (k * k)
	elif k <= 6.6e-8:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2e-310)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 6.6e-8)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2e-310)
		tmp = a / (k * k);
	elseif (k <= 6.6e-8)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2e-310], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6.6e-8], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.999999999999994e-310

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

      3. +-commutative 90.3%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in m around 0 22.4%

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

   5. Taylor expanded in k around inf 35.3%

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

      1. unpow2 35.3%

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

   7. Simplified 35.3%

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

   if 1.999999999999994e-310 < k < 6.59999999999999954e-8

   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-/l* 100.0%

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

      2. associate-+l+ 100.0%

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

      3. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 99.9%

      \[\leadsto \frac{a}{\color{blue}{10 \cdot \frac{k}{e^{\log k \cdot m}} + \frac{1}{e^{\log k \cdot m}}}} \]
   5. Step-by-step derivation

      1. fma-def 99.9%

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

      2. exp-to-pow 99.9%

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

      3. exp-to-pow 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in m around 0 51.9%

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

      1. *-commutative 51.9%

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

   9. Simplified 51.9%

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

   if 6.59999999999999954e-8 < k

   1. Initial program 77.0%

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

      1. associate-*r/ 77.0%

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

      2. associate-+l+ 77.0%

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

      3. +-commutative 77.0%

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

      4. distribute-rgt-out 77.0%

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

      5. fma-def 77.0%

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

      6. +-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in m around 0 61.9%

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

   5. Taylor expanded in k around inf 60.6%

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

      1. unpow2 60.6%

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

   7. Simplified 60.6%

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

      1. clear-num 60.6%

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

      2. inv-pow 60.6%

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

   9. Applied egg-rr 60.6%

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

       1. unpow-1 60.6%

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

       2. *-lft-identity 60.6%

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

       3. times-frac 75.8%

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

       4. /-rgt-identity 75.8%

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

   11. Simplified 75.8%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-310}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 13: 57.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.05:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.05)
   (/ a (* k k))
   (if (<= m 0.95) (/ a (+ 1.0 (* k k))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.05) {
		tmp = a / (k * k);
	} else if (m <= 0.95) {
		tmp = a / (1.0 + (k * k));
	} 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 <= (-0.05d0)) then
        tmp = a / (k * k)
    else if (m <= 0.95d0) then
        tmp = a / (1.0d0 + (k * k))
    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 <= -0.05) {
		tmp = a / (k * k);
	} else if (m <= 0.95) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.05:
		tmp = a / (k * k)
	elif m <= 0.95:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.05)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.95)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	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 <= -0.05)
		tmp = a / (k * k);
	elseif (m <= 0.95)
		tmp = a / (1.0 + (k * k));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.05], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.050000000000000003

   1. Initial program 98.8%

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

      1. associate-*r/ 98.8%

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

      2. associate-+l+ 98.8%

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

      3. +-commutative 98.8%

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

      4. distribute-rgt-out 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 39.0%

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

   5. Taylor expanded in k around inf 64.3%

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

      1. unpow2 64.3%

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

   7. Simplified 64.3%

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

   if -0.050000000000000003 < m < 0.94999999999999996

   1. Initial program 84.2%

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

      1. associate-/l* 84.2%

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

      2. associate-+l+ 84.2%

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

      3. *-commutative 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in k around inf 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

      1. add-cube-cbrt 80.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}}{\frac{1 + k \cdot k}{{k}^{m}}} \]

      2. add-sqr-sqrt 80.9%

         \[\leadsto \frac{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt[3]{a}}{\color{blue}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}} \cdot \sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      3. times-frac 80.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{a} \cdot \sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}}} \]

      4. pow2 80.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a}\right)}^{2}}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      5. sqrt-div 80.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      6. hypot-1-def 80.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\sqrt{\frac{1 + k \cdot k}{{k}^{m}}}} \]

      7. sqrt-div 80.9%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\color{blue}{\frac{\sqrt{1 + k \cdot k}}{\sqrt{{k}^{m}}}}} \]

      8. hypot-1-def 96.3%

         \[\leadsto \frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\color{blue}{\mathsf{hypot}\left(1, k\right)}}{\sqrt{{k}^{m}}}} \]

   8. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{a}\right)}^{2}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 93.8%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{a}\right)}^{2}\right)\right)}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \]

   10. Applied egg-rr 93.8%

       \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{a}\right)}^{2}\right)\right)}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \cdot \frac{\sqrt[3]{a}}{\frac{\mathsf{hypot}\left(1, k\right)}{\sqrt{{k}^{m}}}} \]

   11. Taylor expanded in m around 0 81.7%

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

       1. unpow2 81.7%

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

   13. Simplified 81.7%

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

   if 0.94999999999999996 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 9.8%

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

      1. *-commutative 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in k around inf 26.7%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.05:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 14: 46.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-308} \lor \neg \left(k \leq 6.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 3.4e-308) (not (<= k 6.6e-8))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 3.4e-308) || !(k <= 6.6e-8)) {
		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 <= 3.4d-308) .or. (.not. (k <= 6.6d-8))) 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 <= 3.4e-308) || !(k <= 6.6e-8)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 3.4e-308) or not (k <= 6.6e-8):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 3.4e-308) || !(k <= 6.6e-8))
		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 <= 3.4e-308) || ~((k <= 6.6e-8)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 3.4e-308], N[Not[LessEqual[k, 6.6e-8]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 3.39999999999999999e-308 or 6.59999999999999954e-8 < k

   1. Initial program 82.6%

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

      1. associate-*r/ 82.6%

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

      2. associate-+l+ 82.6%

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

      3. +-commutative 82.6%

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

      4. distribute-rgt-out 83.2%

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

      5. fma-def 83.2%

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

      6. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in m around 0 45.2%

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

   5. Taylor expanded in k around inf 49.9%

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

      1. unpow2 49.9%

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

   7. Simplified 49.9%

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

   if 3.39999999999999999e-308 < k < 6.59999999999999954e-8

   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 51.9%

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

   5. Taylor expanded in k around 0 50.8%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-308} \lor \neg \left(k \leq 6.6 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 15: 31.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.18:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.4e-16) (* (/ a k) 0.1) (if (<= m 0.18) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.4e-16) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.18) {
		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 <= (-7.4d-16)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 0.18d0) 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 <= -7.4e-16) {
		tmp = (a / k) * 0.1;
	} else if (m <= 0.18) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -7.4e-16:
		tmp = (a / k) * 0.1
	elif m <= 0.18:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.4e-16)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 0.18)
		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 <= -7.4e-16)
		tmp = (a / k) * 0.1;
	elseif (m <= 0.18)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.4e-16], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 0.18], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.3999999999999999e-16

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. associate-+l+ 98.8%

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

      3. *-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in k around 0 50.1%

      \[\leadsto \frac{a}{\color{blue}{10 \cdot \frac{k}{e^{\log k \cdot m}} + \frac{1}{e^{\log k \cdot m}}}} \]
   5. Step-by-step derivation

      1. fma-def 50.1%

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

      2. exp-to-pow 50.1%

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

      3. exp-to-pow 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in m around 0 19.8%

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

      1. *-commutative 19.8%

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

   9. Simplified 19.8%

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

   10. Taylor expanded in k around inf 25.0%

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

   if -7.3999999999999999e-16 < m < 0.17999999999999999

   1. Initial program 83.9%

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

      1. associate-*r/ 83.9%

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

      2. associate-+l+ 83.9%

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

      3. +-commutative 83.9%

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

      4. distribute-rgt-out 83.9%

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

      5. fma-def 83.9%

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

      6. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in m around 0 83.9%

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

   5. Taylor expanded in k around 0 43.3%

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

   if 0.17999999999999999 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 9.8%

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

      1. *-commutative 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in k around inf 26.7%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 0.18:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 16: 31.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e-21) (/ a (* k 10.0)) (if (<= m 0.27) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e-21) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.27) {
		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 <= (-1.7d-21)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 0.27d0) 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 <= -1.7e-21) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.27) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.7e-21:
		tmp = a / (k * 10.0)
	elif m <= 0.27:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e-21)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 0.27)
		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 <= -1.7e-21)
		tmp = a / (k * 10.0);
	elseif (m <= 0.27)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e-21], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.27], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.7e-21

   1. Initial program 98.8%

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

      1. associate-/l* 98.8%

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

      2. associate-+l+ 98.8%

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

      3. *-commutative 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in k around 0 50.1%

      \[\leadsto \frac{a}{\color{blue}{10 \cdot \frac{k}{e^{\log k \cdot m}} + \frac{1}{e^{\log k \cdot m}}}} \]
   5. Step-by-step derivation

      1. fma-def 50.1%

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

      2. exp-to-pow 50.1%

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

      3. exp-to-pow 69.6%

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

   6. Simplified 69.6%

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

   7. Taylor expanded in m around 0 19.8%

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

      1. *-commutative 19.8%

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

   9. Simplified 19.8%

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

   10. Taylor expanded in k around inf 26.1%

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

       1. *-commutative 26.1%

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

   12. Simplified 26.1%

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

   if -1.7e-21 < m < 0.27000000000000002

   1. Initial program 83.9%

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

      1. associate-*r/ 83.9%

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

      2. associate-+l+ 83.9%

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

      3. +-commutative 83.9%

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

      4. distribute-rgt-out 83.9%

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

      5. fma-def 83.9%

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

      6. +-commutative 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in m around 0 83.9%

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

   5. Taylor expanded in k around 0 43.3%

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

   if 0.27000000000000002 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 9.8%

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

      1. *-commutative 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in k around inf 26.7%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-21}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.27:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 17: 25.8% accurate, 16.1× speedup

Math

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

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.9) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.9) {
		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 <= 0.9d0) 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 <= 0.9) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.9:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.9)
		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 <= 0.9)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.900000000000000022

   1. Initial program 90.5%

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

      1. associate-*r/ 90.5%

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

      2. associate-+l+ 90.5%

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

      3. +-commutative 90.5%

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

      4. distribute-rgt-out 91.0%

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

      5. fma-def 91.0%

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

      6. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 64.5%

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

   5. Taylor expanded in k around 0 26.0%

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

   if 0.900000000000000022 < m

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-+l+ 83.1%

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

      3. +-commutative 83.1%

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

      4. distribute-rgt-out 83.1%

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

      5. fma-def 83.1%

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

      6. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 9.8%

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

      1. *-commutative 9.8%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in k around inf 26.7%

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

4. Final simplification 26.2%

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

Alternative 18: 20.1% 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 88.5%

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

   1. associate-*r/ 88.4%

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

   2. associate-+l+ 88.4%

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

   3. +-commutative 88.4%

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

   4. distribute-rgt-out 88.8%

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

   5. fma-def 88.8%

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

   6. +-commutative 88.8%

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

3. Simplified 88.8%

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

4. Taylor expanded in m around 0 47.4%

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

5. Taylor expanded in k around 0 19.8%

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

6. Final simplification 19.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023178 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))