Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.7%

Time: 8.4s

Alternatives: 13

Speedup: 1.1×

• 22.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq -2.2 \cdot 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 10^{+76}:\\ \;\;\;\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k -2.2e+168)
     t_0
     (if (<= k 1e+76)
       (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))
       (/ (* a (/ (pow k m) k)) k)))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= -2.2e+168) {
		tmp = t_0;
	} else if (k <= 1e+76) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a * (pow(k, m) / k)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= (-2.2d+168)) then
        tmp = t_0
    else if (k <= 1d+76) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = (a * ((k ** m) / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (k <= -2.2e+168) {
		tmp = t_0;
	} else if (k <= 1e+76) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = (a * (Math.pow(k, m) / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= -2.2e+168:
		tmp = t_0
	elif k <= 1e+76:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = (a * (math.pow(k, m) / k)) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= -2.2e+168)
		tmp = t_0;
	elseif (k <= 1e+76)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= -2.2e+168)
		tmp = t_0;
	elseif (k <= 1e+76)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = (a * ((k ^ m) / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.2e+168], t$95$0, If[LessEqual[k, 1e+76], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.2000000000000002e168

   1. Initial program 61.1%

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

   2. Taylor expanded in k around 0 0.0%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -2.2000000000000002e168 < k < 1e76

   1. Initial program 99.9%

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

   if 1e76 < k

   1. Initial program 79.9%

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

   2. Taylor expanded in k around inf 79.9%

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

      1. unpow2 79.9%

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

      2. times-frac 90.3%

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

      3. mul-1-neg 90.3%

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

      4. distribute-rgt-neg-in 90.3%

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

      5. exp-to-pow 90.3%

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

   4. Simplified 90.3%

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

      1. associate-*l/ 100.0%

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

      2. inv-pow 100.0%

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

      3. pow-pow 100.0%

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

      4. add-sqr-sqrt 56.2%

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

      5. sqrt-unprod 73.8%

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

      6. sqr-neg 73.8%

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

      7. sqrt-unprod 17.6%

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

      8. add-sqr-sqrt 48.1%

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

      9. neg-mul-1 48.1%

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

      10. add-sqr-sqrt 30.5%

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

      11. sqrt-unprod 74.3%

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

      12. sqr-neg 74.3%

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

      13. sqrt-unprod 43.8%

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

      14. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{+168}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 10^{+76}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(k, -10, 1\right) \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.1)
   (* (fma k -10.0 1.0) (* a (pow k m)))
   (/ (* a (/ (pow k m) k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = fma(k, -10.0, 1.0) * (a * pow(k, m));
	} else {
		tmp = (a * (pow(k, m) / k)) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.1)
		tmp = Float64(fma(k, -10.0, 1.0) * Float64(a * (k ^ m)));
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / k)) / k);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 95.6%

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

   2. Taylor expanded in k around 0 42.8%

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

      1. exp-to-pow 42.8%

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

      2. exp-to-pow 42.8%

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

      3. exp-to-pow 42.8%

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

      4. *-commutative 42.8%

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

      5. associate-*r* 42.8%

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

      6. exp-to-pow 89.5%

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

      7. *-commutative 89.5%

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

      8. distribute-lft1-in 99.3%

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

      9. *-commutative 99.3%

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

      10. fma-def 99.3%

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

   4. Simplified 99.3%

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

   if 0.10000000000000001 < k

   1. Initial program 84.1%

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

   2. Taylor expanded in k around inf 82.2%

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

      1. unpow2 82.2%

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

      2. times-frac 90.3%

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

      3. mul-1-neg 90.3%

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

      4. distribute-rgt-neg-in 90.3%

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

      5. exp-to-pow 90.3%

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

   4. Simplified 90.3%

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

      1. associate-*l/ 97.9%

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

      2. inv-pow 97.9%

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

      3. pow-pow 97.9%

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

      4. add-sqr-sqrt 53.5%

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

      5. sqrt-unprod 72.1%

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

      6. sqr-neg 72.1%

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

      7. sqrt-unprod 18.5%

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

      8. add-sqr-sqrt 47.7%

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

      9. neg-mul-1 47.7%

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

      10. add-sqr-sqrt 29.2%

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

      11. sqrt-unprod 73.6%

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

      12. sqr-neg 73.6%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(k, -10, 1\right) \cdot \left(a \cdot {k}^{m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{k}}{k}\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 95.6%

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

   2. Taylor expanded in k around 0 51.8%

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

      1. exp-to-pow 98.4%

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

      2. *-commutative 98.4%

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

   4. Simplified 98.4%

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

   if 1 < k

   1. Initial program 84.1%

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

   2. Taylor expanded in k around inf 82.2%

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

      1. unpow2 82.2%

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

      2. times-frac 90.3%

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

      3. mul-1-neg 90.3%

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

      4. distribute-rgt-neg-in 90.3%

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

      5. exp-to-pow 90.3%

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

   4. Simplified 90.3%

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

      1. associate-*l/ 97.9%

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

      2. inv-pow 97.9%

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

      3. pow-pow 97.9%

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

      4. add-sqr-sqrt 53.5%

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

      5. sqrt-unprod 72.1%

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

      6. sqr-neg 72.1%

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

      7. sqrt-unprod 18.5%

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

      8. add-sqr-sqrt 47.7%

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

      9. neg-mul-1 47.7%

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

      10. add-sqr-sqrt 29.2%

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

      11. sqrt-unprod 73.6%

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

      12. sqr-neg 73.6%

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

      13. sqrt-unprod 44.4%

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

      14. add-sqr-sqrt 97.9%

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

   6. Applied egg-rr 97.9%

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

4. Final simplification 98.2%

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

Alternative 4: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.06 \lor \neg \left(m \leq 0.0031\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.06) (not (<= m 0.0031)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.06) || !(m <= 0.0031)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-0.06d0)) .or. (.not. (m <= 0.0031d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.06) || !(m <= 0.0031)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.06) or not (m <= 0.0031):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.06) || !(m <= 0.0031))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.06) || ~((m <= 0.0031)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.06], N[Not[LessEqual[m, 0.0031]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.059999999999999998 or 0.00309999999999999989 < m

   1. Initial program 90.0%

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

   2. Taylor expanded in k around 0 52.5%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -0.059999999999999998 < m < 0.00309999999999999989

   1. Initial program 93.9%

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

      1. +-commutative 93.9%

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

      2. add-cbrt-cube 91.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{\left(k \cdot k\right) \cdot k}} \cdot k + \left(1 + 10 \cdot k\right)} \]

      3. cbrt-prod 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{k \cdot k} \cdot \sqrt[3]{k}\right)} \cdot k + \left(1 + 10 \cdot k\right)} \]

      4. associate-*l* 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{k \cdot k} \cdot \left(\sqrt[3]{k} \cdot k\right)} + \left(1 + 10 \cdot k\right)} \]

      5. fma-def 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{k \cdot k}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)}} \]

      6. cbrt-prod 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{\sqrt[3]{k} \cdot \sqrt[3]{k}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      7. pow2 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      8. +-commutative 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{10 \cdot k + 1}\right)} \]

      9. *-commutative 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{k \cdot 10} + 1\right)} \]

      10. fma-def 93.7%

          \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]

   3. Applied egg-rr 93.7%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \mathsf{fma}\left(k, 10, 1\right)\right)}} \]
   4. Step-by-step derivation

      1. fma-udef 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right) + \mathsf{fma}\left(k, 10, 1\right)}} \]

      2. +-commutative 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right)}} \]

      3. *-commutative 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \color{blue}{\left(k \cdot \sqrt[3]{k}\right)}} \]

      4. associate-*r* 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot k\right) \cdot \sqrt[3]{k}}} \]

      5. *-commutative 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left(k \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)} \cdot \sqrt[3]{k}} \]

      6. associate-*l* 93.7%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{k \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{k}\right)}} \]

      7. pow-plus 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{\left(2 + 1\right)}}} \]

      8. metadata-eval 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{\color{blue}{3}}} \]

   5. Simplified 93.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{3}}} \]

   6. Taylor expanded in m around 0 93.6%

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

      1. unpow2 93.6%

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

      2. distribute-rgt-in 93.6%

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

   8. Simplified 93.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.06 \lor \neg \left(m \leq 0.0031\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 5: 55.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.315:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 160000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k} + \left(a \cdot -0.01 + 0.001 \cdot \left(k \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.315)
   (/ a (* k k))
   (if (<= m 160000000000.0)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ (* 0.1 (/ a k)) (+ (* a -0.01) (* 0.001 (* k a)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.315) {
		tmp = a / (k * k);
	} else if (m <= 160000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (0.1 * (a / k)) + ((a * -0.01) + (0.001 * (k * a)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.315d0)) then
        tmp = a / (k * k)
    else if (m <= 160000000000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (0.1d0 * (a / k)) + ((a * (-0.01d0)) + (0.001d0 * (k * a)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.315) {
		tmp = a / (k * k);
	} else if (m <= 160000000000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (0.1 * (a / k)) + ((a * -0.01) + (0.001 * (k * a)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.315:
		tmp = a / (k * k)
	elif m <= 160000000000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (0.1 * (a / k)) + ((a * -0.01) + (0.001 * (k * a)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.315)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 160000000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(0.1 * Float64(a / k)) + Float64(Float64(a * -0.01) + Float64(0.001 * Float64(k * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.315)
		tmp = a / (k * k);
	elseif (m <= 160000000000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = (0.1 * (a / k)) + ((a * -0.01) + (0.001 * (k * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.315], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 160000000000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision] + N[(N[(a * -0.01), $MachinePrecision] + N[(0.001 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.315000000000000002

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 43.0%

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

   3. Taylor expanded in k around inf 59.9%

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

      1. unpow2 59.9%

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

   5. Simplified 59.9%

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

   if -0.315000000000000002 < m < 1.6e11

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. add-cbrt-cube 92.1%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{\left(k \cdot k\right) \cdot k}} \cdot k + \left(1 + 10 \cdot k\right)} \]

      3. cbrt-prod 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{k \cdot k} \cdot \sqrt[3]{k}\right)} \cdot k + \left(1 + 10 \cdot k\right)} \]

      4. associate-*l* 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{k \cdot k} \cdot \left(\sqrt[3]{k} \cdot k\right)} + \left(1 + 10 \cdot k\right)} \]

      5. fma-def 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{k \cdot k}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)}} \]

      6. cbrt-prod 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{\sqrt[3]{k} \cdot \sqrt[3]{k}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      7. pow2 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      8. +-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{10 \cdot k + 1}\right)} \]

      9. *-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{k \cdot 10} + 1\right)} \]

      10. fma-def 93.9%

          \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]

   3. Applied egg-rr 93.9%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \mathsf{fma}\left(k, 10, 1\right)\right)}} \]
   4. Step-by-step derivation

      1. fma-udef 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right) + \mathsf{fma}\left(k, 10, 1\right)}} \]

      2. +-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right)}} \]

      3. *-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \color{blue}{\left(k \cdot \sqrt[3]{k}\right)}} \]

      4. associate-*r* 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot k\right) \cdot \sqrt[3]{k}}} \]

      5. *-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left(k \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)} \cdot \sqrt[3]{k}} \]

      6. associate-*l* 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{k \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{k}\right)}} \]

      7. pow-plus 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{\left(2 + 1\right)}}} \]

      8. metadata-eval 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{\color{blue}{3}}} \]

   5. Simplified 93.9%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{3}}} \]

   6. Taylor expanded in m around 0 91.8%

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

      1. unpow2 91.8%

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

      2. distribute-rgt-in 91.8%

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

   8. Simplified 91.8%

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

   if 1.6e11 < m

   1. Initial program 80.7%

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

   2. Taylor expanded in m around 0 3.1%

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

   3. Taylor expanded in k around inf 2.4%

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

      1. unpow2 2.4%

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

      2. distribute-rgt-in 2.4%

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

   5. Simplified 2.4%

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

   6. Taylor expanded in k around 0 9.5%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k} + \left(-0.01 \cdot a + 0.001 \cdot \left(k \cdot a\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.315:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 160000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k} + \left(a \cdot -0.01 + 0.001 \cdot \left(k \cdot a\right)\right)\\ \end{array} \]

Alternative 6: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -3.4e-274) (not (<= k 0.1)))
   (/ a (* k k))
   (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -3.4e-274) || !(k <= 0.1)) {
		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 <= (-3.4d-274)) .or. (.not. (k <= 0.1d0))) 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 <= -3.4e-274) || !(k <= 0.1)) {
		tmp = a / (k * k);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -3.4e-274) or not (k <= 0.1):
		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 <= -3.4e-274) || !(k <= 0.1))
		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 <= -3.4e-274) || ~((k <= 0.1)))
		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, -3.4e-274], N[Not[LessEqual[k, 0.1]], $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 < -3.39999999999999981e-274 or 0.10000000000000001 < k

   1. Initial program 87.1%

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

   2. Taylor expanded in m around 0 43.8%

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

   3. Taylor expanded in k around inf 46.0%

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

      1. unpow2 46.0%

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

   5. Simplified 46.0%

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

   if -3.39999999999999981e-274 < k < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in k around 0 79.3%

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

      1. exp-to-pow 79.3%

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

      2. exp-to-pow 79.3%

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

      3. exp-to-pow 79.3%

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

      4. *-commutative 79.3%

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

      5. associate-*r* 79.3%

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

      6. exp-to-pow 80.5%

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

      7. *-commutative 80.5%

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

      8. distribute-lft1-in 98.6%

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

      9. *-commutative 98.6%

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

      10. fma-def 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in m around 0 56.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]

Alternative 7: 48.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -3.4e-274)
   (/ a (* k k))
   (if (<= k 0.1) (* a (+ 1.0 (* k -10.0))) (* (/ a k) (/ 1.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.4e-274) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) * (1.0 / 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 <= (-3.4d-274)) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) then
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = (a / k) * (1.0d0 / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.4e-274) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a * (1.0 + (k * -10.0));
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -3.4e-274:
		tmp = a / (k * k)
	elif k <= 0.1:
		tmp = a * (1.0 + (k * -10.0))
	else:
		tmp = (a / k) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -3.4e-274)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -3.4e-274)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		tmp = a * (1.0 + (k * -10.0));
	else
		tmp = (a / k) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -3.4e-274], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -3.39999999999999981e-274

   1. Initial program 90.7%

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

   2. Taylor expanded in m around 0 22.0%

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

   3. Taylor expanded in k around inf 29.4%

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

      1. unpow2 29.4%

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

   5. Simplified 29.4%

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

   if -3.39999999999999981e-274 < k < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in k around 0 79.3%

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

      1. exp-to-pow 79.3%

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

      2. exp-to-pow 79.3%

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

      3. exp-to-pow 79.3%

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

      4. *-commutative 79.3%

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

      5. associate-*r* 79.3%

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

      6. exp-to-pow 80.5%

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

      7. *-commutative 80.5%

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

      8. distribute-lft1-in 98.6%

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

      9. *-commutative 98.6%

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

      10. fma-def 98.6%

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

   4. Simplified 98.6%

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

   5. Taylor expanded in m around 0 56.8%

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

   if 0.10000000000000001 < k

   1. Initial program 84.1%

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

   2. Taylor expanded in m around 0 61.4%

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

   3. Taylor expanded in k around inf 59.5%

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

      1. unpow2 59.5%

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

   5. Simplified 59.5%

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

      1. associate-/r* 61.5%

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

      2. div-inv 61.4%

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

   7. Applied egg-rr 61.4%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 8: 48.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -3.4e-274)
   (/ a (* k k))
   (if (<= k 10.0) (/ a (+ 1.0 (* k 10.0))) (* (/ a k) (/ 1.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.4e-274) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) * (1.0 / 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 <= (-3.4d-274)) then
        tmp = a / (k * k)
    else if (k <= 10.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) * (1.0d0 / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -3.4e-274) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -3.4e-274:
		tmp = a / (k * k)
	elif k <= 10.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -3.4e-274)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 10.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -3.4e-274)
		tmp = a / (k * k);
	elseif (k <= 10.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -3.4e-274], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -3.39999999999999981e-274

   1. Initial program 90.7%

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

   2. Taylor expanded in m around 0 22.0%

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

   3. Taylor expanded in k around inf 29.4%

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

      1. unpow2 29.4%

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

   5. Simplified 29.4%

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

   if -3.39999999999999981e-274 < k < 10

   1. Initial program 99.9%

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

   2. Taylor expanded in k around 0 99.1%

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

   3. Taylor expanded in m around 0 57.2%

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

   if 10 < k

   1. Initial program 84.1%

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

   2. Taylor expanded in m around 0 61.4%

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

   3. Taylor expanded in k around inf 59.5%

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

      1. unpow2 59.5%

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

   5. Simplified 59.5%

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

      1. associate-/r* 61.5%

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

      2. div-inv 61.4%

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

   7. Applied egg-rr 61.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 9: 53.2% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.74:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.74) (/ a (* k k)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.74) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.74d0)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.74) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.74:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.74)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.74)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.74], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.73999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 43.0%

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

   3. Taylor expanded in k around inf 59.9%

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

      1. unpow2 59.9%

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

   5. Simplified 59.9%

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

   if -0.73999999999999999 < m

   1. Initial program 87.9%

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

      1. +-commutative 87.9%

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

      2. add-cbrt-cube 80.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{\left(k \cdot k\right) \cdot k}} \cdot k + \left(1 + 10 \cdot k\right)} \]

      3. cbrt-prod 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\sqrt[3]{k \cdot k} \cdot \sqrt[3]{k}\right)} \cdot k + \left(1 + 10 \cdot k\right)} \]

      4. associate-*l* 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\sqrt[3]{k \cdot k} \cdot \left(\sqrt[3]{k} \cdot k\right)} + \left(1 + 10 \cdot k\right)} \]

      5. fma-def 87.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(\sqrt[3]{k \cdot k}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)}} \]

      6. cbrt-prod 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{\sqrt[3]{k} \cdot \sqrt[3]{k}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      7. pow2 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{k}\right)}^{2}}, \sqrt[3]{k} \cdot k, 1 + 10 \cdot k\right)} \]

      8. +-commutative 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{10 \cdot k + 1}\right)} \]

      9. *-commutative 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{k \cdot 10} + 1\right)} \]

      10. fma-def 87.8%

          \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \color{blue}{\mathsf{fma}\left(k, 10, 1\right)}\right)} \]

   3. Applied egg-rr 87.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{k}\right)}^{2}, \sqrt[3]{k} \cdot k, \mathsf{fma}\left(k, 10, 1\right)\right)}} \]
   4. Step-by-step derivation

      1. fma-udef 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{{\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right) + \mathsf{fma}\left(k, 10, 1\right)}} \]

      2. +-commutative 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \left(\sqrt[3]{k} \cdot k\right)}} \]

      3. *-commutative 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + {\left(\sqrt[3]{k}\right)}^{2} \cdot \color{blue}{\left(k \cdot \sqrt[3]{k}\right)}} \]

      4. associate-*r* 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left({\left(\sqrt[3]{k}\right)}^{2} \cdot k\right) \cdot \sqrt[3]{k}}} \]

      5. *-commutative 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{\left(k \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)} \cdot \sqrt[3]{k}} \]

      6. associate-*l* 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + \color{blue}{k \cdot \left({\left(\sqrt[3]{k}\right)}^{2} \cdot \sqrt[3]{k}\right)}} \]

      7. pow-plus 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot \color{blue}{{\left(\sqrt[3]{k}\right)}^{\left(2 + 1\right)}}} \]

      8. metadata-eval 87.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{\color{blue}{3}}} \]

   5. Simplified 87.8%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) + k \cdot {\left(\sqrt[3]{k}\right)}^{3}}} \]

   6. Taylor expanded in m around 0 51.1%

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

      1. unpow2 51.1%

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

      2. distribute-rgt-in 51.1%

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

   8. Simplified 51.1%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.74:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10: 28.7% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{+101} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2.6e+101) (not (<= k 0.1))) (/ a (* k 10.0)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.6e+101) || !(k <= 0.1)) {
		tmp = a / (k * 10.0);
	} 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 <= (-2.6d+101)) .or. (.not. (k <= 0.1d0))) then
        tmp = a / (k * 10.0d0)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.6e+101) || !(k <= 0.1)) {
		tmp = a / (k * 10.0);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2.6e+101) or not (k <= 0.1):
		tmp = a / (k * 10.0)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2.6e+101) || !(k <= 0.1))
		tmp = Float64(a / Float64(k * 10.0));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -2.6e+101) || ~((k <= 0.1)))
		tmp = a / (k * 10.0);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2.6e+101], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -2.6e101 or 0.10000000000000001 < k

   1. Initial program 82.2%

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

   2. Taylor expanded in k around 0 68.1%

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

   3. Taylor expanded in m around 0 27.0%

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

   4. Taylor expanded in k around inf 27.0%

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

      1. *-commutative 27.0%

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

   6. Simplified 27.0%

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

   if -2.6e101 < k < 0.10000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 39.4%

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

   3. Taylor expanded in k around 0 37.5%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{+101} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 47.0% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -3.4e-274) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -3.4e-274) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-3.4d-274)) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -3.4e-274) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -3.4e-274) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -3.4e-274) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -3.4e-274) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -3.4e-274], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -3.39999999999999981e-274 or 1 < k

   1. Initial program 87.1%

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

   2. Taylor expanded in m around 0 43.8%

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

   3. Taylor expanded in k around inf 46.0%

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

      1. unpow2 46.0%

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

   5. Simplified 46.0%

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

   if -3.39999999999999981e-274 < k < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 58.1%

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

   3. Taylor expanded in k around 0 55.2%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{-274} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12: 52.5% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.5:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.5) (/ a (* k k)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.5) {
		tmp = a / (k * 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 (m <= (-0.5d0)) then
        tmp = a / (k * 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 (m <= -0.5) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.5:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.5)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.5)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.5], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.5

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 43.0%

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

   3. Taylor expanded in k around inf 59.9%

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

      1. unpow2 59.9%

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

   5. Simplified 59.9%

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

   if -0.5 < m

   1. Initial program 87.9%

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

   2. Taylor expanded in m around 0 51.1%

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

   3. Taylor expanded in k around 0 48.7%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.5:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 13: 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 91.5%

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

2. Taylor expanded in m around 0 48.7%

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

3. Taylor expanded in k around 0 21.7%

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

4. Final simplification 21.7%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023238 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))