Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 97.9%

Time: 14.7s

Alternatives: 16

Speedup: 1.0×

• 21.5% 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: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.45:\\ \;\;\;\;\frac{t\_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{t\_0}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= m 1.45) (/ t_0 (+ 1.0 (* k (+ k 10.0)))) (pow (sqrt t_0) 2.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= 1.45) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = pow(sqrt(t_0), 2.0);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if (m <= 1.45d0) then
        tmp = t_0 / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = sqrt(t_0) ** 2.0d0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if (m <= 1.45) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = Math.pow(Math.sqrt(t_0), 2.0);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if m <= 1.45:
		tmp = t_0 / (1.0 + (k * (k + 10.0)))
	else:
		tmp = math.pow(math.sqrt(t_0), 2.0)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= 1.45)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = sqrt(t_0) ^ 2.0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if (m <= 1.45)
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	else
		tmp = sqrt(t_0) ^ 2.0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, 1.45], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.44999999999999996

   1. Initial program 97.8%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in a around 0 97.8%

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

   if 1.44999999999999996 < m

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

      2. remove-double-neg 85.7%

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

      3. distribute-frac-neg2 85.7%

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

      4. distribute-neg-frac2 85.7%

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

      5. remove-double-neg 85.7%

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

      6. sqr-neg 85.7%

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

      7. associate-+l+ 85.7%

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

      8. sqr-neg 85.7%

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

      9. distribute-rgt-out 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. add-sqr-sqrt 84.5%

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

      2. pow2 84.5%

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

   9. Applied egg-rr 84.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.45:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{a \cdot {k}^{m}}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.95e-15)
    (* a_m (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
    (* a_m (pow k m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.95e-15) {
		tmp = a_m * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a_m * pow(k, m);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.95d-15) then
        tmp = a_m * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a_m * (k ** m)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 2.95e-15) {
		tmp = a_m * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a_m * Math.pow(k, m);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 2.95e-15:
		tmp = a_m * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a_m * math.pow(k, m)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 2.95e-15)
		tmp = Float64(a_m * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a_m * (k ^ m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 2.95e-15)
		tmp = a_m * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a_m * (k ^ m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.95e-15], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.94999999999999982e-15

   1. Initial program 97.7%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   if 2.94999999999999982e-15 < m

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. remove-double-neg 86.0%

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

      3. distribute-frac-neg2 86.0%

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

      4. distribute-neg-frac2 86.0%

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

      5. remove-double-neg 86.0%

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

      6. sqr-neg 86.0%

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

      7. associate-+l+ 86.0%

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

      8. sqr-neg 86.0%

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

      9. distribute-rgt-out 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{t\_0}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (* a_s (if (<= m 2.95e-15) (/ t_0 (+ 1.0 (* k (+ k 10.0)))) t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if (m <= 2.95e-15) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if (m <= 2.95d-15) then
        tmp = t_0 / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if (m <= 2.95e-15) {
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if m <= 2.95e-15:
		tmp = t_0 / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (m <= 2.95e-15)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if (m <= 2.95e-15)
		tmp = t_0 / (1.0 + (k * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, 2.95e-15], N[(t$95$0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.94999999999999982e-15

   1. Initial program 97.7%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in a around 0 97.7%

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

   if 2.94999999999999982e-15 < m

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. remove-double-neg 86.0%

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

      3. distribute-frac-neg2 86.0%

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

      4. distribute-neg-frac2 86.0%

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

      5. remove-double-neg 86.0%

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

      6. sqr-neg 86.0%

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

      7. associate-+l+ 86.0%

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

      8. sqr-neg 86.0%

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

      9. distribute-rgt-out 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-11} \lor \neg \left(m \leq 2.95 \cdot 10^{-15}\right):\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -3.5e-11) (not (<= m 2.95e-15)))
    (* a_m (pow k m))
    (/ a_m (+ 1.0 (* k (+ k 10.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -3.5e-11) || !(m <= 2.95e-15)) {
		tmp = a_m * pow(k, m);
	} else {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3.5d-11)) .or. (.not. (m <= 2.95d-15))) then
        tmp = a_m * (k ** m)
    else
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -3.5e-11) || !(m <= 2.95e-15)) {
		tmp = a_m * Math.pow(k, m);
	} else {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (m <= -3.5e-11) or not (m <= 2.95e-15):
		tmp = a_m * math.pow(k, m)
	else:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((m <= -3.5e-11) || !(m <= 2.95e-15))
		tmp = Float64(a_m * (k ^ m));
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((m <= -3.5e-11) || ~((m <= 2.95e-15)))
		tmp = a_m * (k ^ m);
	else
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[m, -3.5e-11], N[Not[LessEqual[m, 2.95e-15]], $MachinePrecision]], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -3.50000000000000019e-11 or 2.94999999999999982e-15 < m

   1. Initial program 92.7%

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

      1. associate-/l* 92.7%

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

      2. remove-double-neg 92.7%

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

      3. distribute-frac-neg2 92.7%

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

      4. distribute-neg-frac2 92.7%

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

      5. remove-double-neg 92.7%

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

      6. sqr-neg 92.7%

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

      7. associate-+l+ 92.7%

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

      8. sqr-neg 92.7%

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

      9. distribute-rgt-out 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -3.50000000000000019e-11 < m < 2.94999999999999982e-15

   1. Initial program 95.8%

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

      1. associate-/l* 95.7%

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

      2. remove-double-neg 95.7%

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

      3. distribute-frac-neg2 95.7%

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

      4. distribute-neg-frac2 95.7%

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

      5. remove-double-neg 95.7%

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

      6. sqr-neg 95.7%

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

      7. associate-+l+ 95.7%

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

      8. sqr-neg 95.7%

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

      9. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in m around 0 95.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-11} \lor \neg \left(m \leq 2.95 \cdot 10^{-15}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{+176} \lor \neg \left(m \leq 2.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\frac{a\_m}{k \cdot \left(\left(--1\right) - \frac{\frac{-1}{k} + -10}{k}\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + a\_m \cdot \left(k \cdot \left(k \cdot \left(99 + k \cdot -980\right) - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -11500000000.0)
    (/ (/ a_m k) k)
    (if (<= m 0.65)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (if (or (<= m 1.85e+176) (not (<= m 2.2e+233)))
        (/ (/ a_m (* k (- (- -1.0) (/ (+ (/ -1.0 k) -10.0) k)))) k)
        (+ a_m (* a_m (* k (- (* k (+ 99.0 (* k -980.0))) 10.0)))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 0.65) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if ((m <= 1.85e+176) || !(m <= 2.2e+233)) {
		tmp = (a_m / (k * (-(-1.0) - (((-1.0 / k) + -10.0) / k)))) / k;
	} else {
		tmp = a_m + (a_m * (k * ((k * (99.0 + (k * -980.0))) - 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-11500000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 0.65d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else if ((m <= 1.85d+176) .or. (.not. (m <= 2.2d+233))) then
        tmp = (a_m / (k * (-(-1.0d0) - ((((-1.0d0) / k) + (-10.0d0)) / k)))) / k
    else
        tmp = a_m + (a_m * (k * ((k * (99.0d0 + (k * (-980.0d0)))) - 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 0.65) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if ((m <= 1.85e+176) || !(m <= 2.2e+233)) {
		tmp = (a_m / (k * (-(-1.0) - (((-1.0 / k) + -10.0) / k)))) / k;
	} else {
		tmp = a_m + (a_m * (k * ((k * (99.0 + (k * -980.0))) - 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -11500000000.0:
		tmp = (a_m / k) / k
	elif m <= 0.65:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	elif (m <= 1.85e+176) or not (m <= 2.2e+233):
		tmp = (a_m / (k * (-(-1.0) - (((-1.0 / k) + -10.0) / k)))) / k
	else:
		tmp = a_m + (a_m * (k * ((k * (99.0 + (k * -980.0))) - 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -11500000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 0.65)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif ((m <= 1.85e+176) || !(m <= 2.2e+233))
		tmp = Float64(Float64(a_m / Float64(k * Float64(Float64(-(-1.0)) - Float64(Float64(Float64(-1.0 / k) + -10.0) / k)))) / k);
	else
		tmp = Float64(a_m + Float64(a_m * Float64(k * Float64(Float64(k * Float64(99.0 + Float64(k * -980.0))) - 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -11500000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 0.65)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	elseif ((m <= 1.85e+176) || ~((m <= 2.2e+233)))
		tmp = (a_m / (k * (-(-1.0) - (((-1.0 / k) + -10.0) / k)))) / k;
	else
		tmp = a_m + (a_m * (k * ((k * (99.0 + (k * -980.0))) - 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -11500000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.65], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[m, 1.85e+176], N[Not[LessEqual[m, 2.2e+233]], $MachinePrecision]], N[(N[(a$95$m / N[(k * N[((--1.0) - N[(N[(N[(-1.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(a$95$m + N[(a$95$m * N[(k * N[(N[(k * N[(99.0 + N[(k * -980.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if m < -1.15e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.15e10 < m < 0.650000000000000022

   1. Initial program 96.0%

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

      1. associate-/l* 95.9%

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

      2. remove-double-neg 95.9%

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

      3. distribute-frac-neg2 95.9%

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

      4. distribute-neg-frac2 95.9%

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

      5. remove-double-neg 95.9%

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

      6. sqr-neg 95.9%

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

      7. associate-+l+ 95.9%

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

      8. sqr-neg 95.9%

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

      9. distribute-rgt-out 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in m around 0 93.4%

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

   if 0.650000000000000022 < m < 1.8499999999999999e176 or 2.19999999999999999e233 < m

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around inf 3.2%

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

      1. *-un-lft-identity 3.2%

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

      2. distribute-lft-out 3.2%

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

      3. times-frac 12.7%

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

   8. Applied egg-rr 12.7%

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

      1. associate-*l/ 12.7%

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

      2. *-lft-identity 12.7%

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

      3. +-commutative 12.7%

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

      4. associate-+l+ 12.7%

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

      5. +-commutative 12.7%

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

   10. Simplified 12.7%

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

   11. Taylor expanded in k around -inf 32.7%

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

       1. mul-1-neg 32.7%

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

       2. distribute-rgt-neg-in 32.7%

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

       3. sub-neg 32.7%

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

       4. metadata-eval 32.7%

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

       5. +-commutative 32.7%

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

       6. associate-*r/ 32.7%

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

       7. +-commutative 32.7%

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

       8. distribute-lft-in 32.7%

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

       9. neg-mul-1 32.7%

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

       10. distribute-neg-frac 32.7%

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

       11. metadata-eval 32.7%

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

       12. metadata-eval 32.7%

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

   13. Simplified 32.7%

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

   if 1.8499999999999999e176 < m < 2.19999999999999999e233

   1. Initial program 68.8%

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

      1. associate-/l* 68.8%

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

      2. remove-double-neg 68.8%

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

      3. distribute-frac-neg2 68.8%

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

      4. distribute-neg-frac2 68.8%

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

      5. remove-double-neg 68.8%

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

      6. sqr-neg 68.8%

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

      7. associate-+l+ 68.8%

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

      8. sqr-neg 68.8%

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

      9. distribute-rgt-out 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 34.1%

      \[\leadsto \color{blue}{a + k \cdot \left(k \cdot \left(-1 \cdot \left(k \cdot \left(-10 \cdot a + -10 \cdot \left(a + -100 \cdot a\right)\right)\right) - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

   7. Taylor expanded in a around 0 34.1%

      \[\leadsto a + k \cdot \left(k \cdot \left(-1 \cdot \color{blue}{\left(980 \cdot \left(a \cdot k\right)\right)} - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]
   8. Step-by-step derivation

      1. *-commutative 34.1%

         \[\leadsto a + k \cdot \left(k \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot 980\right)} - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

      2. associate-*l* 34.1%

         \[\leadsto a + k \cdot \left(k \cdot \left(-1 \cdot \color{blue}{\left(a \cdot \left(k \cdot 980\right)\right)} - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   9. Simplified 34.1%

      \[\leadsto a + k \cdot \left(k \cdot \left(-1 \cdot \color{blue}{\left(a \cdot \left(k \cdot 980\right)\right)} - \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \]

   10. Taylor expanded in a around 0 45.9%

       \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(k \cdot \left(99 + -980 \cdot k\right) - 10\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.65:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.85 \cdot 10^{+176} \lor \neg \left(m \leq 2.2 \cdot 10^{+233}\right):\\ \;\;\;\;\frac{\frac{a}{k \cdot \left(\left(--1\right) - \frac{\frac{-1}{k} + -10}{k}\right)}}{k}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot \left(99 + k \cdot -980\right) - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{-39}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{k \cdot \left(a\_m - k \cdot \left(k \cdot \left(a\_m + a\_m \cdot -100\right) + a\_m \cdot 10\right)\right)}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -11500000000.0)
    (/ (/ a_m k) k)
    (if (<= m 2.85e-39)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (/
       (* k (- a_m (* k (+ (* k (+ a_m (* a_m -100.0))) (* a_m 10.0)))))
       k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.85e-39) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * (a_m - (k * ((k * (a_m + (a_m * -100.0))) + (a_m * 10.0))))) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-11500000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 2.85d-39) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (k * (a_m - (k * ((k * (a_m + (a_m * (-100.0d0)))) + (a_m * 10.0d0))))) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.85e-39) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * (a_m - (k * ((k * (a_m + (a_m * -100.0))) + (a_m * 10.0))))) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -11500000000.0:
		tmp = (a_m / k) / k
	elif m <= 2.85e-39:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (k * (a_m - (k * ((k * (a_m + (a_m * -100.0))) + (a_m * 10.0))))) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -11500000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 2.85e-39)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(k * Float64(a_m - Float64(k * Float64(Float64(k * Float64(a_m + Float64(a_m * -100.0))) + Float64(a_m * 10.0))))) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -11500000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 2.85e-39)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = (k * (a_m - (k * ((k * (a_m + (a_m * -100.0))) + (a_m * 10.0))))) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -11500000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 2.85e-39], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * N[(a$95$m - N[(k * N[(N[(k * N[(a$95$m + N[(a$95$m * -100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a$95$m * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.15e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.15e10 < m < 2.8499999999999998e-39

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 93.9%

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

   if 2.8499999999999998e-39 < m

   1. Initial program 86.3%

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

      1. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in m around 0 6.8%

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

   6. Taylor expanded in k around inf 6.8%

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

      1. *-un-lft-identity 6.8%

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

      2. distribute-lft-out 6.8%

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

      3. times-frac 15.1%

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

   8. Applied egg-rr 15.1%

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

      1. associate-*l/ 15.1%

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

      2. *-lft-identity 15.1%

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

      3. +-commutative 15.1%

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

      4. associate-+l+ 15.1%

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

      5. +-commutative 15.1%

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

   10. Simplified 15.1%

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

   11. Taylor expanded in k around 0 35.3%

       \[\leadsto \frac{\color{blue}{k \cdot \left(a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)\right)}}{k} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 2.85 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{k \cdot \left(a - k \cdot \left(k \cdot \left(a + a \cdot -100\right) + a \cdot 10\right)\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -13000000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 185000000000:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{k \cdot \left(a\_m + -10 \cdot \left(a\_m \cdot k\right)\right)}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -13000000000.0)
    (/ (/ a_m k) k)
    (if (<= m 185000000000.0)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (/ (* k (+ a_m (* -10.0 (* a_m k)))) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -13000000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 185000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * (a_m + (-10.0 * (a_m * k)))) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-13000000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 185000000000.0d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (k * (a_m + ((-10.0d0) * (a_m * k)))) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -13000000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 185000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (k * (a_m + (-10.0 * (a_m * k)))) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -13000000000.0:
		tmp = (a_m / k) / k
	elif m <= 185000000000.0:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (k * (a_m + (-10.0 * (a_m * k)))) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -13000000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 185000000000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(k * Float64(a_m + Float64(-10.0 * Float64(a_m * k)))) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -13000000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 185000000000.0)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = (k * (a_m + (-10.0 * (a_m * k)))) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -13000000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 185000000000.0], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(k * N[(a$95$m + N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.3e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.3e10 < m < 1.85e11

   1. Initial program 96.1%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in m around 0 89.8%

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

   if 1.85e11 < m

   1. Initial program 85.0%

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

      1. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in m around 0 3.1%

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

   6. Taylor expanded in k around inf 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. distribute-lft-out 3.1%

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

      3. times-frac 12.3%

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

   8. Applied egg-rr 12.3%

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

      1. associate-*l/ 12.3%

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

      2. *-lft-identity 12.3%

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

      3. +-commutative 12.3%

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

      4. associate-+l+ 12.3%

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

      5. +-commutative 12.3%

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

   10. Simplified 12.3%

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

   11. Taylor expanded in k around 0 21.9%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -13000000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 185000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{k \cdot \left(a + -10 \cdot \left(a \cdot k\right)\right)}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{\frac{1}{k}}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -11500000000.0)
    (/ (/ a_m k) k)
    (if (<= m 2.95e-15)
      (/ a_m (+ 1.0 (* k (+ k 10.0))))
      (/ (/ a_m (/ 1.0 k)) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.95e-15) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (a_m / (1.0 / k)) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-11500000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 2.95d-15) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (a_m / (1.0d0 / k)) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.95e-15) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = (a_m / (1.0 / k)) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -11500000000.0:
		tmp = (a_m / k) / k
	elif m <= 2.95e-15:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = (a_m / (1.0 / k)) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -11500000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 2.95e-15)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(a_m / Float64(1.0 / k)) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -11500000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 2.95e-15)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = (a_m / (1.0 / k)) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -11500000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 2.95e-15], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / N[(1.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.15e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.15e10 < m < 2.94999999999999982e-15

   1. Initial program 95.9%

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

      1. associate-/l* 95.8%

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

      2. remove-double-neg 95.8%

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

      3. distribute-frac-neg2 95.8%

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

      4. distribute-neg-frac2 95.8%

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

      5. remove-double-neg 95.8%

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

      6. sqr-neg 95.8%

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

      7. associate-+l+ 95.8%

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

      8. sqr-neg 95.8%

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

      9. distribute-rgt-out 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 94.0%

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

   if 2.94999999999999982e-15 < m

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in m around 0 4.6%

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

   6. Taylor expanded in k around inf 4.6%

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

      1. *-un-lft-identity 4.6%

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

      2. distribute-lft-out 4.6%

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

      3. times-frac 13.1%

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

   8. Applied egg-rr 13.1%

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

      1. associate-*l/ 13.1%

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

      2. *-lft-identity 13.1%

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

      3. +-commutative 13.1%

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

      4. associate-+l+ 13.1%

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

      5. +-commutative 13.1%

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

   10. Simplified 13.1%

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

   11. Taylor expanded in k around 0 21.2%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k}}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + -10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= k -4e-265) (not (<= k 0.1)))
    (/ (/ a_m k) k)
    (+ a_m (* -10.0 (* a_m k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 0.1)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m + (-10.0 * (a_m * k));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-4d-265)) .or. (.not. (k <= 0.1d0))) then
        tmp = (a_m / k) / k
    else
        tmp = a_m + ((-10.0d0) * (a_m * k))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 0.1)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m + (-10.0 * (a_m * k));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (k <= -4e-265) or not (k <= 0.1):
		tmp = (a_m / k) / k
	else:
		tmp = a_m + (-10.0 * (a_m * k))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((k <= -4e-265) || !(k <= 0.1))
		tmp = Float64(Float64(a_m / k) / k);
	else
		tmp = Float64(a_m + Float64(-10.0 * Float64(a_m * k)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((k <= -4e-265) || ~((k <= 0.1)))
		tmp = (a_m / k) / k;
	else
		tmp = a_m + (-10.0 * (a_m * k));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[k, -4e-265], N[Not[LessEqual[k, 0.1]], $MachinePrecision]], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], N[(a$95$m + N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < -3.99999999999999994e-265 or 0.10000000000000001 < k

   1. Initial program 90.0%

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

      1. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in m around 0 44.1%

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

   6. Taylor expanded in k around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. distribute-lft-out 44.1%

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

      3. times-frac 45.4%

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

   8. Applied egg-rr 45.4%

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

      1. associate-*l/ 45.4%

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

      2. *-lft-identity 45.4%

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

      3. +-commutative 45.4%

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

      4. associate-+l+ 45.4%

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

      5. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   11. Taylor expanded in k around inf 45.5%

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

   if -3.99999999999999994e-265 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 52.7%

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

   6. Taylor expanded in k around 0 52.3%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 0.1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 10\right):\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= k -4e-265) (not (<= k 10.0)))
    (/ (/ a_m k) k)
    (/ a_m (+ 1.0 (* k 10.0))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 10.0)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m / (1.0 + (k * 10.0));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-4d-265)) .or. (.not. (k <= 10.0d0))) then
        tmp = (a_m / k) / k
    else
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 10.0)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m / (1.0 + (k * 10.0));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (k <= -4e-265) or not (k <= 10.0):
		tmp = (a_m / k) / k
	else:
		tmp = a_m / (1.0 + (k * 10.0))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((k <= -4e-265) || !(k <= 10.0))
		tmp = Float64(Float64(a_m / k) / k);
	else
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((k <= -4e-265) || ~((k <= 10.0)))
		tmp = (a_m / k) / k;
	else
		tmp = a_m / (1.0 + (k * 10.0));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[k, -4e-265], N[Not[LessEqual[k, 10.0]], $MachinePrecision]], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < -3.99999999999999994e-265 or 10 < k

   1. Initial program 90.0%

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

      1. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in m around 0 44.1%

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

   6. Taylor expanded in k around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. distribute-lft-out 44.1%

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

      3. times-frac 45.4%

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

   8. Applied egg-rr 45.4%

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

      1. associate-*l/ 45.4%

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

      2. *-lft-identity 45.4%

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

      3. +-commutative 45.4%

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

      4. associate-+l+ 45.4%

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

      5. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   11. Taylor expanded in k around inf 45.5%

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

   if -3.99999999999999994e-265 < k < 10

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 52.7%

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

   6. Taylor expanded in k around 0 52.4%

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

      1. *-commutative 52.4%

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

   8. Simplified 52.4%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 10\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.0% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 0.76:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m \cdot k}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -11500000000.0)
    (/ (/ a_m k) k)
    (if (<= m 0.76) (/ a_m (+ 1.0 (* k k))) (/ (* a_m k) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 0.76) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = (a_m * k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-11500000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 0.76d0) then
        tmp = a_m / (1.0d0 + (k * k))
    else
        tmp = (a_m * k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 0.76) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = (a_m * k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -11500000000.0:
		tmp = (a_m / k) / k
	elif m <= 0.76:
		tmp = a_m / (1.0 + (k * k))
	else:
		tmp = (a_m * k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -11500000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 0.76)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(Float64(a_m * k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -11500000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 0.76)
		tmp = a_m / (1.0 + (k * k));
	else
		tmp = (a_m * k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -11500000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.76], N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m * k), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.15e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.15e10 < m < 0.76000000000000001

   1. Initial program 96.0%

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

      1. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in m around 0 93.4%

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

   6. Taylor expanded in k around inf 93.3%

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

   7. Taylor expanded in k around 0 90.2%

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

   if 0.76000000000000001 < m

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around inf 3.2%

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

      1. *-un-lft-identity 3.2%

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

      2. distribute-lft-out 3.2%

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

      3. times-frac 11.9%

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

   8. Applied egg-rr 11.9%

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

      1. associate-*l/ 11.9%

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

      2. *-lft-identity 11.9%

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

      3. +-commutative 11.9%

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

      4. associate-+l+ 11.9%

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

      5. +-commutative 11.9%

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

   10. Simplified 11.9%

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

   11. Taylor expanded in k around 0 20.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 0.76:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot k}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.7% accurate, 6.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{elif}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a\_m}{\frac{1}{k}}}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -11500000000.0)
    (/ (/ a_m k) k)
    (if (<= m 2.95e-15) (/ a_m (+ 1.0 (* k k))) (/ (/ a_m (/ 1.0 k)) k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.95e-15) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = (a_m / (1.0 / k)) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-11500000000.0d0)) then
        tmp = (a_m / k) / k
    else if (m <= 2.95d-15) then
        tmp = a_m / (1.0d0 + (k * k))
    else
        tmp = (a_m / (1.0d0 / k)) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -11500000000.0) {
		tmp = (a_m / k) / k;
	} else if (m <= 2.95e-15) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = (a_m / (1.0 / k)) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -11500000000.0:
		tmp = (a_m / k) / k
	elif m <= 2.95e-15:
		tmp = a_m / (1.0 + (k * k))
	else:
		tmp = (a_m / (1.0 / k)) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -11500000000.0)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (m <= 2.95e-15)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(Float64(a_m / Float64(1.0 / k)) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -11500000000.0)
		tmp = (a_m / k) / k;
	elseif (m <= 2.95e-15)
		tmp = a_m / (1.0 + (k * k));
	else
		tmp = (a_m / (1.0 / k)) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -11500000000.0], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 2.95e-15], N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / N[(1.0 / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.15e10

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 38.2%

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

   6. Taylor expanded in k around inf 38.2%

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

      1. *-un-lft-identity 38.2%

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

      2. distribute-lft-out 38.2%

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

      3. times-frac 33.6%

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

   8. Applied egg-rr 33.6%

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

      1. associate-*l/ 33.6%

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

      2. *-lft-identity 33.6%

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

      3. +-commutative 33.6%

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

      4. associate-+l+ 33.6%

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

      5. +-commutative 33.6%

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

   10. Simplified 33.6%

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

   11. Taylor expanded in k around inf 51.7%

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

   if -1.15e10 < m < 2.94999999999999982e-15

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in m around 0 94.0%

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

   6. Taylor expanded in k around inf 93.9%

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

   7. Taylor expanded in k around 0 90.7%

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

   if 2.94999999999999982e-15 < m

   1. Initial program 86.0%

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

      1. *-commutative 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in m around 0 4.6%

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

   6. Taylor expanded in k around inf 4.6%

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

      1. *-un-lft-identity 4.6%

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

      2. distribute-lft-out 4.6%

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

      3. times-frac 13.1%

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

   8. Applied egg-rr 13.1%

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

      1. associate-*l/ 13.1%

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

      2. *-lft-identity 13.1%

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

      3. +-commutative 13.1%

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

      4. associate-+l+ 13.1%

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

      5. +-commutative 13.1%

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

   10. Simplified 13.1%

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

   11. Taylor expanded in k around 0 21.2%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -11500000000:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 2.95 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\frac{1}{k}}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.2% accurate, 7.6× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{\frac{a\_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (or (<= k -4e-265) (not (<= k 1.0))) (/ (/ a_m k) k) a_m)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 1.0)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-4d-265)) .or. (.not. (k <= 1.0d0))) then
        tmp = (a_m / k) / k
    else
        tmp = a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((k <= -4e-265) || !(k <= 1.0)) {
		tmp = (a_m / k) / k;
	} else {
		tmp = a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (k <= -4e-265) or not (k <= 1.0):
		tmp = (a_m / k) / k
	else:
		tmp = a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((k <= -4e-265) || !(k <= 1.0))
		tmp = Float64(Float64(a_m / k) / k);
	else
		tmp = a_m;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((k <= -4e-265) || ~((k <= 1.0)))
		tmp = (a_m / k) / k;
	else
		tmp = a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[k, -4e-265], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], a$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < -3.99999999999999994e-265 or 1 < k

   1. Initial program 90.0%

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

      1. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in m around 0 44.1%

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

   6. Taylor expanded in k around inf 44.1%

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

      1. *-un-lft-identity 44.1%

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

      2. distribute-lft-out 44.1%

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

      3. times-frac 45.4%

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

   8. Applied egg-rr 45.4%

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

      1. associate-*l/ 45.4%

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

      2. *-lft-identity 45.4%

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

      3. +-commutative 45.4%

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

      4. associate-+l+ 45.4%

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

      5. +-commutative 45.4%

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

   10. Simplified 45.4%

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

   11. Taylor expanded in k around inf 45.5%

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

   if -3.99999999999999994e-265 < k < 1

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 52.7%

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

   6. Taylor expanded in k around 0 51.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4 \cdot 10^{-265} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 25.5% accurate, 11.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.5:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 1.5) a_m (* -10.0 (* a_m k)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.5) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.5d0) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.5) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.5:
		tmp = a_m
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.5)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.5)
		tmp = a_m;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.5], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.5

   1. Initial program 97.8%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in m around 0 68.6%

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

   6. Taylor expanded in k around 0 31.0%

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

   if 1.5 < m

   1. Initial program 85.5%

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

      1. associate-/l* 85.5%

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

      2. remove-double-neg 85.5%

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

      3. distribute-frac-neg2 85.5%

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

      4. distribute-neg-frac2 85.5%

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

      5. remove-double-neg 85.5%

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

      6. sqr-neg 85.5%

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

      7. associate-+l+ 85.5%

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

      8. sqr-neg 85.5%

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

      9. distribute-rgt-out 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around 0 7.8%

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

   7. Taylor expanded in k around inf 17.4%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.5:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.5:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m \cdot k}{k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 1.5) a_m (/ (* a_m k) k))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.5) {
		tmp = a_m;
	} else {
		tmp = (a_m * k) / k;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.5d0) then
        tmp = a_m
    else
        tmp = (a_m * k) / k
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.5) {
		tmp = a_m;
	} else {
		tmp = (a_m * k) / k;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.5:
		tmp = a_m
	else:
		tmp = (a_m * k) / k
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.5)
		tmp = a_m;
	else
		tmp = Float64(Float64(a_m * k) / k);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.5)
		tmp = a_m;
	else
		tmp = (a_m * k) / k;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.5], a$95$m, N[(N[(a$95$m * k), $MachinePrecision] / k), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.5

   1. Initial program 97.8%

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

      1. associate-/l* 97.7%

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

      2. remove-double-neg 97.7%

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

      3. distribute-frac-neg2 97.7%

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

      4. distribute-neg-frac2 97.7%

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

      5. remove-double-neg 97.7%

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

      6. sqr-neg 97.7%

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

      7. associate-+l+ 97.7%

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

      8. sqr-neg 97.7%

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

      9. distribute-rgt-out 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in m around 0 68.6%

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

   6. Taylor expanded in k around 0 31.0%

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

   if 1.5 < m

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in m around 0 3.2%

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

   6. Taylor expanded in k around inf 3.2%

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

      1. *-un-lft-identity 3.2%

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

      2. distribute-lft-out 3.2%

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

      3. times-frac 12.0%

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

   8. Applied egg-rr 12.0%

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

      1. associate-*l/ 12.0%

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

      2. *-lft-identity 12.0%

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

      3. +-commutative 12.0%

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

      4. associate-+l+ 12.0%

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

      5. +-commutative 12.0%

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

   10. Simplified 12.0%

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

   11. Taylor expanded in k around 0 20.4%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.5:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot k}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.1% accurate, 114.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot a\_m \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s a_m))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * a_m
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * a_m

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * a_m)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * a_m;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * a$95$m), $MachinePrecision]

Derivation

1. Initial program 93.8%

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

   1. associate-/l* 93.8%

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

   2. remove-double-neg 93.8%

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

   3. distribute-frac-neg2 93.8%

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

   4. distribute-neg-frac2 93.8%

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

   5. remove-double-neg 93.8%

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

   6. sqr-neg 93.8%

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

   7. associate-+l+ 93.8%

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

   8. sqr-neg 93.8%

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

   9. distribute-rgt-out 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in m around 0 47.4%

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

6. Taylor expanded in k around 0 22.2%

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

7. Final simplification 22.2%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))