Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 98.0%

Time: 10.1s

Alternatives: 9

Speedup: 1.0×

• 22.5% 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: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+274}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (* a_s (if (<= t_1 5e+274) t_1 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 t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+274) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 5d+274) then
        tmp = t_1
    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 t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 5e+274) {
		tmp = t_1;
	} 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)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 5e+274:
		tmp = t_1
	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))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e+274)
		tmp = t_1;
	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);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 5e+274)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 5e+274], t$95$1, t$95$0]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 4.9999999999999998e274

   1. Initial program 99.1%

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

   if 4.9999999999999998e274 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 70.7%

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

      1. associate-/l* 70.7%

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

      2. remove-double-neg 70.7%

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

      3. distribute-frac-neg2 70.7%

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

      4. distribute-neg-frac2 70.7%

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

      5. remove-double-neg 70.7%

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

      6. sqr-neg 70.7%

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

      7. associate-+l+ 70.7%

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

      8. sqr-neg 70.7%

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

      9. distribute-rgt-out 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in k around 0 98.3%

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

      1. *-commutative 98.3%

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

   7. Simplified 98.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 5 \cdot 10^{+274}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.95)
    (* 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 <= 2.95) {
		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 <= 2.95d0) 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 <= 2.95) {
		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 <= 2.95:
		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 <= 2.95)
		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 <= 2.95)
		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, 2.95], 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 < 2.9500000000000002

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. remove-double-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. distribute-neg-frac2 98.9%

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

      5. remove-double-neg 98.9%

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

      6. sqr-neg 98.9%

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

      7. associate-+l+ 98.9%

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

      8. sqr-neg 98.9%

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

      9. distribute-rgt-out 98.9%

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

   3. Simplified 98.9%

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

   if 2.9500000000000002 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.9%

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

      2. remove-double-neg 80.9%

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

      3. distribute-frac-neg2 80.9%

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

      4. distribute-neg-frac2 80.9%

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

      5. remove-double-neg 80.9%

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

      6. sqr-neg 80.9%

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

      7. associate-+l+ 80.9%

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

      8. sqr-neg 80.9%

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

      9. distribute-rgt-out 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in k around 0 97.8%

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

      1. *-commutative 97.8%

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

   7. Simplified 97.8%

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

4. Final simplification 98.5%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-14} \lor \neg \left(m \leq 0.15\right):\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -3.5e-14) (not (<= m 0.15)))
    (* 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-14) || !(m <= 0.15)) {
		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-14)) .or. (.not. (m <= 0.15d0))) 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-14) || !(m <= 0.15)) {
		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-14) or not (m <= 0.15):
		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-14) || !(m <= 0.15))
		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-14) || ~((m <= 0.15)))
		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-14], N[Not[LessEqual[m, 0.15]], $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 < -3.5000000000000002e-14 or 0.149999999999999994 < m

   1. Initial program 90.8%

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

      1. associate-/l* 90.8%

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

      2. remove-double-neg 90.8%

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

      3. distribute-frac-neg2 90.8%

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

      4. distribute-neg-frac2 90.8%

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

      5. remove-double-neg 90.8%

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

      6. sqr-neg 90.8%

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

      7. associate-+l+ 90.8%

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

      8. sqr-neg 90.8%

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

      9. distribute-rgt-out 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in k around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   if -3.5000000000000002e-14 < m < 0.149999999999999994

   1. Initial program 97.5%

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

      1. associate-/l* 97.4%

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

      2. remove-double-neg 97.4%

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

      3. distribute-frac-neg2 97.4%

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

      4. distribute-neg-frac2 97.4%

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

      5. remove-double-neg 97.4%

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

      6. sqr-neg 97.4%

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

      7. associate-+l+ 97.4%

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

      8. sqr-neg 97.4%

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

      9. distribute-rgt-out 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in m around 0 97.4%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.5 \cdot 10^{-14} \lor \neg \left(m \leq 0.15\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.0% accurate, 4.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{1}{\frac{k \cdot \left(k + 10\right)}{a\_m}}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -5.7 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-303}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \mathbf{elif}\;k \leq 0.12:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ (* k (+ k 10.0)) a_m))))
   (*
    a_s
    (if (<= k -5.7e+109)
      t_0
      (if (<= k 6.5e-303)
        (* -10.0 (* a_m k))
        (if (<= k 0.12) (/ a_m (+ 1.0 (* 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 = 1.0 / ((k * (k + 10.0)) / a_m);
	double tmp;
	if (k <= -5.7e+109) {
		tmp = t_0;
	} else if (k <= 6.5e-303) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.12) {
		tmp = a_m / (1.0 + (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 = 1.0d0 / ((k * (k + 10.0d0)) / a_m)
    if (k <= (-5.7d+109)) then
        tmp = t_0
    else if (k <= 6.5d-303) then
        tmp = (-10.0d0) * (a_m * k)
    else if (k <= 0.12d0) then
        tmp = a_m / (1.0d0 + (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 = 1.0 / ((k * (k + 10.0)) / a_m);
	double tmp;
	if (k <= -5.7e+109) {
		tmp = t_0;
	} else if (k <= 6.5e-303) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.12) {
		tmp = a_m / (1.0 + (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 = 1.0 / ((k * (k + 10.0)) / a_m)
	tmp = 0
	if k <= -5.7e+109:
		tmp = t_0
	elif k <= 6.5e-303:
		tmp = -10.0 * (a_m * k)
	elif k <= 0.12:
		tmp = a_m / (1.0 + (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(1.0 / Float64(Float64(k * Float64(k + 10.0)) / a_m))
	tmp = 0.0
	if (k <= -5.7e+109)
		tmp = t_0;
	elseif (k <= 6.5e-303)
		tmp = Float64(-10.0 * Float64(a_m * k));
	elseif (k <= 0.12)
		tmp = Float64(a_m / Float64(1.0 + 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 = 1.0 / ((k * (k + 10.0)) / a_m);
	tmp = 0.0;
	if (k <= -5.7e+109)
		tmp = t_0;
	elseif (k <= 6.5e-303)
		tmp = -10.0 * (a_m * k);
	elseif (k <= 0.12)
		tmp = a_m / (1.0 + (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[(1.0 / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -5.7e+109], t$95$0, If[LessEqual[k, 6.5e-303], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.12], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < -5.7000000000000002e109 or 0.12 < k

   1. Initial program 82.1%

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

      1. associate-/l* 82.1%

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

      2. remove-double-neg 82.1%

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

      3. distribute-frac-neg2 82.1%

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

      4. distribute-neg-frac2 82.1%

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

      5. remove-double-neg 82.1%

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

      6. sqr-neg 82.1%

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

      7. associate-+l+ 82.1%

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

      8. sqr-neg 82.1%

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

      9. distribute-rgt-out 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in m around 0 62.1%

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

      1. div-inv 62.1%

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

      2. clear-num 62.1%

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

      3. +-commutative 62.1%

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

      4. +-commutative 62.1%

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

      5. fma-undefine 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in k around inf 61.9%

      \[\leadsto \frac{1}{\frac{\color{blue}{10 \cdot k + {k}^{2}}}{a}} \]
   9. Step-by-step derivation

      1. +-commutative 61.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 61.9%

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

      3. distribute-rgt-in 61.9%

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

   10. Simplified 61.9%

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

   if -5.7000000000000002e109 < k < 6.50000000000000028e-303

   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 3.2%

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

   6. Taylor expanded in k around 0 4.8%

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

   7. Taylor expanded in k around inf 12.2%

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

   if 6.50000000000000028e-303 < k < 0.12

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. remove-double-neg 99.9%

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

      6. sqr-neg 99.9%

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

      7. associate-+l+ 99.9%

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

      8. sqr-neg 99.9%

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

      9. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in m around 0 39.7%

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

   6. Taylor expanded in k around 0 39.3%

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

      1. *-commutative 39.3%

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

   8. Simplified 39.3%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.7 \cdot 10^{+109}:\\ \;\;\;\;\frac{1}{\frac{k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-303}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.12:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot \left(k + 10\right)}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{a\_m}}\\ \mathbf{elif}\;m \leq 90:\\ \;\;\;\;\frac{a\_m}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (*
    a_s
    (if (<= m -8e+37)
      (/ 1.0 (/ t_0 a_m))
      (if (<= m 90.0) (/ a_m (+ 1.0 t_0)) (* -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 t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -8e+37) {
		tmp = 1.0 / (t_0 / a_m);
	} else if (m <= 90.0) {
		tmp = a_m / (1.0 + t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-8d+37)) then
        tmp = 1.0d0 / (t_0 / a_m)
    else if (m <= 90.0d0) then
        tmp = a_m / (1.0d0 + t_0)
    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 t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -8e+37) {
		tmp = 1.0 / (t_0 / a_m);
	} else if (m <= 90.0) {
		tmp = a_m / (1.0 + t_0);
	} 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):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -8e+37:
		tmp = 1.0 / (t_0 / a_m)
	elif m <= 90.0:
		tmp = a_m / (1.0 + t_0)
	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)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -8e+37)
		tmp = Float64(1.0 / Float64(t_0 / a_m));
	elseif (m <= 90.0)
		tmp = Float64(a_m / Float64(1.0 + t_0));
	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)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -8e+37)
		tmp = 1.0 / (t_0 / a_m);
	elseif (m <= 90.0)
		tmp = a_m / (1.0 + t_0);
	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_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -8e+37], N[(1.0 / N[(t$95$0 / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 90.0], N[(a$95$m / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -7.99999999999999963e37

   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 31.7%

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

      1. div-inv 31.7%

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

      2. clear-num 31.7%

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

      3. +-commutative 31.7%

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

      4. +-commutative 31.7%

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

      5. fma-undefine 31.7%

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

   7. Applied egg-rr 31.7%

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

   8. Taylor expanded in k around inf 40.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{10 \cdot k + {k}^{2}}}{a}} \]
   9. Step-by-step derivation

      1. +-commutative 40.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

      2. unpow2 40.0%

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

      3. distribute-rgt-in 40.0%

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

   10. Simplified 40.0%

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

   if -7.99999999999999963e37 < m < 90

   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 m around 0 90.9%

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

   if 90 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.9%

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

      2. remove-double-neg 80.9%

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

      3. distribute-frac-neg2 80.9%

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

      4. distribute-neg-frac2 80.9%

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

      5. remove-double-neg 80.9%

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

      6. sqr-neg 80.9%

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

      7. associate-+l+ 80.9%

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

      8. sqr-neg 80.9%

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

      9. distribute-rgt-out 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 9.6%

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

   7. Taylor expanded in k around inf 22.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\frac{k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;m \leq 90:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{+36}:\\ \;\;\;\;0.1 \cdot \frac{a\_m}{k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -2.15e+36)
    (* 0.1 (/ a_m k))
    (if (<= m 1.1) (/ a_m (+ 1.0 (* k 10.0))) (* -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 <= -2.15e+36) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 1.1) {
		tmp = a_m / (1.0 + (k * 10.0));
	} 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 <= (-2.15d+36)) then
        tmp = 0.1d0 * (a_m / k)
    else if (m <= 1.1d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    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 <= -2.15e+36) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 1.1) {
		tmp = a_m / (1.0 + (k * 10.0));
	} 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 <= -2.15e+36:
		tmp = 0.1 * (a_m / k)
	elif m <= 1.1:
		tmp = a_m / (1.0 + (k * 10.0))
	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 <= -2.15e+36)
		tmp = Float64(0.1 * Float64(a_m / k));
	elseif (m <= 1.1)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	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 <= -2.15e+36)
		tmp = 0.1 * (a_m / k);
	elseif (m <= 1.1)
		tmp = a_m / (1.0 + (k * 10.0));
	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, -2.15e+36], N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.1], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -2.15000000000000002e36

   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 31.7%

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

   6. Taylor expanded in k around 0 14.1%

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

      1. *-commutative 14.1%

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

   8. Simplified 14.1%

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

   9. Taylor expanded in k around inf 22.4%

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

   if -2.15000000000000002e36 < m < 1.1000000000000001

   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 m around 0 90.9%

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

   6. Taylor expanded in k around 0 61.7%

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

      1. *-commutative 61.7%

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

   8. Simplified 61.7%

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

   if 1.1000000000000001 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.9%

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

      2. remove-double-neg 80.9%

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

      3. distribute-frac-neg2 80.9%

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

      4. distribute-neg-frac2 80.9%

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

      5. remove-double-neg 80.9%

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

      6. sqr-neg 80.9%

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

      7. associate-+l+ 80.9%

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

      8. sqr-neg 80.9%

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

      9. distribute-rgt-out 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 9.6%

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

   7. Taylor expanded in k around inf 22.2%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{+36}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 1.1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.00054:\\ \;\;\;\;0.1 \cdot \frac{a\_m}{k}\\ \mathbf{elif}\;m \leq 0.215:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.00054)
    (* 0.1 (/ a_m k))
    (if (<= m 0.215) 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 <= -0.00054) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 0.215) {
		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 <= (-0.00054d0)) then
        tmp = 0.1d0 * (a_m / k)
    else if (m <= 0.215d0) 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 <= -0.00054) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 0.215) {
		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 <= -0.00054:
		tmp = 0.1 * (a_m / k)
	elif m <= 0.215:
		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 <= -0.00054)
		tmp = Float64(0.1 * Float64(a_m / k));
	elseif (m <= 0.215)
		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 <= -0.00054)
		tmp = 0.1 * (a_m / k);
	elseif (m <= 0.215)
		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, -0.00054], N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.215], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -5.40000000000000007e-4

   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 32.7%

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

   6. Taylor expanded in k around 0 15.4%

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

      1. *-commutative 15.4%

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

   8. Simplified 15.4%

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

   9. Taylor expanded in k around inf 23.0%

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

   if -5.40000000000000007e-4 < m < 0.214999999999999997

   1. Initial program 97.5%

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

      1. associate-/l* 97.5%

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

      2. remove-double-neg 97.5%

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

      3. distribute-frac-neg2 97.5%

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

      4. distribute-neg-frac2 97.5%

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

      5. remove-double-neg 97.5%

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

      6. sqr-neg 97.5%

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

      7. associate-+l+ 97.5%

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

      8. sqr-neg 97.5%

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

      9. distribute-rgt-out 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in m around 0 95.3%

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

   6. Taylor expanded in k around 0 47.7%

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

   if 0.214999999999999997 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.9%

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

      2. remove-double-neg 80.9%

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

      3. distribute-frac-neg2 80.9%

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

      4. distribute-neg-frac2 80.9%

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

      5. remove-double-neg 80.9%

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

      6. sqr-neg 80.9%

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

      7. associate-+l+ 80.9%

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

      8. sqr-neg 80.9%

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

      9. distribute-rgt-out 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 9.6%

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

   7. Taylor expanded in k around inf 22.2%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00054:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.215:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 25.8% accurate, 11.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 200:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a\_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 200.0) 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 <= 200.0) {
		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 <= 200.0d0) 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 <= 200.0) {
		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 <= 200.0:
		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 <= 200.0)
		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 <= 200.0)
		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, 200.0], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 200

   1. Initial program 98.9%

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

      1. associate-/l* 98.9%

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

      2. remove-double-neg 98.9%

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

      3. distribute-frac-neg2 98.9%

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

      4. distribute-neg-frac2 98.9%

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

      5. remove-double-neg 98.9%

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

      6. sqr-neg 98.9%

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

      7. associate-+l+ 98.9%

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

      8. sqr-neg 98.9%

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

      9. distribute-rgt-out 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in m around 0 60.4%

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

   6. Taylor expanded in k around 0 23.2%

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

   if 200 < m

   1. Initial program 80.9%

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

      1. associate-/l* 80.9%

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

      2. remove-double-neg 80.9%

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

      3. distribute-frac-neg2 80.9%

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

      4. distribute-neg-frac2 80.9%

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

      5. remove-double-neg 80.9%

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

      6. sqr-neg 80.9%

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

      7. associate-+l+ 80.9%

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

      8. sqr-neg 80.9%

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

      9. distribute-rgt-out 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 9.6%

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

   7. Taylor expanded in k around inf 22.2%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 200:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 1 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 92.6%

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

   1. associate-/l* 92.6%

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

   2. remove-double-neg 92.6%

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

   3. distribute-frac-neg2 92.6%

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

   4. distribute-neg-frac2 92.6%

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

   5. remove-double-neg 92.6%

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

   6. sqr-neg 92.6%

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

   7. associate-+l+ 92.6%

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

   8. sqr-neg 92.6%

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

   9. distribute-rgt-out 92.6%

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

3. Simplified 92.6%

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

5. Taylor expanded in m around 0 40.5%

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

6. Taylor expanded in k around 0 16.4%

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

7. Final simplification 16.4%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))