Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 97.6%

Time: 11.4s

Alternatives: 17

Speedup: 1.1×

• 23.5% 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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a_m \cdot {k}^{m}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+271}:\\ \;\;\;\;{k}^{m} \cdot \frac{a_m}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

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))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+271)
      (* (pow k m) (/ a_m (fma k (+ k 10.0) 1.0)))
      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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+271) {
		tmp = pow(k, m) * (a_m / fma(k, (k + 10.0), 1.0));
	} 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 (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+271)
		tmp = Float64((k ^ m) * Float64(a_m / fma(k, Float64(k + 10.0), 1.0)));
	else
		tmp = t_0;
	end
	return Float64(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[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+271], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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))) < 9.99999999999999953e270

   1. Initial program 95.7%

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

      1. associate-*l/ 94.7%

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

      2. *-commutative 94.7%

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

      3. sqr-neg 94.7%

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

      4. associate-+l+ 94.7%

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

      5. +-commutative 94.7%

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

      6. sqr-neg 94.7%

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

      7. distribute-rgt-out 94.7%

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

      8. fma-def 94.7%

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

      9. +-commutative 94.7%

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

   3. Simplified 94.7%

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

   if 9.99999999999999953e270 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 55.3%

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

      1. associate-*l/ 53.2%

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

      2. *-commutative 53.2%

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

      3. sqr-neg 53.2%

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

      4. associate-+l+ 53.2%

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

      5. +-commutative 53.2%

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

      6. sqr-neg 53.2%

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

      7. distribute-rgt-out 53.2%

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

      8. fma-def 53.2%

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

      9. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+271}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a_m \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 10^{+271}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

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 1e+271) 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 <= 1e+271) {
		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 <= 1d+271) 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 <= 1e+271) {
		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 <= 1e+271:
		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 <= 1e+271)
		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 <= 1e+271)
		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, 1e+271], 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))) < 9.99999999999999953e270

   1. Initial program 95.7%

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

   if 9.99999999999999953e270 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 55.3%

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

      1. associate-*l/ 53.2%

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

      2. *-commutative 53.2%

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

      3. sqr-neg 53.2%

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

      4. associate-+l+ 53.2%

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

      5. +-commutative 53.2%

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

      6. sqr-neg 53.2%

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

      7. distribute-rgt-out 53.2%

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

      8. fma-def 53.2%

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

      9. +-commutative 53.2%

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

   3. Simplified 53.2%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+271}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 3.35:\\ \;\;\;\;\frac{a_m}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot {k}^{m}\\ \end{array} \end{array} \]

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 3.35)
    (/ a_m (/ (+ 1.0 (* k (+ k 10.0))) (pow k m)))
    (* 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 <= 3.35) {
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} 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 <= 3.35d0) then
        tmp = a_m / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    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 <= 3.35) {
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} 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 <= 3.35:
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	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 <= 3.35)
		tmp = Float64(a_m / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	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 <= 3.35)
		tmp = a_m / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	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, 3.35], N[(a$95$m / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 3.35000000000000009

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. sqr-neg 95.1%

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

      3. associate-+l+ 95.1%

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

      4. sqr-neg 95.1%

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

      5. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   if 3.35000000000000009 < m

   1. Initial program 71.2%

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

      1. associate-*l/ 67.1%

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

      2. *-commutative 67.1%

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

      3. sqr-neg 67.1%

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

      4. associate-+l+ 67.1%

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

      5. +-commutative 67.1%

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

      6. sqr-neg 67.1%

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

      7. distribute-rgt-out 67.1%

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

      8. fma-def 67.1%

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

      9. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.35:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a_m}}\\ \mathbf{elif}\;m \leq 0.0008:\\ \;\;\;\;\frac{a_m}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot {k}^{m}\\ \end{array} \end{array} \]

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 -4.1e-19)
    (/ (pow k m) (/ 1.0 a_m))
    (if (<= m 0.0008) (/ a_m (fma (+ k 10.0) k 1.0)) (* 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 <= -4.1e-19) {
		tmp = pow(k, m) / (1.0 / a_m);
	} else if (m <= 0.0008) {
		tmp = a_m / fma((k + 10.0), k, 1.0);
	} else {
		tmp = a_m * 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 <= -4.1e-19)
		tmp = Float64((k ^ m) / Float64(1.0 / a_m));
	elseif (m <= 0.0008)
		tmp = Float64(a_m / fma(Float64(k + 10.0), k, 1.0));
	else
		tmp = Float64(a_m * (k ^ m));
	end
	return Float64(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, -4.1e-19], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0008], N[(a$95$m / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.09999999999999985e-19

   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%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -4.09999999999999985e-19 < m < 8.00000000000000038e-4

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 90.3%

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

      1. unpow2 90.3%

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

      2. distribute-rgt-in 90.3%

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

      3. +-commutative 90.3%

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

      4. *-commutative 90.3%

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

      5. fma-def 90.3%

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

      6. +-commutative 90.3%

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

   6. Applied egg-rr 90.3%

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

   if 8.00000000000000038e-4 < m

   1. Initial program 71.2%

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

      1. associate-*l/ 67.1%

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

      2. *-commutative 67.1%

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

      3. sqr-neg 67.1%

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

      4. associate-+l+ 67.1%

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

      5. +-commutative 67.1%

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

      6. sqr-neg 67.1%

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

      7. distribute-rgt-out 67.1%

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

      8. fma-def 67.1%

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

      9. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \mathbf{elif}\;m \leq 0.0008:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k + 10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19} \lor \neg \left(m \leq 0.027\right):\\ \;\;\;\;a_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

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 -4.1e-19) (not (<= m 0.027)))
    (* 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 <= -4.1e-19) || !(m <= 0.027)) {
		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 <= (-4.1d-19)) .or. (.not. (m <= 0.027d0))) 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 <= -4.1e-19) || !(m <= 0.027)) {
		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 <= -4.1e-19) or not (m <= 0.027):
		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 <= -4.1e-19) || !(m <= 0.027))
		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 <= -4.1e-19) || ~((m <= 0.027)))
		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, -4.1e-19], N[Not[LessEqual[m, 0.027]], $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 < -4.09999999999999985e-19 or 0.0269999999999999997 < m

   1. Initial program 86.5%

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

      1. associate-*l/ 84.6%

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

      2. *-commutative 84.6%

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

      3. sqr-neg 84.6%

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

      4. associate-+l+ 84.6%

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

      5. +-commutative 84.6%

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

      6. sqr-neg 84.6%

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

      7. distribute-rgt-out 84.6%

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

      8. fma-def 84.6%

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

      9. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -4.09999999999999985e-19 < m < 0.0269999999999999997

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 90.3%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19} \lor \neg \left(m \leq 0.027\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 6: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a_m}}\\ \mathbf{elif}\;m \leq 0.0016:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot {k}^{m}\\ \end{array} \end{array} \]

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 -4.1e-19)
    (/ (pow k m) (/ 1.0 a_m))
    (if (<= m 0.0016) (/ a_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 <= -4.1e-19) {
		tmp = pow(k, m) / (1.0 / a_m);
	} else if (m <= 0.0016) {
		tmp = a_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 <= (-4.1d-19)) then
        tmp = (k ** m) / (1.0d0 / a_m)
    else if (m <= 0.0016d0) then
        tmp = a_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 <= -4.1e-19) {
		tmp = Math.pow(k, m) / (1.0 / a_m);
	} else if (m <= 0.0016) {
		tmp = a_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 <= -4.1e-19:
		tmp = math.pow(k, m) / (1.0 / a_m)
	elif m <= 0.0016:
		tmp = a_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 <= -4.1e-19)
		tmp = Float64((k ^ m) / Float64(1.0 / a_m));
	elseif (m <= 0.0016)
		tmp = Float64(a_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 <= -4.1e-19)
		tmp = (k ^ m) / (1.0 / a_m);
	elseif (m <= 0.0016)
		tmp = a_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, -4.1e-19], N[(N[Power[k, m], $MachinePrecision] / N[(1.0 / a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0016], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.09999999999999985e-19

   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%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -4.09999999999999985e-19 < m < 0.00160000000000000008

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 90.3%

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

   if 0.00160000000000000008 < m

   1. Initial program 71.2%

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

      1. associate-*l/ 67.1%

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

      2. *-commutative 67.1%

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

      3. sqr-neg 67.1%

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

      4. associate-+l+ 67.1%

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

      5. +-commutative 67.1%

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

      6. sqr-neg 67.1%

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

      7. distribute-rgt-out 67.1%

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

      8. fma-def 67.1%

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

      9. +-commutative 67.1%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a}}\\ \mathbf{elif}\;m \leq 0.0016:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 7: 46.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := -10 \cdot \left(a_m \cdot k\right)\\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{a_m}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-269}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-282}:\\ \;\;\;\;\frac{\frac{a_m}{k}}{k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a_m + t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* -10.0 (* a_m k))))
   (*
    a_s
    (if (<= k -2.7e+71)
      (/ a_m (* k (+ k 10.0)))
      (if (<= k -1e-269)
        t_0
        (if (<= k 4e-282)
          (/ (/ a_m k) k)
          (if (<= k 0.098) (+ a_m t_0) (* (/ 1.0 k) (/ 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 = -10.0 * (a_m * k);
	double tmp;
	if (k <= -2.7e+71) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= -1e-269) {
		tmp = t_0;
	} else if (k <= 4e-282) {
		tmp = (a_m / k) / k;
	} else if (k <= 0.098) {
		tmp = a_m + t_0;
	} else {
		tmp = (1.0 / k) * (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 = (-10.0d0) * (a_m * k)
    if (k <= (-2.7d+71)) then
        tmp = a_m / (k * (k + 10.0d0))
    else if (k <= (-1d-269)) then
        tmp = t_0
    else if (k <= 4d-282) then
        tmp = (a_m / k) / k
    else if (k <= 0.098d0) then
        tmp = a_m + t_0
    else
        tmp = (1.0d0 / k) * (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 = -10.0 * (a_m * k);
	double tmp;
	if (k <= -2.7e+71) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= -1e-269) {
		tmp = t_0;
	} else if (k <= 4e-282) {
		tmp = (a_m / k) / k;
	} else if (k <= 0.098) {
		tmp = a_m + t_0;
	} else {
		tmp = (1.0 / k) * (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 = -10.0 * (a_m * k)
	tmp = 0
	if k <= -2.7e+71:
		tmp = a_m / (k * (k + 10.0))
	elif k <= -1e-269:
		tmp = t_0
	elif k <= 4e-282:
		tmp = (a_m / k) / k
	elif k <= 0.098:
		tmp = a_m + t_0
	else:
		tmp = (1.0 / k) * (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(-10.0 * Float64(a_m * k))
	tmp = 0.0
	if (k <= -2.7e+71)
		tmp = Float64(a_m / Float64(k * Float64(k + 10.0)));
	elseif (k <= -1e-269)
		tmp = t_0;
	elseif (k <= 4e-282)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (k <= 0.098)
		tmp = Float64(a_m + t_0);
	else
		tmp = Float64(Float64(1.0 / k) * 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 = -10.0 * (a_m * k);
	tmp = 0.0;
	if (k <= -2.7e+71)
		tmp = a_m / (k * (k + 10.0));
	elseif (k <= -1e-269)
		tmp = t_0;
	elseif (k <= 4e-282)
		tmp = (a_m / k) / k;
	elseif (k <= 0.098)
		tmp = a_m + t_0;
	else
		tmp = (1.0 / k) * (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[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -2.7e+71], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-269], t$95$0, If[LessEqual[k, 4e-282], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 0.098], N[(a$95$m + t$95$0), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if k < -2.69999999999999997e71

   1. Initial program 73.9%

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

      1. associate-*l/ 73.9%

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

      2. *-commutative 73.9%

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

      3. sqr-neg 73.9%

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

      4. associate-+l+ 73.9%

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

      5. +-commutative 73.9%

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

      6. sqr-neg 73.9%

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

      7. distribute-rgt-out 73.9%

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

      8. fma-def 73.9%

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

      9. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in k around inf 73.9%

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

      1. unpow2 73.9%

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

      2. distribute-rgt-in 73.9%

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

      3. +-commutative 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in m around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

   if -2.69999999999999997e71 < k < -9.9999999999999996e-270

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 3.7%

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

   6. Taylor expanded in k around inf 15.1%

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

   if -9.9999999999999996e-270 < k < 4.0000000000000001e-282

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 13.1%

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

   5. Taylor expanded in k around inf 61.0%

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

      1. *-un-lft-identity 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 61.0%

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

   7. Applied egg-rr 61.0%

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

      1. associate-*l/ 61.0%

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

      2. *-un-lft-identity 61.0%

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

   9. Applied egg-rr 61.0%

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

   if 4.0000000000000001e-282 < k < 0.098000000000000004

   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%

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

      2. sqr-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 54.9%

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

   5. Taylor expanded in k around 0 54.2%

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

   if 0.098000000000000004 < k

   1. Initial program 75.3%

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

      1. associate-/l* 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l+ 75.3%

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

      4. sqr-neg 75.3%

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

      5. distribute-rgt-out 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in m around 0 56.9%

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

   5. Taylor expanded in k around inf 54.9%

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

      1. *-un-lft-identity 54.9%

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

      2. unpow2 54.9%

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

      3. times-frac 62.2%

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

   7. Applied egg-rr 62.2%

      \[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-269}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-282}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 8: 47.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{a_m}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-269}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{a_m}{k}}{k}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -9.6e+70)
    (/ a_m (* k (+ k 10.0)))
    (if (<= k -1e-269)
      (* -10.0 (* a_m k))
      (if (<= k 6e-279)
        (/ (/ a_m k) k)
        (if (<= k 48000.0)
          (/ a_m (+ 1.0 (* k 10.0)))
          (* (/ 1.0 k) (/ 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 <= -9.6e+70) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= -1e-269) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 6e-279) {
		tmp = (a_m / k) / k;
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (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 <= (-9.6d+70)) then
        tmp = a_m / (k * (k + 10.0d0))
    else if (k <= (-1d-269)) then
        tmp = (-10.0d0) * (a_m * k)
    else if (k <= 6d-279) then
        tmp = (a_m / k) / k
    else if (k <= 48000.0d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = (1.0d0 / k) * (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 <= -9.6e+70) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= -1e-269) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 6e-279) {
		tmp = (a_m / k) / k;
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (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 <= -9.6e+70:
		tmp = a_m / (k * (k + 10.0))
	elif k <= -1e-269:
		tmp = -10.0 * (a_m * k)
	elif k <= 6e-279:
		tmp = (a_m / k) / k
	elif k <= 48000.0:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = (1.0 / k) * (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 <= -9.6e+70)
		tmp = Float64(a_m / Float64(k * Float64(k + 10.0)));
	elseif (k <= -1e-269)
		tmp = Float64(-10.0 * Float64(a_m * k));
	elseif (k <= 6e-279)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (k <= 48000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(1.0 / k) * 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 <= -9.6e+70)
		tmp = a_m / (k * (k + 10.0));
	elseif (k <= -1e-269)
		tmp = -10.0 * (a_m * k);
	elseif (k <= 6e-279)
		tmp = (a_m / k) / k;
	elseif (k <= 48000.0)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = (1.0 / k) * (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[k, -9.6e+70], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-269], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e-279], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 48000.0], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if k < -9.59999999999999947e70

   1. Initial program 73.9%

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

      1. associate-*l/ 73.9%

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

      2. *-commutative 73.9%

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

      3. sqr-neg 73.9%

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

      4. associate-+l+ 73.9%

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

      5. +-commutative 73.9%

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

      6. sqr-neg 73.9%

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

      7. distribute-rgt-out 73.9%

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

      8. fma-def 73.9%

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

      9. +-commutative 73.9%

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

   3. Simplified 73.9%

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

   4. Taylor expanded in k around inf 73.9%

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

      1. unpow2 73.9%

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

      2. distribute-rgt-in 73.9%

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

      3. +-commutative 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in m around 0 61.6%

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

      1. +-commutative 61.6%

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

   9. Simplified 61.6%

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

   if -9.59999999999999947e70 < k < -9.9999999999999996e-270

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 3.7%

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

   6. Taylor expanded in k around inf 15.1%

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

   if -9.9999999999999996e-270 < k < 5.9999999999999999e-279

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 13.1%

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

   5. Taylor expanded in k around inf 61.0%

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

      1. *-un-lft-identity 61.0%

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

      2. unpow2 61.0%

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

      3. times-frac 61.0%

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

   7. Applied egg-rr 61.0%

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

      1. associate-*l/ 61.0%

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

      2. *-un-lft-identity 61.0%

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

   9. Applied egg-rr 61.0%

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

   if 5.9999999999999999e-279 < k < 48000

   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. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 53.8%

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

   5. Taylor expanded in k around 0 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   if 48000 < k

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-+l+ 74.3%

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

      4. sqr-neg 74.3%

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

      5. distribute-rgt-out 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in m around 0 58.1%

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

   5. Taylor expanded in k around inf 57.0%

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

      1. *-un-lft-identity 57.0%

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

      2. unpow2 57.0%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

      \[\leadsto \color{blue}{\frac{1}{k} \cdot \frac{a}{k}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-269}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 9: 47.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-281}:\\ \;\;\;\;\frac{a_m}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k + 10\right) \cdot \frac{k}{a_m}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 1.7e-281)
    (/ a_m (/ k (/ 1.0 k)))
    (if (<= k 48000.0)
      (/ a_m (+ 1.0 (* k 10.0)))
      (/ 1.0 (* (+ k 10.0) (/ 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 <= 1.7e-281) {
		tmp = a_m / (k / (1.0 / k));
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / ((k + 10.0) * (k / 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 <= 1.7d-281) then
        tmp = a_m / (k / (1.0d0 / k))
    else if (k <= 48000.0d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = 1.0d0 / ((k + 10.0d0) * (k / 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 <= 1.7e-281) {
		tmp = a_m / (k / (1.0 / k));
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / ((k + 10.0) * (k / 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 <= 1.7e-281:
		tmp = a_m / (k / (1.0 / k))
	elif k <= 48000.0:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = 1.0 / ((k + 10.0) * (k / 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 <= 1.7e-281)
		tmp = Float64(a_m / Float64(k / Float64(1.0 / k)));
	elseif (k <= 48000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(1.0 / Float64(Float64(k + 10.0) * Float64(k / 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 <= 1.7e-281)
		tmp = a_m / (k / (1.0 / k));
	elseif (k <= 48000.0)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = 1.0 / ((k + 10.0) * (k / 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[LessEqual[k, 1.7e-281], N[(a$95$m / N[(k / N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 48000.0], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k + 10.0), $MachinePrecision] * N[(k / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.7e-281

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 24.9%

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

   5. Taylor expanded in k around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. unpow2 34.6%

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

      3. times-frac 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-*l/ 28.5%

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

      3. associate-/l* 34.6%

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

   9. Applied egg-rr 34.6%

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

   if 1.7e-281 < k < 48000

   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. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 53.8%

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

   5. Taylor expanded in k around 0 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   if 48000 < k

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-+l+ 74.3%

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

      4. sqr-neg 74.3%

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

      5. distribute-rgt-out 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in m around 0 58.1%

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

      1. clear-num 58.0%

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

      2. distribute-rgt-in 58.0%

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

      3. unpow2 58.0%

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

      4. inv-pow 58.0%

         \[\leadsto \color{blue}{{\left(\frac{1 + \left(10 \cdot k + {k}^{2}\right)}{a}\right)}^{-1}} \]

      5. unpow2 58.0%

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

      6. distribute-rgt-in 58.0%

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

      7. +-commutative 58.0%

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

      8. fma-def 58.0%

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

      9. +-commutative 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. unpow-1 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in k around inf 58.0%

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

       1. unpow2 72.1%

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

       2. distribute-rgt-in 72.1%

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

       3. +-commutative 72.1%

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

   11. Simplified 58.0%

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

       1. associate-/l* 65.0%

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

       2. associate-/r/ 65.0%

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

   13. Applied egg-rr 65.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-281}:\\ \;\;\;\;\frac{a}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(k + 10\right) \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 10: 58.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -650:\\ \;\;\;\;\frac{a_m}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;m \leq 3.7:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

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 -650.0)
    (/ a_m (/ k (/ 1.0 k)))
    (if (<= m 3.7) (/ a_m (+ 1.0 (* k (+ 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 <= -650.0) {
		tmp = a_m / (k / (1.0 / k));
	} else if (m <= 3.7) {
		tmp = a_m / (1.0 + (k * (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 <= (-650.0d0)) then
        tmp = a_m / (k / (1.0d0 / k))
    else if (m <= 3.7d0) then
        tmp = a_m / (1.0d0 + (k * (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 <= -650.0) {
		tmp = a_m / (k / (1.0 / k));
	} else if (m <= 3.7) {
		tmp = a_m / (1.0 + (k * (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 <= -650.0:
		tmp = a_m / (k / (1.0 / k))
	elif m <= 3.7:
		tmp = a_m / (1.0 + (k * (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 <= -650.0)
		tmp = Float64(a_m / Float64(k / Float64(1.0 / k)));
	elseif (m <= 3.7)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 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 <= -650.0)
		tmp = a_m / (k / (1.0 / k));
	elseif (m <= 3.7)
		tmp = a_m / (1.0 + (k * (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, -650.0], N[(a$95$m / N[(k / N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.7], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -650

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 34.4%

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

   5. Taylor expanded in k around inf 58.5%

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

      1. *-un-lft-identity 58.5%

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

      2. unpow2 58.5%

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

      3. times-frac 49.5%

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

   7. Applied egg-rr 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l/ 49.5%

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

      3. associate-/l* 58.5%

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

   9. Applied egg-rr 58.5%

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

   if -650 < m < 3.7000000000000002

   1. Initial program 91.6%

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

      1. associate-/l* 91.6%

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

      2. sqr-neg 91.6%

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

      3. associate-+l+ 91.6%

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

      4. sqr-neg 91.6%

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

      5. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in m around 0 87.9%

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

   if 3.7000000000000002 < m

   1. Initial program 71.2%

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

      1. associate-/l* 71.2%

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

      2. sqr-neg 71.2%

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

      3. associate-+l+ 71.2%

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

      4. sqr-neg 71.2%

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

      5. distribute-rgt-out 71.2%

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

   3. Simplified 71.2%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 5.5%

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

   6. Taylor expanded in k around inf 17.5%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -650:\\ \;\;\;\;\frac{a}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;m \leq 3.7:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 45.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{a_m}{k}}{k}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 3e-279)
    (/ (/ a_m k) k)
    (if (<= k 48000.0) a_m (* (/ 1.0 k) (/ 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 <= 3e-279) {
		tmp = (a_m / k) / k;
	} else if (k <= 48000.0) {
		tmp = a_m;
	} else {
		tmp = (1.0 / k) * (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 <= 3d-279) then
        tmp = (a_m / k) / k
    else if (k <= 48000.0d0) then
        tmp = a_m
    else
        tmp = (1.0d0 / k) * (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 <= 3e-279) {
		tmp = (a_m / k) / k;
	} else if (k <= 48000.0) {
		tmp = a_m;
	} else {
		tmp = (1.0 / k) * (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 <= 3e-279:
		tmp = (a_m / k) / k
	elif k <= 48000.0:
		tmp = a_m
	else:
		tmp = (1.0 / k) * (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 <= 3e-279)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (k <= 48000.0)
		tmp = a_m;
	else
		tmp = Float64(Float64(1.0 / k) * 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 <= 3e-279)
		tmp = (a_m / k) / k;
	elseif (k <= 48000.0)
		tmp = a_m;
	else
		tmp = (1.0 / k) * (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[k, 3e-279], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 48000.0], a$95$m, N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3e-279

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 24.9%

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

   5. Taylor expanded in k around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. unpow2 34.6%

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

      3. times-frac 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. associate-*l/ 28.5%

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

      2. *-un-lft-identity 28.5%

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

   9. Applied egg-rr 28.5%

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

   if 3e-279 < k < 48000

   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%

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

      2. sqr-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 53.8%

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

   5. Taylor expanded in k around 0 52.0%

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

   if 48000 < k

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-+l+ 74.3%

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

      4. sqr-neg 74.3%

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

      5. distribute-rgt-out 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in m around 0 58.1%

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

   5. Taylor expanded in k around inf 57.0%

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

      1. *-un-lft-identity 57.0%

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

      2. unpow2 57.0%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-279}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 12: 46.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{\frac{a_m}{k}}{k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a_m + -10 \cdot \left(a_m \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 6.8e-282)
    (/ (/ a_m k) k)
    (if (<= k 0.098) (+ a_m (* -10.0 (* a_m k))) (* (/ 1.0 k) (/ 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 <= 6.8e-282) {
		tmp = (a_m / k) / k;
	} else if (k <= 0.098) {
		tmp = a_m + (-10.0 * (a_m * k));
	} else {
		tmp = (1.0 / k) * (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 <= 6.8d-282) then
        tmp = (a_m / k) / k
    else if (k <= 0.098d0) then
        tmp = a_m + ((-10.0d0) * (a_m * k))
    else
        tmp = (1.0d0 / k) * (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 <= 6.8e-282) {
		tmp = (a_m / k) / k;
	} else if (k <= 0.098) {
		tmp = a_m + (-10.0 * (a_m * k));
	} else {
		tmp = (1.0 / k) * (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 <= 6.8e-282:
		tmp = (a_m / k) / k
	elif k <= 0.098:
		tmp = a_m + (-10.0 * (a_m * k))
	else:
		tmp = (1.0 / k) * (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 <= 6.8e-282)
		tmp = Float64(Float64(a_m / k) / k);
	elseif (k <= 0.098)
		tmp = Float64(a_m + Float64(-10.0 * Float64(a_m * k)));
	else
		tmp = Float64(Float64(1.0 / k) * 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 <= 6.8e-282)
		tmp = (a_m / k) / k;
	elseif (k <= 0.098)
		tmp = a_m + (-10.0 * (a_m * k));
	else
		tmp = (1.0 / k) * (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[k, 6.8e-282], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[k, 0.098], N[(a$95$m + N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 6.79999999999999997e-282

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 24.9%

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

   5. Taylor expanded in k around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. unpow2 34.6%

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

      3. times-frac 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. associate-*l/ 28.5%

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

      2. *-un-lft-identity 28.5%

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

   9. Applied egg-rr 28.5%

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

   if 6.79999999999999997e-282 < k < 0.098000000000000004

   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%

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

      2. sqr-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 54.9%

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

   5. Taylor expanded in k around 0 54.2%

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

   if 0.098000000000000004 < k

   1. Initial program 75.3%

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

      1. associate-/l* 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l+ 75.3%

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

      4. sqr-neg 75.3%

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

      5. distribute-rgt-out 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in m around 0 56.9%

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

   5. Taylor expanded in k around inf 54.9%

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

      1. *-un-lft-identity 54.9%

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

      2. unpow2 54.9%

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

      3. times-frac 62.2%

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

   7. Applied egg-rr 62.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.8 \cdot 10^{-282}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 13: 46.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a_m}}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a_m + -10 \cdot \left(a_m \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 1.7e-280)
    (/ 1.0 (* k (/ k a_m)))
    (if (<= k 0.098) (+ a_m (* -10.0 (* a_m k))) (* (/ 1.0 k) (/ 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 <= 1.7e-280) {
		tmp = 1.0 / (k * (k / a_m));
	} else if (k <= 0.098) {
		tmp = a_m + (-10.0 * (a_m * k));
	} else {
		tmp = (1.0 / k) * (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 <= 1.7d-280) then
        tmp = 1.0d0 / (k * (k / a_m))
    else if (k <= 0.098d0) then
        tmp = a_m + ((-10.0d0) * (a_m * k))
    else
        tmp = (1.0d0 / k) * (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 <= 1.7e-280) {
		tmp = 1.0 / (k * (k / a_m));
	} else if (k <= 0.098) {
		tmp = a_m + (-10.0 * (a_m * k));
	} else {
		tmp = (1.0 / k) * (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 <= 1.7e-280:
		tmp = 1.0 / (k * (k / a_m))
	elif k <= 0.098:
		tmp = a_m + (-10.0 * (a_m * k))
	else:
		tmp = (1.0 / k) * (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 <= 1.7e-280)
		tmp = Float64(1.0 / Float64(k * Float64(k / a_m)));
	elseif (k <= 0.098)
		tmp = Float64(a_m + Float64(-10.0 * Float64(a_m * k)));
	else
		tmp = Float64(Float64(1.0 / k) * 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 <= 1.7e-280)
		tmp = 1.0 / (k * (k / a_m));
	elseif (k <= 0.098)
		tmp = a_m + (-10.0 * (a_m * k));
	else
		tmp = (1.0 / k) * (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[k, 1.7e-280], N[(1.0 / N[(k * N[(k / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.098], N[(a$95$m + N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.6999999999999999e-280

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 24.9%

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

   5. Taylor expanded in k around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. unpow2 34.6%

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

      3. times-frac 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. *-commutative 28.5%

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

      2. clear-num 28.5%

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

      3. frac-times 29.0%

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

      4. metadata-eval 29.0%

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

   9. Applied egg-rr 29.0%

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

   if 1.6999999999999999e-280 < k < 0.098000000000000004

   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%

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

      2. sqr-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 54.9%

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

   5. Taylor expanded in k around 0 54.2%

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

   if 0.098000000000000004 < k

   1. Initial program 75.3%

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

      1. associate-/l* 75.3%

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

      2. sqr-neg 75.3%

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

      3. associate-+l+ 75.3%

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

      4. sqr-neg 75.3%

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

      5. distribute-rgt-out 75.3%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in m around 0 56.9%

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

   5. Taylor expanded in k around inf 54.9%

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

      1. *-un-lft-identity 54.9%

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

      2. unpow2 54.9%

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

      3. times-frac 62.2%

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

   7. Applied egg-rr 62.2%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \mathbf{elif}\;k \leq 0.098:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 14: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-281}:\\ \;\;\;\;\frac{a_m}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 2.3e-281)
    (/ a_m (/ k (/ 1.0 k)))
    (if (<= k 48000.0) (/ a_m (+ 1.0 (* k 10.0))) (* (/ 1.0 k) (/ 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 <= 2.3e-281) {
		tmp = a_m / (k / (1.0 / k));
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (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 <= 2.3d-281) then
        tmp = a_m / (k / (1.0d0 / k))
    else if (k <= 48000.0d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = (1.0d0 / k) * (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 <= 2.3e-281) {
		tmp = a_m / (k / (1.0 / k));
	} else if (k <= 48000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (1.0 / k) * (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 <= 2.3e-281:
		tmp = a_m / (k / (1.0 / k))
	elif k <= 48000.0:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = (1.0 / k) * (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 <= 2.3e-281)
		tmp = Float64(a_m / Float64(k / Float64(1.0 / k)));
	elseif (k <= 48000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(1.0 / k) * 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 <= 2.3e-281)
		tmp = a_m / (k / (1.0 / k));
	elseif (k <= 48000.0)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = (1.0 / k) * (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[k, 2.3e-281], N[(a$95$m / N[(k / N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 48000.0], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 2.29999999999999989e-281

   1. Initial program 91.0%

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

      1. associate-/l* 91.0%

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

      2. sqr-neg 91.0%

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

      3. associate-+l+ 91.0%

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

      4. sqr-neg 91.0%

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

      5. distribute-rgt-out 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 24.9%

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

   5. Taylor expanded in k around inf 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. unpow2 34.6%

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

      3. times-frac 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. *-commutative 28.5%

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

      2. associate-*l/ 28.5%

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

      3. associate-/l* 34.6%

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

   9. Applied egg-rr 34.6%

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

   if 2.29999999999999989e-281 < k < 48000

   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. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 53.8%

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

   5. Taylor expanded in k around 0 52.4%

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

      1. *-commutative 52.4%

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

   7. Simplified 52.4%

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

   if 48000 < k

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. sqr-neg 74.3%

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

      3. associate-+l+ 74.3%

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

      4. sqr-neg 74.3%

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

      5. distribute-rgt-out 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in m around 0 58.1%

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

   5. Taylor expanded in k around inf 57.0%

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

      1. *-un-lft-identity 57.0%

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

      2. unpow2 57.0%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{-281}:\\ \;\;\;\;\frac{a}{\frac{k}{\frac{1}{k}}}\\ \mathbf{elif}\;k \leq 48000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 15: 45.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 10^{-280} \lor \neg \left(k \leq 48000\right):\\ \;\;\;\;\frac{\frac{a_m}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a_m\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (or (<= k 1e-280) (not (<= k 48000.0))) (/ (/ 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 <= 1e-280) || !(k <= 48000.0)) {
		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 <= 1d-280) .or. (.not. (k <= 48000.0d0))) 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 <= 1e-280) || !(k <= 48000.0)) {
		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 <= 1e-280) or not (k <= 48000.0):
		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 <= 1e-280) || !(k <= 48000.0))
		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 <= 1e-280) || ~((k <= 48000.0)))
		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, 1e-280], N[Not[LessEqual[k, 48000.0]], $MachinePrecision]], N[(N[(a$95$m / k), $MachinePrecision] / k), $MachinePrecision], a$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999996e-281 or 48000 < k

   1. Initial program 81.3%

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

      1. associate-/l* 81.3%

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

      2. sqr-neg 81.3%

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

      3. associate-+l+ 81.3%

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

      4. sqr-neg 81.3%

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

      5. distribute-rgt-out 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in m around 0 44.2%

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

   5. Taylor expanded in k around inf 47.6%

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

      1. *-un-lft-identity 47.6%

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

      2. unpow2 47.6%

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

      3. times-frac 49.4%

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

   7. Applied egg-rr 49.4%

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

      1. associate-*l/ 49.4%

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

      2. *-un-lft-identity 49.4%

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

   9. Applied egg-rr 49.4%

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

   if 9.9999999999999996e-281 < k < 48000

   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%

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

      2. sqr-neg 99.9%

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

      3. associate-+l+ 99.9%

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

      4. sqr-neg 99.9%

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

      5. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in m around 0 53.8%

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

   5. Taylor expanded in k around 0 52.0%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-280} \lor \neg \left(k \leq 48000\right):\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 16: 25.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

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 2.8e+36) 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 <= 2.8e+36) {
		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 <= 2.8d+36) 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 <= 2.8e+36) {
		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 <= 2.8e+36:
		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 <= 2.8e+36)
		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 <= 2.8e+36)
		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, 2.8e+36], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.8000000000000001e36

   1. Initial program 94.6%

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

      1. associate-/l* 94.6%

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

      2. sqr-neg 94.6%

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

      3. associate-+l+ 94.6%

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

      4. sqr-neg 94.6%

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

      5. distribute-rgt-out 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in m around 0 65.0%

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

   5. Taylor expanded in k around 0 29.5%

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

   if 2.8000000000000001e36 < m

   1. Initial program 71.8%

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

      1. associate-/l* 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-+l+ 71.8%

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

      4. sqr-neg 71.8%

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

      5. distribute-rgt-out 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in m around 0 3.0%

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

   5. Taylor expanded in k around 0 5.6%

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

   6. Taylor expanded in k around inf 17.9%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 17: 20.4% 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 88.3%

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

   1. associate-/l* 88.3%

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

   2. sqr-neg 88.3%

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

   3. associate-+l+ 88.3%

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

   4. sqr-neg 88.3%

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

   5. distribute-rgt-out 88.3%

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

3. Simplified 88.3%

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

4. Taylor expanded in m around 0 47.8%

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

5. Taylor expanded in k around 0 22.3%

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

6. Final simplification 22.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023319 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))