Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.8%

Time: 12.9s

Alternatives: 12

Speedup: 1.0×

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a_m \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a_m}{k + 10}\\ \mathbf{elif}\;t_1 \leq 10^{+255}:\\ \;\;\;\;{k}^{m} \cdot \frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 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 0.0)
      (* (/ (pow k m) k) (/ a_m (+ k 10.0)))
      (if (<= t_1 1e+255)
        (* (pow k m) (/ a_m (+ 1.0 (* k (+ k 10.0)))))
        t_0)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (pow(k, m) / k) * (a_m / (k + 10.0));
	} else if (t_1 <= 1e+255) {
		tmp = pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: 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 <= 0.0d0) then
        tmp = ((k ** m) / k) * (a_m / (k + 10.0d0))
    else if (t_1 <= 1d+255) then
        tmp = (k ** m) * (a_m / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (Math.pow(k, m) / k) * (a_m / (k + 10.0));
	} else if (t_1 <= 1e+255) {
		tmp = Math.pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (math.pow(k, m) / k) * (a_m / (k + 10.0))
	elif t_1 <= 1e+255:
		tmp = math.pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64((k ^ m) / k) * Float64(a_m / Float64(k + 10.0)));
	elseif (t_1 <= 1e+255)
		tmp = Float64((k ^ m) * Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = ((k ^ m) / k) * (a_m / (k + 10.0));
	elseif (t_1 <= 1e+255)
		tmp = (k ^ m) * (a_m / (1.0 + (k * (k + 10.0))));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, 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, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a$95$m / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+255], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 0.0

   1. Initial program 97.5%

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

      1. associate-*l/ 96.5%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. *-commutative 96.5%

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

      2. clear-num 96.4%

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

      3. un-div-inv 96.4%

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

      4. +-commutative 96.4%

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

      5. fma-def 96.4%

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

      6. +-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in k around inf 72.9%

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

      1. +-commutative 72.9%

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

      2. unpow2 72.9%

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

      3. distribute-rgt-in 72.9%

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

   9. Simplified 72.9%

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

       1. associate-/l* 75.3%

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

       2. associate-/r/ 82.3%

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

   11. Applied egg-rr 82.3%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 9.99999999999999988e254

   1. Initial program 99.8%

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

      1. associate-*l/ 99.8%

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

      2. sqr-neg 99.8%

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

      3. associate-+l+ 99.8%

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

      4. sqr-neg 99.8%

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

      5. distribute-rgt-out 99.8%

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

   3. Simplified 99.8%

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

   if 9.99999999999999988e254 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 66.7%

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

      1. associate-*l/ 62.2%

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

      2. sqr-neg 62.2%

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

      3. associate-+l+ 62.2%

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

      4. sqr-neg 62.2%

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

      5. distribute-rgt-out 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+255}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.2× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (cbrt (* a_m (pow k m)))))
   (* a_s (* (/ (pow t_0 2.0) (hypot 1.0 k)) (/ t_0 (hypot 1.0 k))))))

C

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

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 = Math.cbrt((a_m * Math.pow(k, m)));
	return a_s * ((Math.pow(t_0, 2.0) / Math.hypot(1.0, k)) * (t_0 / Math.hypot(1.0, k)));
}

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = cbrt(Float64(a_m * (k ^ m)))
	return Float64(a_s * Float64(Float64((t_0 ^ 2.0) / hypot(1.0, k)) * Float64(t_0 / hypot(1.0, k))))
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[Power[N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(a$95$s * N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 92.3%

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

   1. *-commutative 92.3%

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

3. Simplified 92.3%

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

5. Taylor expanded in k around 0 91.6%

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

   1. add-cube-cbrt 91.2%

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

   2. add-sqr-sqrt 91.2%

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

   3. times-frac 91.2%

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

   4. pow2 91.2%

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

   5. hypot-1-def 91.2%

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

   6. hypot-1-def 98.8%

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

7. Applied egg-rr 98.8%

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

8. Final simplification 98.8%

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

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a_m \cdot {k}^{m}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+255}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a_m}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

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+255)
      (* (/ (pow k m) (hypot 1.0 k)) (/ a_m (hypot 1.0 k)))
      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+255) {
		tmp = (pow(k, m) / hypot(1.0, k)) * (a_m / hypot(1.0, k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+255) {
		tmp = (Math.pow(k, m) / Math.hypot(1.0, k)) * (a_m / Math.hypot(1.0, k));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+255:
		tmp = (math.pow(k, m) / math.hypot(1.0, k)) * (a_m / math.hypot(1.0, k))
	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+255)
		tmp = Float64(Float64((k ^ m) / hypot(1.0, k)) * Float64(a_m / hypot(1.0, k)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+255)
		tmp = ((k ^ m) / hypot(1.0, k)) * (a_m / hypot(1.0, k));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+255], N[(N[(N[Power[k, m], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a$95$m / N[Sqrt[1.0 ^ 2 + k ^ 2], $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))) < 9.99999999999999988e254

   1. Initial program 97.8%

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

      1. *-commutative 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in k around 0 96.9%

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

      1. *-commutative 96.9%

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

      2. add-sqr-sqrt 96.9%

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

      3. times-frac 95.9%

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

      4. hypot-1-def 95.9%

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

      5. hypot-1-def 98.1%

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

   7. Applied egg-rr 98.1%

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

   if 9.99999999999999988e254 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 66.7%

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

      1. associate-*l/ 62.2%

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

      2. sqr-neg 62.2%

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

      3. associate-+l+ 62.2%

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

      4. sqr-neg 62.2%

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

      5. distribute-rgt-out 62.2%

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

   3. Simplified 62.2%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+255}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.0× speedup

Math

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

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 0.1) (* a_m (pow k m)) (* (/ (pow k m) k) (/ a_m (+ k 10.0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = a_m * pow(k, m);
	} else {
		tmp = (pow(k, m) / k) * (a_m / (k + 10.0));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.1d0) then
        tmp = a_m * (k ** m)
    else
        tmp = ((k ** m) / k) * (a_m / (k + 10.0d0))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = a_m * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) / k) * (a_m / (k + 10.0));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= 0.1:
		tmp = a_m * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) / k) * (a_m / (k + 10.0))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= 0.1)
		tmp = Float64(a_m * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) / k) * Float64(a_m / Float64(k + 10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= 0.1)
		tmp = a_m * (k ^ m);
	else
		tmp = ((k ^ m) / k) * (a_m / (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[LessEqual[k, 0.1], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / k), $MachinePrecision] * N[(a$95$m / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 94.9%

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

      1. associate-*l/ 93.2%

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

      2. sqr-neg 93.2%

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

      3. associate-+l+ 93.2%

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

      4. sqr-neg 93.2%

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

      5. distribute-rgt-out 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in k around 0 99.0%

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

   if 0.10000000000000001 < k

   1. Initial program 86.7%

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

      1. associate-*l/ 85.4%

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

      2. sqr-neg 85.4%

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

      3. associate-+l+ 85.4%

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

      4. sqr-neg 85.4%

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

      5. distribute-rgt-out 85.5%

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

   3. Simplified 85.5%

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

      1. *-commutative 85.5%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

      4. +-commutative 84.2%

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

      5. fma-def 84.2%

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

      6. +-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

   7. Taylor expanded in k around inf 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. distribute-rgt-in 84.2%

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

   9. Simplified 84.2%

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

       1. associate-/l* 89.9%

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

       2. associate-/r/ 94.9%

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

   11. Applied egg-rr 94.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{k} \cdot \frac{a}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-10} \lor \neg \left(m \leq 2.5 \cdot 10^{-16}\right):\\ \;\;\;\;a_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

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 -1.02e-10) (not (<= m 2.5e-16)))
    (* 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 <= -1.02e-10) || !(m <= 2.5e-16)) {
		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 <= (-1.02d-10)) .or. (.not. (m <= 2.5d-16))) 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 <= -1.02e-10) || !(m <= 2.5e-16)) {
		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 <= -1.02e-10) or not (m <= 2.5e-16):
		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 <= -1.02e-10) || !(m <= 2.5e-16))
		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 <= -1.02e-10) || ~((m <= 2.5e-16)))
		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, -1.02e-10], N[Not[LessEqual[m, 2.5e-16]], $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 < -1.01999999999999997e-10 or 2.5000000000000002e-16 < m

   1. Initial program 90.7%

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

      1. associate-*l/ 88.3%

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

      2. sqr-neg 88.3%

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

      3. associate-+l+ 88.3%

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

      4. sqr-neg 88.3%

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

      5. distribute-rgt-out 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -1.01999999999999997e-10 < m < 2.5000000000000002e-16

   1. Initial program 95.0%

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

      1. associate-*l/ 95.0%

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

      2. sqr-neg 95.0%

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

      3. associate-+l+ 95.0%

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

      4. sqr-neg 95.0%

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

      5. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in m around 0 94.6%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.02 \cdot 10^{-10} \lor \neg \left(m \leq 2.5 \cdot 10^{-16}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a_m}{k \cdot \left(k + 10\right)}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 3.05 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a_m \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

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 (* k (+ k 10.0)))))
   (*
    a_s
    (if (<= k -1.55e+32)
      t_0
      (if (<= k 3.05e-294)
        (* -10.0 (* a_m k))
        (if (<= k 0.075) (* a_m (+ 1.0 (* k -10.0))) t_0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * (k + 10.0));
	double tmp;
	if (k <= -1.55e+32) {
		tmp = t_0;
	} else if (k <= 3.05e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.075) {
		tmp = a_m * (1.0 + (k * -10.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m / (k * (k + 10.0d0))
    if (k <= (-1.55d+32)) then
        tmp = t_0
    else if (k <= 3.05d-294) then
        tmp = (-10.0d0) * (a_m * k)
    else if (k <= 0.075d0) then
        tmp = a_m * (1.0d0 + (k * (-10.0d0)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * (k + 10.0));
	double tmp;
	if (k <= -1.55e+32) {
		tmp = t_0;
	} else if (k <= 3.05e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.075) {
		tmp = a_m * (1.0 + (k * -10.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m / (k * (k + 10.0))
	tmp = 0
	if k <= -1.55e+32:
		tmp = t_0
	elif k <= 3.05e-294:
		tmp = -10.0 * (a_m * k)
	elif k <= 0.075:
		tmp = a_m * (1.0 + (k * -10.0))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * Float64(k + 10.0)))
	tmp = 0.0
	if (k <= -1.55e+32)
		tmp = t_0;
	elseif (k <= 3.05e-294)
		tmp = Float64(-10.0 * Float64(a_m * k));
	elseif (k <= 0.075)
		tmp = Float64(a_m * Float64(1.0 + Float64(k * -10.0)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m / (k * (k + 10.0));
	tmp = 0.0;
	if (k <= -1.55e+32)
		tmp = t_0;
	elseif (k <= 3.05e-294)
		tmp = -10.0 * (a_m * k);
	elseif (k <= 0.075)
		tmp = a_m * (1.0 + (k * -10.0));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -1.55e+32], t$95$0, If[LessEqual[k, 3.05e-294], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a$95$m * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < -1.54999999999999997e32 or 0.0749999999999999972 < k

   1. Initial program 83.3%

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

      1. associate-*l/ 80.0%

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

      2. sqr-neg 80.0%

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

      3. associate-+l+ 80.0%

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

      4. sqr-neg 80.0%

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

      5. distribute-rgt-out 80.0%

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

   3. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.1%

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

      4. +-commutative 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in k around inf 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. distribute-rgt-in 79.1%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in m around 0 54.4%

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

   if -1.54999999999999997e32 < k < 3.0500000000000001e-294

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 3.7%

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

   7. Taylor expanded in k around inf 22.7%

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

   if 3.0500000000000001e-294 < k < 0.0749999999999999972

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 51.5%

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

   6. Taylor expanded in k around 0 51.1%

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

   7. Taylor expanded in a around 0 51.1%

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

      1. *-commutative 51.1%

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

   9. Simplified 51.1%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.55 \cdot 10^{+32}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 3.05 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a_m}{k \cdot \left(k + 10\right)}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -6.1 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \mathbf{elif}\;k \leq 0.65:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k (+ k 10.0)))))
   (*
    a_s
    (if (<= k -6.1e+43)
      t_0
      (if (<= k 2.15e-294)
        (* -10.0 (* a_m k))
        (if (<= k 0.65) (/ a_m (+ 1.0 (* k 10.0))) t_0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * (k + 10.0));
	double tmp;
	if (k <= -6.1e+43) {
		tmp = t_0;
	} else if (k <= 2.15e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.65) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m / (k * (k + 10.0d0))
    if (k <= (-6.1d+43)) then
        tmp = t_0
    else if (k <= 2.15d-294) then
        tmp = (-10.0d0) * (a_m * k)
    else if (k <= 0.65d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * (k + 10.0));
	double tmp;
	if (k <= -6.1e+43) {
		tmp = t_0;
	} else if (k <= 2.15e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.65) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m / (k * (k + 10.0))
	tmp = 0
	if k <= -6.1e+43:
		tmp = t_0
	elif k <= 2.15e-294:
		tmp = -10.0 * (a_m * k)
	elif k <= 0.65:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * Float64(k + 10.0)))
	tmp = 0.0
	if (k <= -6.1e+43)
		tmp = t_0;
	elseif (k <= 2.15e-294)
		tmp = Float64(-10.0 * Float64(a_m * k));
	elseif (k <= 0.65)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m / (k * (k + 10.0));
	tmp = 0.0;
	if (k <= -6.1e+43)
		tmp = t_0;
	elseif (k <= 2.15e-294)
		tmp = -10.0 * (a_m * k);
	elseif (k <= 0.65)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[k, -6.1e+43], t$95$0, If[LessEqual[k, 2.15e-294], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.65], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if k < -6.0999999999999998e43 or 0.650000000000000022 < k

   1. Initial program 83.3%

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

      1. associate-*l/ 80.0%

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

      2. sqr-neg 80.0%

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

      3. associate-+l+ 80.0%

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

      4. sqr-neg 80.0%

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

      5. distribute-rgt-out 80.0%

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

   3. Simplified 80.0%

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

      1. *-commutative 80.0%

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

      2. clear-num 79.1%

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

      3. un-div-inv 79.1%

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

      4. +-commutative 79.1%

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

      5. fma-def 79.1%

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

      6. +-commutative 79.1%

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

   6. Applied egg-rr 79.1%

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

   7. Taylor expanded in k around inf 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. distribute-rgt-in 79.1%

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

   9. Simplified 79.1%

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

   10. Taylor expanded in m around 0 54.4%

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

   if -6.0999999999999998e43 < k < 2.1500000000000001e-294

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 3.7%

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

   7. Taylor expanded in k around inf 22.7%

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

   if 2.1500000000000001e-294 < k < 0.650000000000000022

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 51.5%

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

   6. Taylor expanded in k around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.65:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{+24}:\\ \;\;\;\;\frac{a_m}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \mathbf{elif}\;k \leq 0.65:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a_m}{k}}{k + 10}\\ \end{array} \end{array} \]

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 -1.75e+24)
    (/ a_m (* k (+ k 10.0)))
    (if (<= k 2.15e-294)
      (* -10.0 (* a_m k))
      (if (<= k 0.65) (/ a_m (+ 1.0 (* k 10.0))) (/ (/ a_m 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 (k <= -1.75e+24) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= 2.15e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.65) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (a_m / 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 (k <= (-1.75d+24)) then
        tmp = a_m / (k * (k + 10.0d0))
    else if (k <= 2.15d-294) then
        tmp = (-10.0d0) * (a_m * k)
    else if (k <= 0.65d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a_m / 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 (k <= -1.75e+24) {
		tmp = a_m / (k * (k + 10.0));
	} else if (k <= 2.15e-294) {
		tmp = -10.0 * (a_m * k);
	} else if (k <= 0.65) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = (a_m / 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 k <= -1.75e+24:
		tmp = a_m / (k * (k + 10.0))
	elif k <= 2.15e-294:
		tmp = -10.0 * (a_m * k)
	elif k <= 0.65:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = (a_m / 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 (k <= -1.75e+24)
		tmp = Float64(a_m / Float64(k * Float64(k + 10.0)));
	elseif (k <= 2.15e-294)
		tmp = Float64(-10.0 * Float64(a_m * k));
	elseif (k <= 0.65)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a_m / 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 (k <= -1.75e+24)
		tmp = a_m / (k * (k + 10.0));
	elseif (k <= 2.15e-294)
		tmp = -10.0 * (a_m * k);
	elseif (k <= 0.65)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = (a_m / 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[LessEqual[k, -1.75e+24], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.15e-294], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.65], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a$95$m / k), $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if k < -1.7500000000000001e24

   1. Initial program 76.3%

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

      1. associate-*l/ 68.4%

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

      2. sqr-neg 68.4%

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

      3. associate-+l+ 68.4%

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

      4. sqr-neg 68.4%

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

      5. distribute-rgt-out 68.4%

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

   3. Simplified 68.4%

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

      1. *-commutative 68.4%

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

      2. clear-num 68.4%

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

      3. un-div-inv 68.4%

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

      4. +-commutative 68.4%

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

      5. fma-def 68.4%

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

      6. +-commutative 68.4%

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

   6. Applied egg-rr 68.4%

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

   7. Taylor expanded in k around inf 68.4%

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

      1. +-commutative 68.4%

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

      2. unpow2 68.4%

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

      3. distribute-rgt-in 68.4%

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

   9. Simplified 68.4%

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

   10. Taylor expanded in m around 0 33.3%

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

   if -1.7500000000000001e24 < k < 2.1500000000000001e-294

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 3.7%

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

   6. Taylor expanded in k around 0 3.7%

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

   7. Taylor expanded in k around inf 22.7%

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

   if 2.1500000000000001e-294 < k < 0.650000000000000022

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 51.5%

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

   6. Taylor expanded in k around 0 51.2%

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

      1. *-commutative 51.2%

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

   8. Simplified 51.2%

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

   if 0.650000000000000022 < k

   1. Initial program 86.7%

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

      1. associate-*l/ 85.4%

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

      2. sqr-neg 85.4%

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

      3. associate-+l+ 85.4%

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

      4. sqr-neg 85.4%

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

      5. distribute-rgt-out 85.5%

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

   3. Simplified 85.5%

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

      1. *-commutative 85.5%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

      4. +-commutative 84.2%

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

      5. fma-def 84.2%

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

      6. +-commutative 84.2%

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

   6. Applied egg-rr 84.2%

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

   7. Taylor expanded in k around inf 84.2%

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

      1. +-commutative 84.2%

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

      2. unpow2 84.2%

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

      3. distribute-rgt-in 84.2%

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

   9. Simplified 84.2%

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

   10. Taylor expanded in m around 0 64.5%

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

       1. associate-/r* 67.5%

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

       2. +-commutative 67.5%

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

   12. Simplified 67.5%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k + 10}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-294}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \mathbf{elif}\;k \leq 0.65:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{a_m}{t_0}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{a_m}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (*
    a_s
    (if (<= m -0.39)
      (/ a_m t_0)
      (if (<= m 1.25e+43) (/ a_m (+ 1.0 t_0)) (* a_m (* 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 t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.39) {
		tmp = a_m / t_0;
	} else if (m <= 1.25e+43) {
		tmp = a_m / (1.0 + t_0);
	} else {
		tmp = a_m * (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) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-0.39d0)) then
        tmp = a_m / t_0
    else if (m <= 1.25d+43) then
        tmp = a_m / (1.0d0 + t_0)
    else
        tmp = a_m * (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 t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.39) {
		tmp = a_m / t_0;
	} else if (m <= 1.25e+43) {
		tmp = a_m / (1.0 + t_0);
	} else {
		tmp = a_m * (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):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.39:
		tmp = a_m / t_0
	elif m <= 1.25e+43:
		tmp = a_m / (1.0 + t_0)
	else:
		tmp = a_m * (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)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.39)
		tmp = Float64(a_m / t_0);
	elseif (m <= 1.25e+43)
		tmp = Float64(a_m / Float64(1.0 + t_0));
	else
		tmp = Float64(a_m * 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)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.39)
		tmp = a_m / t_0;
	elseif (m <= 1.25e+43)
		tmp = a_m / (1.0 + t_0);
	else
		tmp = a_m * (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_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -0.39], N[(a$95$m / t$95$0), $MachinePrecision], If[LessEqual[m, 1.25e+43], N[(a$95$m / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -0.39000000000000001

   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%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in k around inf 87.7%

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

      1. +-commutative 87.7%

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

      2. unpow2 87.7%

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

      3. distribute-rgt-in 87.7%

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

   9. Simplified 87.7%

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

   10. Taylor expanded in m around 0 41.8%

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

   if -0.39000000000000001 < m < 1.2500000000000001e43

   1. Initial program 94.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. sqr-neg 94.7%

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

      3. associate-+l+ 94.7%

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

      4. sqr-neg 94.7%

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

      5. distribute-rgt-out 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in m around 0 84.8%

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

   if 1.2500000000000001e43 < m

   1. Initial program 81.6%

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

      1. associate-*l/ 76.3%

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

      2. sqr-neg 76.3%

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

      3. associate-+l+ 76.3%

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

      4. sqr-neg 76.3%

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

      5. distribute-rgt-out 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 10.9%

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

   7. Taylor expanded in a around 0 10.9%

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

      1. *-commutative 10.9%

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

   9. Simplified 10.9%

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

   10. Taylor expanded in k around inf 27.6%

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

       1. *-commutative 27.6%

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

   12. Simplified 27.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.39:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 24.3% accurate, 11.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

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 1.25e+43) a_m (* -10.0 (* a_m k)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.25e+43) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.25d+43) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.25e+43) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.25e+43:
		tmp = a_m
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.25e+43)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.25e+43)
		tmp = a_m;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.25e+43], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.2500000000000001e43

   1. Initial program 96.8%

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

      1. associate-*l/ 96.9%

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

      2. sqr-neg 96.9%

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

      3. associate-+l+ 96.9%

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

      4. sqr-neg 96.9%

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

      5. distribute-rgt-out 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in m around 0 63.2%

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

   6. Taylor expanded in k around 0 28.9%

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

   if 1.2500000000000001e43 < m

   1. Initial program 81.6%

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

      1. associate-*l/ 76.3%

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

      2. sqr-neg 76.3%

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

      3. associate-+l+ 76.3%

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

      4. sqr-neg 76.3%

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

      5. distribute-rgt-out 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 10.9%

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

   7. Taylor expanded in k around inf 27.6%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.25 \cdot 10^{+43}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 24.2% accurate, 11.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 3.2e+43) a_m (* a_m (* k -10.0)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 3.2e+43) {
		tmp = a_m;
	} else {
		tmp = a_m * (k * -10.0);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 3.2d+43) then
        tmp = a_m
    else
        tmp = a_m * (k * (-10.0d0))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 3.2e+43) {
		tmp = a_m;
	} else {
		tmp = a_m * (k * -10.0);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 3.2e+43:
		tmp = a_m
	else:
		tmp = a_m * (k * -10.0)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 3.2e+43)
		tmp = a_m;
	else
		tmp = Float64(a_m * Float64(k * -10.0));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 3.2e+43)
		tmp = a_m;
	else
		tmp = a_m * (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[LessEqual[m, 3.2e+43], a$95$m, N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 3.20000000000000014e43

   1. Initial program 96.8%

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

      1. associate-*l/ 96.9%

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

      2. sqr-neg 96.9%

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

      3. associate-+l+ 96.9%

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

      4. sqr-neg 96.9%

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

      5. distribute-rgt-out 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in m around 0 63.2%

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

   6. Taylor expanded in k around 0 28.9%

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

   if 3.20000000000000014e43 < m

   1. Initial program 81.6%

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

      1. associate-*l/ 76.3%

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

      2. sqr-neg 76.3%

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

      3. associate-+l+ 76.3%

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

      4. sqr-neg 76.3%

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

      5. distribute-rgt-out 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 10.9%

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

   7. Taylor expanded in a around 0 10.9%

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

      1. *-commutative 10.9%

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

   9. Simplified 10.9%

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

   10. Taylor expanded in k around inf 27.6%

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

       1. *-commutative 27.6%

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

   12. Simplified 27.6%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.2 \cdot 10^{+43}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 19.5% accurate, 114.0× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s a_m))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.3%

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

   1. associate-*l/ 90.8%

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

   2. sqr-neg 90.8%

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

   3. associate-+l+ 90.8%

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

   4. sqr-neg 90.8%

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

   5. distribute-rgt-out 90.8%

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

3. Simplified 90.8%

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

5. Taylor expanded in m around 0 45.4%

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

6. Taylor expanded in k around 0 21.4%

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

7. Final simplification 21.4%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))