Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.6%

Time: 13.1s

Alternatives: 13

Speedup: 1.0×

• 21.9% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+256}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m} + \frac{k + 10}{\frac{a\_m}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+256)
      (/ (pow k m) (+ (/ 1.0 a_m) (/ (+ k 10.0) (/ a_m 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+256) {
		tmp = pow(k, m) / ((1.0 / a_m) + ((k + 10.0) / (a_m / k)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 1d+256) then
        tmp = (k ** m) / ((1.0d0 / a_m) + ((k + 10.0d0) / (a_m / k)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+256) {
		tmp = Math.pow(k, m) / ((1.0 / a_m) + ((k + 10.0) / (a_m / 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+256:
		tmp = math.pow(k, m) / ((1.0 / a_m) + ((k + 10.0) / (a_m / 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+256)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 / a_m) + Float64(Float64(k + 10.0) / Float64(a_m / 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+256)
		tmp = (k ^ m) / ((1.0 / a_m) + ((k + 10.0) / (a_m / 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+256], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(N[(k + 10.0), $MachinePrecision] / N[(a$95$m / k), $MachinePrecision]), $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 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1e256

   1. Initial program 97.0%

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

      1. associate-/l* 97.0%

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

      2. remove-double-neg 97.0%

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

      3. distribute-frac-neg2 97.0%

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

      4. distribute-neg-frac2 97.0%

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

      5. remove-double-neg 97.0%

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

      6. sqr-neg 97.0%

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

      7. associate-+l+ 97.0%

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

      8. sqr-neg 97.0%

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

      9. distribute-rgt-out 97.0%

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

   3. Simplified 97.0%

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

      1. distribute-lft-in 97.0%

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

      2. associate-+l+ 97.0%

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

      3. associate-*r/ 97.0%

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

      4. clear-num 97.0%

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

      5. associate-/r* 96.0%

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

      6. associate-+l+ 96.0%

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

      7. distribute-lft-in 96.0%

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

      8. +-commutative 96.0%

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

      9. +-commutative 96.0%

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

      10. fma-undefine 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in k around 0 98.8%

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

   8. Taylor expanded in m around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-in 97.3%

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

      3. associate-*r/ 97.3%

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

      4. metadata-eval 97.3%

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

      5. *-commutative 97.3%

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

      6. associate-*l/ 97.3%

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

      7. associate-*r/ 97.3%

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

      8. distribute-rgt-out 98.8%

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

   10. Simplified 98.8%

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

       1. *-commutative 98.8%

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

       2. clear-num 98.8%

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

       3. un-div-inv 98.8%

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

       4. +-commutative 98.8%

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

   12. Applied egg-rr 98.8%

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

   if 1e256 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 64.9%

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

      1. associate-/l* 64.9%

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

      2. remove-double-neg 64.9%

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

      3. distribute-frac-neg2 64.9%

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

      4. distribute-neg-frac2 64.9%

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

      5. remove-double-neg 64.9%

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

      6. sqr-neg 64.9%

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

      7. associate-+l+ 64.9%

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

      8. sqr-neg 64.9%

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

      9. distribute-rgt-out 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+256}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a} + \frac{k + 10}{\frac{a}{k}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a\_m \cdot {k}^{m}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+256}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a\_m} + \frac{k}{a\_m} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+256)
      (/ (pow k m) (+ (/ 1.0 a_m) (* (/ k a_m) (+ k 10.0))))
      t_0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+256) {
		tmp = pow(k, m) / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a_m * (k ** m)
    if ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 1d+256) then
        tmp = (k ** m) / ((1.0d0 / a_m) + ((k / a_m) * (k + 10.0d0)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m * Math.pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+256) {
		tmp = Math.pow(k, m) / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = a_m * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+256:
		tmp = math.pow(k, m) / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+256)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 / a_m) + Float64(Float64(k / a_m) * Float64(k + 10.0))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = a_m * (k ^ m);
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+256)
		tmp = (k ^ m) / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+256], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(N[(k / a$95$m), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $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 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1e256

   1. Initial program 97.0%

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

      1. associate-/l* 97.0%

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

      2. remove-double-neg 97.0%

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

      3. distribute-frac-neg2 97.0%

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

      4. distribute-neg-frac2 97.0%

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

      5. remove-double-neg 97.0%

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

      6. sqr-neg 97.0%

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

      7. associate-+l+ 97.0%

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

      8. sqr-neg 97.0%

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

      9. distribute-rgt-out 97.0%

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

   3. Simplified 97.0%

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

      1. distribute-lft-in 97.0%

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

      2. associate-+l+ 97.0%

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

      3. associate-*r/ 97.0%

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

      4. clear-num 97.0%

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

      5. associate-/r* 96.0%

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

      6. associate-+l+ 96.0%

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

      7. distribute-lft-in 96.0%

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

      8. +-commutative 96.0%

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

      9. +-commutative 96.0%

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

      10. fma-undefine 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in k around 0 98.8%

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

   8. Taylor expanded in m around inf 98.8%

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

      1. +-commutative 98.8%

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

      2. distribute-lft-in 97.3%

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

      3. associate-*r/ 97.3%

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

      4. metadata-eval 97.3%

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

      5. *-commutative 97.3%

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

      6. associate-*l/ 97.3%

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

      7. associate-*r/ 97.3%

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

      8. distribute-rgt-out 98.8%

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

   10. Simplified 98.8%

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

   if 1e256 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 64.9%

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

      1. associate-/l* 64.9%

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

      2. remove-double-neg 64.9%

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

      3. distribute-frac-neg2 64.9%

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

      4. distribute-neg-frac2 64.9%

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

      5. remove-double-neg 64.9%

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

      6. sqr-neg 64.9%

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

      7. associate-+l+ 64.9%

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

      8. sqr-neg 64.9%

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

      9. distribute-rgt-out 64.9%

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

   3. Simplified 64.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+256}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-7} \lor \neg \left(m \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a\_m} + \frac{k}{a\_m} \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -8.8e-7) (not (<= m 7.2e-9)))
    (* a_m (pow k m))
    (/ 1.0 (+ (/ 1.0 a_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 ((m <= -8.8e-7) || !(m <= 7.2e-9)) {
		tmp = a_m * pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a_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 ((m <= (-8.8d-7)) .or. (.not. (m <= 7.2d-9))) then
        tmp = a_m * (k ** m)
    else
        tmp = 1.0d0 / ((1.0d0 / a_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 ((m <= -8.8e-7) || !(m <= 7.2e-9)) {
		tmp = a_m * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a_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 (m <= -8.8e-7) or not (m <= 7.2e-9):
		tmp = a_m * math.pow(k, m)
	else:
		tmp = 1.0 / ((1.0 / a_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 ((m <= -8.8e-7) || !(m <= 7.2e-9))
		tmp = Float64(a_m * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(Float64(k / 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 <= -8.8e-7) || ~((m <= 7.2e-9)))
		tmp = a_m * (k ^ m);
	else
		tmp = 1.0 / ((1.0 / a_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[Or[LessEqual[m, -8.8e-7], N[Not[LessEqual[m, 7.2e-9]], $MachinePrecision]], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(N[(k / a$95$m), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -8.8000000000000004e-7 or 7.2e-9 < m

   1. Initial program 88.1%

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

      1. associate-/l* 88.1%

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

      2. remove-double-neg 88.1%

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

      3. distribute-frac-neg2 88.1%

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

      4. distribute-neg-frac2 88.1%

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

      5. remove-double-neg 88.1%

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

      6. sqr-neg 88.1%

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

      7. associate-+l+ 88.1%

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

      8. sqr-neg 88.1%

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

      9. distribute-rgt-out 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -8.8000000000000004e-7 < m < 7.2e-9

   1. Initial program 93.3%

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

      1. associate-/l* 93.3%

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

      2. remove-double-neg 93.3%

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

      3. distribute-frac-neg2 93.3%

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

      4. distribute-neg-frac2 93.3%

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

      5. remove-double-neg 93.3%

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

      6. sqr-neg 93.3%

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

      7. associate-+l+ 93.3%

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

      8. sqr-neg 93.3%

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

      9. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

      1. distribute-lft-in 93.3%

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

      2. associate-+l+ 93.3%

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

      3. associate-*r/ 93.3%

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

      4. clear-num 93.3%

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

      5. associate-/r* 93.3%

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

      6. associate-+l+ 93.3%

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

      7. distribute-lft-in 93.3%

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

      8. +-commutative 93.3%

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

      9. +-commutative 93.3%

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

      10. fma-undefine 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in k around 0 99.5%

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

   8. Taylor expanded in m around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. associate-*r/ 99.3%

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

      4. metadata-eval 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l/ 99.3%

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

      7. associate-*r/ 99.3%

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

      8. distribute-rgt-out 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8.8 \cdot 10^{-7} \lor \neg \left(m \leq 7.2 \cdot 10^{-9}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{a\_m}}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{1}{a\_m} + \frac{k}{a\_m} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -3.6e-7)
    (/ 1.0 (/ (/ 1.0 a_m) (pow k m)))
    (if (<= m 2.65e-8)
      (/ 1.0 (+ (/ 1.0 a_m) (* (/ k a_m) (+ 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 <= -3.6e-7) {
		tmp = 1.0 / ((1.0 / a_m) / pow(k, m));
	} else if (m <= 2.65e-8) {
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (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 <= (-3.6d-7)) then
        tmp = 1.0d0 / ((1.0d0 / a_m) / (k ** m))
    else if (m <= 2.65d-8) then
        tmp = 1.0d0 / ((1.0d0 / a_m) + ((k / a_m) * (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 <= -3.6e-7) {
		tmp = 1.0 / ((1.0 / a_m) / Math.pow(k, m));
	} else if (m <= 2.65e-8) {
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (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 <= -3.6e-7:
		tmp = 1.0 / ((1.0 / a_m) / math.pow(k, m))
	elif m <= 2.65e-8:
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (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 <= -3.6e-7)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) / (k ^ m)));
	elseif (m <= 2.65e-8)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(Float64(k / a_m) * 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 <= -3.6e-7)
		tmp = 1.0 / ((1.0 / a_m) / (k ^ m));
	elseif (m <= 2.65e-8)
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (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, -3.6e-7], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.65e-8], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(N[(k / a$95$m), $MachinePrecision] * 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 < -3.59999999999999994e-7

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. associate-+l+ 100.0%

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

      3. associate-*r/ 100.0%

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

      4. clear-num 100.0%

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

      5. associate-/r* 100.0%

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

      6. associate-+l+ 100.0%

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

      7. distribute-lft-in 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in k around 0 100.0%

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

   if -3.59999999999999994e-7 < m < 2.6499999999999999e-8

   1. Initial program 93.3%

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

      1. associate-/l* 93.3%

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

      2. remove-double-neg 93.3%

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

      3. distribute-frac-neg2 93.3%

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

      4. distribute-neg-frac2 93.3%

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

      5. remove-double-neg 93.3%

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

      6. sqr-neg 93.3%

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

      7. associate-+l+ 93.3%

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

      8. sqr-neg 93.3%

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

      9. distribute-rgt-out 93.3%

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

   3. Simplified 93.3%

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

      1. distribute-lft-in 93.3%

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

      2. associate-+l+ 93.3%

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

      3. associate-*r/ 93.3%

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

      4. clear-num 93.3%

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

      5. associate-/r* 93.3%

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

      6. associate-+l+ 93.3%

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

      7. distribute-lft-in 93.3%

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

      8. +-commutative 93.3%

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

      9. +-commutative 93.3%

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

      10. fma-undefine 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in k around 0 99.5%

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

   8. Taylor expanded in m around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. associate-*r/ 99.3%

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

      4. metadata-eval 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*l/ 99.3%

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

      7. associate-*r/ 99.3%

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

      8. distribute-rgt-out 99.3%

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

   10. Simplified 99.3%

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

   if 2.6499999999999999e-8 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{a}}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 2.65 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.45:\\ \;\;\;\;\frac{1}{\frac{1}{a\_m} + \frac{k}{a\_m} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -5e+71)
    (/ a_m (+ 1.0 (* k (+ k 10.0))))
    (if (<= m 2.45)
      (/ 1.0 (+ (/ 1.0 a_m) (* (/ k a_m) (+ k 10.0))))
      (* a_m (+ 1.0 (* k (- (* k 99.0) 10.0))))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -5e+71) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if (m <= 2.45) {
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	} else {
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5d+71)) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= 2.45d0) then
        tmp = 1.0d0 / ((1.0d0 / a_m) + ((k / a_m) * (k + 10.0d0)))
    else
        tmp = a_m * (1.0d0 + (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -5e+71) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if (m <= 2.45) {
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	} else {
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -5e+71:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	elif m <= 2.45:
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)))
	else:
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -5e+71)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= 2.45)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(Float64(k / a_m) * Float64(k + 10.0))));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -5e+71)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	elseif (m <= 2.45)
		tmp = 1.0 / ((1.0 / a_m) + ((k / a_m) * (k + 10.0)));
	else
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -5e+71], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.45], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(N[(k / a$95$m), $MachinePrecision] * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.99999999999999972e71

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 37.5%

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

   if -4.99999999999999972e71 < m < 2.4500000000000002

   1. Initial program 94.5%

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

      1. associate-/l* 94.5%

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

      2. remove-double-neg 94.5%

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

      3. distribute-frac-neg2 94.5%

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

      4. distribute-neg-frac2 94.5%

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

      5. remove-double-neg 94.5%

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

      6. sqr-neg 94.5%

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

      7. associate-+l+ 94.5%

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

      8. sqr-neg 94.5%

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

      9. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

      1. distribute-lft-in 94.5%

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

      2. associate-+l+ 94.5%

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

      3. associate-*r/ 94.5%

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

      4. clear-num 94.4%

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

      5. associate-/r* 94.4%

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

      6. associate-+l+ 94.4%

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

      7. distribute-lft-in 94.5%

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

      8. +-commutative 94.5%

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

      9. +-commutative 94.5%

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

      10. fma-undefine 94.5%

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

   6. Applied egg-rr 94.5%

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

   7. Taylor expanded in k around 0 99.6%

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

   8. Taylor expanded in m around 0 89.8%

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

      1. +-commutative 89.8%

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

      2. distribute-lft-in 88.9%

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

      3. associate-*r/ 88.9%

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

      4. metadata-eval 88.9%

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

      5. *-commutative 88.9%

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

      6. associate-*l/ 88.9%

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

      7. associate-*r/ 88.9%

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

      8. distribute-rgt-out 89.8%

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

   10. Simplified 89.8%

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

   if 2.4500000000000002 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 25.7%

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

   7. Taylor expanded in a around 0 33.1%

      \[\leadsto \color{blue}{a \cdot \left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.45:\\ \;\;\;\;\frac{1}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 1.95)
    (/ a_m (+ 1.0 (* k (+ k 10.0))))
    (* a_m (+ 1.0 (* k (- (* k 99.0) 10.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.95d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m * (1.0d0 + (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.95) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.95:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.95)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.95)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m * (1.0 + (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.95], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.94999999999999996

   1. Initial program 96.5%

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

      1. associate-/l* 96.5%

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

      2. remove-double-neg 96.5%

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

      3. distribute-frac-neg2 96.5%

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

      4. distribute-neg-frac2 96.5%

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

      5. remove-double-neg 96.5%

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

      6. sqr-neg 96.5%

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

      7. associate-+l+ 96.5%

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

      8. sqr-neg 96.5%

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

      9. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in m around 0 67.3%

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

   if 1.94999999999999996 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 25.7%

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

   7. Taylor expanded in a around 0 33.1%

      \[\leadsto \color{blue}{a \cdot \left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m + k \cdot \left(99 \cdot \left(a\_m \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.1) (/ a_m (+ 1.0 (* k k))) (+ a_m (* k (* 99.0 (* 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 <= 2.1) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m + (k * (99.0 * (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 <= 2.1d0) then
        tmp = a_m / (1.0d0 + (k * k))
    else
        tmp = a_m + (k * (99.0d0 * (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 <= 2.1) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m + (k * (99.0 * (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 <= 2.1:
		tmp = a_m / (1.0 + (k * k))
	else:
		tmp = a_m + (k * (99.0 * (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 <= 2.1)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a_m + Float64(k * Float64(99.0 * 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 <= 2.1)
		tmp = a_m / (1.0 + (k * k));
	else
		tmp = a_m + (k * (99.0 * (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, 2.1], N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(k * N[(99.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.10000000000000009

   1. Initial program 96.5%

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

      1. *-commutative 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in m around 0 67.3%

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

   6. Taylor expanded in k around 0 65.8%

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

   if 2.10000000000000009 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 25.7%

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

   7. Taylor expanded in a around 0 25.7%

      \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(-99 \cdot \left(a \cdot k\right)\right)} - 10 \cdot a\right) \]
   8. Step-by-step derivation

      1. *-commutative 25.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot -99\right)} - 10 \cdot a\right) \]

   9. Simplified 25.7%

      \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot -99\right)} - 10 \cdot a\right) \]

   10. Taylor expanded in k around inf 25.7%

       \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.96:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a\_m + k \cdot \left(99 \cdot \left(a\_m \cdot k\right)\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 1.96)
    (/ a_m (+ 1.0 (* k (+ k 10.0))))
    (+ a_m (* k (* 99.0 (* 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 <= 1.96) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (k * (99.0 * (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 <= 1.96d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a_m + (k * (99.0d0 * (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 <= 1.96) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a_m + (k * (99.0 * (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 <= 1.96:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a_m + (k * (99.0 * (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 <= 1.96)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a_m + Float64(k * Float64(99.0 * 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 <= 1.96)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = a_m + (k * (99.0 * (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, 1.96], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m + N[(k * N[(99.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.96

   1. Initial program 96.5%

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

      1. associate-/l* 96.5%

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

      2. remove-double-neg 96.5%

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

      3. distribute-frac-neg2 96.5%

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

      4. distribute-neg-frac2 96.5%

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

      5. remove-double-neg 96.5%

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

      6. sqr-neg 96.5%

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

      7. associate-+l+ 96.5%

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

      8. sqr-neg 96.5%

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

      9. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in m around 0 67.3%

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

   if 1.96 < m

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 25.7%

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

   7. Taylor expanded in a around 0 25.7%

      \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(-99 \cdot \left(a \cdot k\right)\right)} - 10 \cdot a\right) \]
   8. Step-by-step derivation

      1. *-commutative 25.7%

         \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot -99\right)} - 10 \cdot a\right) \]

   9. Simplified 25.7%

      \[\leadsto a + k \cdot \left(-1 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot -99\right)} - 10 \cdot a\right) \]

   10. Taylor expanded in k around inf 25.7%

       \[\leadsto a + k \cdot \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.96:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + k \cdot \left(99 \cdot \left(a \cdot k\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 29.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 5.2e+16) (/ a_m (+ 1.0 (* k 10.0))) (* a_m (+ 1.0 (* k -10.0))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 5.2e+16) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = a_m * (1.0 + (k * -10.0));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.2d+16) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = a_m * (1.0d0 + (k * (-10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 5.2e+16) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = a_m * (1.0 + (k * -10.0));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 5.2e+16:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = a_m * (1.0 + (k * -10.0))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 5.2e+16)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * -10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 5.2e+16)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = a_m * (1.0 + (k * -10.0));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 5.2e+16], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 5.2e16

   1. Initial program 95.9%

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

      1. associate-/l* 95.9%

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

      2. remove-double-neg 95.9%

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

      3. distribute-frac-neg2 95.9%

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

      4. distribute-neg-frac2 95.9%

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

      5. remove-double-neg 95.9%

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

      6. sqr-neg 95.9%

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

      7. associate-+l+ 95.9%

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

      8. sqr-neg 95.9%

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

      9. distribute-rgt-out 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in m around 0 66.2%

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

   6. Taylor expanded in k around 0 38.5%

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

      1. *-commutative 38.5%

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

   8. Simplified 38.5%

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

   if 5.2e16 < m

   1. Initial program 77.9%

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

      1. associate-/l* 77.9%

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

      2. remove-double-neg 77.9%

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

      3. distribute-frac-neg2 77.9%

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

      4. distribute-neg-frac2 77.9%

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

      5. remove-double-neg 77.9%

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

      6. sqr-neg 77.9%

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

      7. associate-+l+ 77.9%

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

      8. sqr-neg 77.9%

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

      9. distribute-rgt-out 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in k around 0 84.9%

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

      1. associate-*r* 84.9%

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

      2. distribute-lft1-in 84.9%

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

      3. *-commutative 84.9%

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

      4. fma-define 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in m around 0 10.0%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a\_m}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\_m \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 5.2e+16) (/ a_m (+ 1.0 (* k k))) (* a_m (+ 1.0 (* k -10.0))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 5.2e+16) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m * (1.0 + (k * -10.0));
	}
	return a_s * tmp;
}

Fortran

a\_m = abs(a)
a\_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 5.2d+16) then
        tmp = a_m / (1.0d0 + (k * k))
    else
        tmp = a_m * (1.0d0 + (k * (-10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 5.2e+16) {
		tmp = a_m / (1.0 + (k * k));
	} else {
		tmp = a_m * (1.0 + (k * -10.0));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 5.2e+16:
		tmp = a_m / (1.0 + (k * k))
	else:
		tmp = a_m * (1.0 + (k * -10.0))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 5.2e+16)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a_m * Float64(1.0 + Float64(k * -10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 5.2e+16)
		tmp = a_m / (1.0 + (k * k));
	else
		tmp = a_m * (1.0 + (k * -10.0));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 5.2e+16], N[(a$95$m / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 5.2e16

   1. Initial program 95.9%

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

      1. *-commutative 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in m around 0 66.2%

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

   6. Taylor expanded in k around 0 64.7%

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

   if 5.2e16 < m

   1. Initial program 77.9%

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

      1. associate-/l* 77.9%

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

      2. remove-double-neg 77.9%

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

      3. distribute-frac-neg2 77.9%

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

      4. distribute-neg-frac2 77.9%

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

      5. remove-double-neg 77.9%

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

      6. sqr-neg 77.9%

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

      7. associate-+l+ 77.9%

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

      8. sqr-neg 77.9%

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

      9. distribute-rgt-out 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in k around 0 84.9%

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

      1. associate-*r* 84.9%

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

      2. distribute-lft1-in 84.9%

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

      3. *-commutative 84.9%

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

      4. fma-define 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in m around 0 10.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 5.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 20.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \left(a\_m \cdot \left(1 + k \cdot -10\right)\right) \end{array} \]

FPCore

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

C

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

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 * (1.0d0 + (k * (-10.0d0))))
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 * (1.0 + (k * -10.0)));
}

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 * (1.0 + (k * -10.0)))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m * Float64(1.0 + Float64(k * -10.0))))
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 * (1.0 + (k * -10.0)));
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 * N[(a$95$m * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in k around 0 64.9%

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

   1. associate-*r* 64.9%

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

   2. distribute-lft1-in 75.1%

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

   3. *-commutative 75.1%

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

   4. fma-define 75.1%

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

7. Simplified 75.1%

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

8. Taylor expanded in m around 0 19.9%

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

9. Final simplification 19.9%

   \[\leadsto a \cdot \left(1 + k \cdot -10\right) \]
10. Add Preprocessing

Alternative 12: 20.6% accurate, 16.3× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \left(a\_m + a\_m \cdot \left(k \cdot -10\right)\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (+ a_m (* 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) {
	return a_s * (a_m + (a_m * (k * -10.0)));
}

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 + (a_m * (k * (-10.0d0))))
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 + (a_m * (k * -10.0)));
}

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 + (a_m * (k * -10.0)))

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * Float64(a_m + Float64(a_m * Float64(k * -10.0))))
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 + (a_m * (k * -10.0)));
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 * N[(a$95$m + N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in m around 0 44.9%

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

6. Taylor expanded in k around 0 25.4%

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

7. Taylor expanded in k around 0 19.9%

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

   1. *-commutative 19.9%

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

   2. associate-*l* 19.9%

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

9. Simplified 19.9%

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

10. Final simplification 19.9%

    \[\leadsto a + a \cdot \left(k \cdot -10\right) \]
11. Add Preprocessing

Alternative 13: 19.6% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in m around 0 44.9%

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

6. Taylor expanded in k around 0 18.3%

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

7. Final simplification 18.3%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))