Falkner and Boettcher, Appendix A

Percentage Accurate: 89.6% → 99.3%

Time: 10.9s

Alternatives: 11

Speedup: 1.1×

• 24.1% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := a_m \cdot {k}^{m}\\ a_s \cdot \begin{array}{l} \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a_m} + \frac{k}{a_m} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* a_m (pow k m))))
   (*
    a_s
    (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+164)
      (/ (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+164) {
		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+164) 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+164) {
		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+164:
		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+164)
		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+164)
		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+164], 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 1 (*.f64 10 k)) (*.f64 k k))) < 1e164

   1. Initial program 97.2%

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

      1. *-commutative 97.2%

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

      2. associate-/l* 95.9%

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

      3. sqr-neg 95.9%

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

      4. associate-+l+ 95.9%

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

      5. +-commutative 95.9%

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

      6. sqr-neg 95.9%

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

      7. distribute-rgt-out 95.9%

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

      8. fma-def 95.9%

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

      9. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in a around 0 95.9%

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

      1. flip-+ 69.4%

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

      2. associate-/l/ 67.4%

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

      3. metadata-eval 67.4%

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

      4. *-un-lft-identity 67.4%

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

      5. metadata-eval 67.4%

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

      6. frac-times 69.4%

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

      7. flip-+ 95.9%

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

      8. distribute-lft-in 95.9%

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

      9. associate-/r/ 95.9%

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

      10. clear-num 95.9%

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

      11. +-commutative 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in k around 0 93.5%

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

      1. *-rgt-identity 93.5%

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

      2. associate-*r/ 93.5%

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

      3. *-commutative 93.5%

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

      4. *-commutative 93.5%

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

      5. fma-def 93.5%

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

      6. *-lft-identity 93.5%

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

      7. associate-*l/ 93.5%

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

      8. unpow2 93.5%

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

      9. associate-*l* 96.2%

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

      10. fma-def 96.2%

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

      11. distribute-lft-out 98.6%

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

      12. +-commutative 98.6%

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

      13. associate-*l/ 98.6%

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

      14. *-lft-identity 98.6%

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

   9. Simplified 98.6%

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

   if 1e164 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 67.3%

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

      1. associate-/l* 67.3%

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

      2. sqr-neg 67.3%

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

      3. associate-+l+ 67.3%

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

      4. sqr-neg 67.3%

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

      5. distribute-rgt-out 67.3%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+164}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1}{a} + \frac{k}{a} \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{a_m}{1 + k \cdot 10}}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 0.00052:\\ \;\;\;\;\frac{1}{\frac{1}{a_m} + k \cdot \frac{k + 10}{a_m}}\\ \mathbf{else}:\\ \;\;\;\;a_m \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -6e-12)
    (/ (/ a_m (+ 1.0 (* k 10.0))) (pow k (- m)))
    (if (<= m 0.00052)
      (/ 1.0 (+ (/ 1.0 a_m) (* k (/ (+ k 10.0) a_m))))
      (* a_m (pow k m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -6e-12) {
		tmp = (a_m / (1.0 + (k * 10.0))) / pow(k, -m);
	} else if (m <= 0.00052) {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	} else {
		tmp = a_m * pow(k, m);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-6d-12)) then
        tmp = (a_m / (1.0d0 + (k * 10.0d0))) / (k ** -m)
    else if (m <= 0.00052d0) then
        tmp = 1.0d0 / ((1.0d0 / a_m) + (k * ((k + 10.0d0) / a_m)))
    else
        tmp = a_m * (k ** m)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -6e-12) {
		tmp = (a_m / (1.0 + (k * 10.0))) / Math.pow(k, -m);
	} else if (m <= 0.00052) {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	} else {
		tmp = a_m * Math.pow(k, m);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -6e-12:
		tmp = (a_m / (1.0 + (k * 10.0))) / math.pow(k, -m)
	elif m <= 0.00052:
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)))
	else:
		tmp = a_m * math.pow(k, m)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -6e-12)
		tmp = Float64(Float64(a_m / Float64(1.0 + Float64(k * 10.0))) / (k ^ Float64(-m)));
	elseif (m <= 0.00052)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(k * Float64(Float64(k + 10.0) / a_m))));
	else
		tmp = Float64(a_m * (k ^ m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -6e-12)
		tmp = (a_m / (1.0 + (k * 10.0))) / (k ^ -m);
	elseif (m <= 0.00052)
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	else
		tmp = a_m * (k ^ m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -6e-12], N[(N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00052], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $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 < -6.0000000000000003e-12

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 20.0%

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

   6. Simplified 100.0%

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

      1. add-sqr-sqrt 48.1%

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

      2. div-inv 48.1%

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

      3. times-frac 48.1%

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

      4. +-commutative 48.1%

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

      5. fma-def 48.1%

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

      6. pow-flip 48.1%

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

   8. Applied egg-rr 48.1%

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

      1. associate-*r/ 48.1%

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

      2. associate-*l/ 48.1%

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

      3. rem-square-sqrt 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in a around 0 100.0%

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

   if -6.0000000000000003e-12 < m < 5.19999999999999954e-4

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-/l* 92.2%

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

      3. sqr-neg 92.2%

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

      4. associate-+l+ 92.2%

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

      5. +-commutative 92.2%

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

      6. sqr-neg 92.2%

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

      7. distribute-rgt-out 92.2%

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

      8. fma-def 92.2%

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

      9. +-commutative 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. flip-+ 57.4%

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

      2. associate-/l/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. *-un-lft-identity 57.2%

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

      5. metadata-eval 57.2%

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

      6. frac-times 57.4%

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

      7. flip-+ 92.2%

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

      8. distribute-lft-in 92.2%

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

      9. associate-/r/ 92.2%

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

      10. clear-num 92.2%

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

      11. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in m around 0 92.2%

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

      1. associate-/l* 98.9%

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

      2. +-commutative 98.9%

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

   9. Simplified 98.9%

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

       1. clear-num 99.0%

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

       2. associate-/r/ 99.0%

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

       3. clear-num 99.0%

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

   11. Applied egg-rr 99.0%

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

   if 5.19999999999999954e-4 < m

   1. Initial program 82.6%

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

      1. associate-/l* 82.6%

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

      2. sqr-neg 82.6%

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

      3. associate-+l+ 82.6%

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

      4. sqr-neg 82.6%

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

      5. distribute-rgt-out 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{a}{1 + k \cdot 10}}{{k}^{\left(-m\right)}}\\ \mathbf{elif}\;m \leq 0.00052:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-13} \lor \neg \left(m \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;a_m \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a_m} + k \cdot \frac{k + 10}{a_m}}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -1.45e-13) (not (<= m 2.2e-11)))
    (* a_m (pow k m))
    (/ 1.0 (+ (/ 1.0 a_m) (* k (/ (+ k 10.0) a_m)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -1.45e-13) || !(m <= 2.2e-11)) {
		tmp = a_m * pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-1.45d-13)) .or. (.not. (m <= 2.2d-11))) then
        tmp = a_m * (k ** m)
    else
        tmp = 1.0d0 / ((1.0d0 / a_m) + (k * ((k + 10.0d0) / a_m)))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -1.45e-13) || !(m <= 2.2e-11)) {
		tmp = a_m * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (m <= -1.45e-13) or not (m <= 2.2e-11):
		tmp = a_m * math.pow(k, m)
	else:
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((m <= -1.45e-13) || !(m <= 2.2e-11))
		tmp = Float64(a_m * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(k * Float64(Float64(k + 10.0) / a_m))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((m <= -1.45e-13) || ~((m <= 2.2e-11)))
		tmp = a_m * (k ^ m);
	else
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[m, -1.45e-13], N[Not[LessEqual[m, 2.2e-11]], $MachinePrecision]], N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -1.4499999999999999e-13 or 2.2000000000000002e-11 < m

   1. Initial program 90.7%

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

      1. associate-/l* 90.7%

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

      2. sqr-neg 90.7%

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

      3. associate-+l+ 90.7%

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

      4. sqr-neg 90.7%

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

      5. distribute-rgt-out 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -1.4499999999999999e-13 < m < 2.2000000000000002e-11

   1. Initial program 93.0%

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

      1. *-commutative 93.0%

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

      2. associate-/l* 92.2%

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

      3. sqr-neg 92.2%

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

      4. associate-+l+ 92.2%

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

      5. +-commutative 92.2%

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

      6. sqr-neg 92.2%

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

      7. distribute-rgt-out 92.2%

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

      8. fma-def 92.2%

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

      9. +-commutative 92.2%

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

   3. Simplified 92.2%

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

   4. Taylor expanded in a around 0 92.2%

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

      1. flip-+ 57.4%

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

      2. associate-/l/ 57.2%

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

      3. metadata-eval 57.2%

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

      4. *-un-lft-identity 57.2%

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

      5. metadata-eval 57.2%

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

      6. frac-times 57.4%

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

      7. flip-+ 92.2%

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

      8. distribute-lft-in 92.2%

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

      9. associate-/r/ 92.2%

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

      10. clear-num 92.2%

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

      11. +-commutative 92.2%

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

   6. Applied egg-rr 92.2%

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

   7. Taylor expanded in m around 0 92.2%

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

      1. associate-/l* 98.9%

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

      2. +-commutative 98.9%

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

   9. Simplified 98.9%

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

       1. clear-num 99.0%

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

       2. associate-/r/ 99.0%

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

       3. clear-num 99.0%

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

   11. Applied egg-rr 99.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.45 \cdot 10^{-13} \lor \neg \left(m \leq 2.2 \cdot 10^{-11}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \end{array} \]

Alternative 4: 51.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 880000000:\\ \;\;\;\;\frac{1}{\frac{1}{a_m} + k \cdot \frac{k + 10}{a_m}}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -5e+40)
    (/ a_m (+ 1.0 (* k (+ k 10.0))))
    (if (<= m 880000000.0)
      (/ 1.0 (+ (/ 1.0 a_m) (* k (/ (+ k 10.0) a_m))))
      (* -10.0 (* a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -5e+40) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if (m <= 880000000.0) {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-5d+40)) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else if (m <= 880000000.0d0) then
        tmp = 1.0d0 / ((1.0d0 / a_m) + (k * ((k + 10.0d0) / a_m)))
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -5e+40) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else if (m <= 880000000.0) {
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -5e+40:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	elif m <= 880000000.0:
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)))
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -5e+40)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	elseif (m <= 880000000.0)
		tmp = Float64(1.0 / Float64(Float64(1.0 / a_m) + Float64(k * Float64(Float64(k + 10.0) / a_m))));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -5e+40)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	elseif (m <= 880000000.0)
		tmp = 1.0 / ((1.0 / a_m) + (k * ((k + 10.0) / a_m)));
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -5e+40], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 880000000.0], N[(1.0 / N[(N[(1.0 / a$95$m), $MachinePrecision] + N[(k * N[(N[(k + 10.0), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -5.00000000000000003e40

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 41.3%

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

   if -5.00000000000000003e40 < m < 8.8e8

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. associate-/l* 93.5%

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

      3. sqr-neg 93.5%

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

      4. associate-+l+ 93.5%

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

      5. +-commutative 93.5%

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

      6. sqr-neg 93.5%

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

      7. distribute-rgt-out 93.5%

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

      8. fma-def 93.5%

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

      9. +-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in a around 0 93.5%

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

      1. flip-+ 63.2%

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

      2. associate-/l/ 63.1%

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

      3. metadata-eval 63.1%

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

      4. *-un-lft-identity 63.1%

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

      5. metadata-eval 63.1%

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

      6. frac-times 63.2%

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

      7. flip-+ 93.4%

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

      8. distribute-lft-in 93.5%

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

      9. associate-/r/ 93.5%

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

      10. clear-num 93.5%

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

      11. +-commutative 93.5%

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in m around 0 80.1%

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

      1. associate-/l* 85.8%

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

      2. +-commutative 85.8%

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

   9. Simplified 85.8%

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

       1. clear-num 85.8%

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

       2. associate-/r/ 85.8%

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

       3. clear-num 85.8%

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

   11. Applied egg-rr 85.8%

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

   if 8.8e8 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+40}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 880000000:\\ \;\;\;\;\frac{1}{\frac{1}{a} + k \cdot \frac{k + 10}{a}}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 5: 49.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 600000000:\\ \;\;\;\;a_m \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 600000000.0)
    (* a_m (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))
    (* -10.0 (* a_m k)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 600000000.0) {
		tmp = a_m * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 600000000.0d0) then
        tmp = a_m * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 600000000.0) {
		tmp = a_m * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 600000000.0:
		tmp = a_m * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 600000000.0)
		tmp = Float64(a_m * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 600000000.0)
		tmp = a_m * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 600000000.0], N[(a$95$m * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 6e8

   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%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

      1. clear-num 96.1%

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

      2. associate-/r/ 96.5%

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

      3. clear-num 96.5%

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

      4. +-commutative 96.5%

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

      5. +-commutative 96.5%

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

      6. fma-udef 96.5%

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

   5. Applied egg-rr 96.5%

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

   6. Taylor expanded in m around 0 65.0%

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

   if 6e8 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 600000000:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 6: 35.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+106}:\\ \;\;\;\;0.1 \cdot \frac{a_m}{k}\\ \mathbf{elif}\;m \leq 960000000:\\ \;\;\;\;\frac{a_m}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -1.7e+106)
    (* 0.1 (/ a_m k))
    (if (<= m 960000000.0) (/ a_m (+ 1.0 (* k 10.0))) (* -10.0 (* a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -1.7e+106) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 960000000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.7d+106)) then
        tmp = 0.1d0 * (a_m / k)
    else if (m <= 960000000.0d0) then
        tmp = a_m / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -1.7e+106) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 960000000.0) {
		tmp = a_m / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -1.7e+106:
		tmp = 0.1 * (a_m / k)
	elif m <= 960000000.0:
		tmp = a_m / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -1.7e+106)
		tmp = Float64(0.1 * Float64(a_m / k));
	elseif (m <= 960000000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -1.7e+106)
		tmp = 0.1 * (a_m / k);
	elseif (m <= 960000000.0)
		tmp = a_m / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -1.7e+106], N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 960000000.0], N[(a$95$m / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -1.69999999999999997e106

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 36.8%

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

   5. Taylor expanded in k around 0 23.4%

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

      1. *-commutative 23.4%

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

   7. Simplified 23.4%

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

   8. Taylor expanded in k around inf 36.6%

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

   if -1.69999999999999997e106 < m < 9.6e8

   1. Initial program 95.0%

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

      1. associate-/l* 95.0%

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

      2. sqr-neg 95.0%

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

      3. associate-+l+ 95.0%

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

      4. sqr-neg 95.0%

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

      5. distribute-rgt-out 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in m around 0 77.3%

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

   5. Taylor expanded in k around 0 49.7%

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

      1. *-commutative 49.7%

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

   7. Simplified 49.7%

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

   if 9.6e8 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+106}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 960000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 7: 49.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 49000000000:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 49000000000.0)
    (/ a_m (+ 1.0 (* k (+ k 10.0))))
    (* -10.0 (* a_m k)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 49000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 49000000000.0d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 49000000000.0) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 49000000000.0:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 49000000000.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 49000000000.0)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 49000000000.0], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 4.9e10

   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%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 65.0%

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

   if 4.9e10 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 49000000000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 30.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;0.1 \cdot \frac{a_m}{k}\\ \mathbf{elif}\;m \leq 600000000:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -4.6e+30)
    (* 0.1 (/ a_m k))
    (if (<= m 600000000.0) a_m (* -10.0 (* a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.6e+30) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 600000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.6d+30)) then
        tmp = 0.1d0 * (a_m / k)
    else if (m <= 600000000.0d0) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.6e+30) {
		tmp = 0.1 * (a_m / k);
	} else if (m <= 600000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -4.6e+30:
		tmp = 0.1 * (a_m / k)
	elif m <= 600000000.0:
		tmp = a_m
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -4.6e+30)
		tmp = Float64(0.1 * Float64(a_m / k));
	elseif (m <= 600000000.0)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -4.6e+30)
		tmp = 0.1 * (a_m / k);
	elseif (m <= 600000000.0)
		tmp = a_m;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -4.6e+30], N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 600000000.0], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.6e30

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.8%

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

   5. Taylor expanded in k around 0 21.8%

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

      1. *-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in k around inf 29.8%

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

   if -4.6e30 < m < 6e8

   1. Initial program 94.1%

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

      1. associate-/l* 94.1%

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

      2. sqr-neg 94.1%

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

      3. associate-+l+ 94.1%

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

      4. sqr-neg 94.1%

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

      5. distribute-rgt-out 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 81.6%

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

   5. Taylor expanded in k around 0 41.9%

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

   if 6e8 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 600000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 9: 30.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a_m}{k \cdot 10}\\ \mathbf{elif}\;m \leq 960000000:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -4.6e+30)
    (/ a_m (* k 10.0))
    (if (<= m 960000000.0) a_m (* -10.0 (* a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.6e+30) {
		tmp = a_m / (k * 10.0);
	} else if (m <= 960000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-4.6d+30)) then
        tmp = a_m / (k * 10.0d0)
    else if (m <= 960000000.0d0) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4.6e+30) {
		tmp = a_m / (k * 10.0);
	} else if (m <= 960000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -4.6e+30:
		tmp = a_m / (k * 10.0)
	elif m <= 960000000.0:
		tmp = a_m
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -4.6e+30)
		tmp = Float64(a_m / Float64(k * 10.0));
	elseif (m <= 960000000.0)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -4.6e+30)
		tmp = a_m / (k * 10.0);
	elseif (m <= 960000000.0)
		tmp = a_m;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -4.6e+30], N[(a$95$m / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 960000000.0], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.6e30

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.8%

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

   5. Taylor expanded in k around 0 21.8%

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

      1. *-commutative 21.8%

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

   7. Simplified 21.8%

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

   8. Taylor expanded in k around inf 31.2%

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

      1. *-commutative 31.2%

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

   10. Simplified 31.2%

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

   if -4.6e30 < m < 9.6e8

   1. Initial program 94.1%

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

      1. associate-/l* 94.1%

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

      2. sqr-neg 94.1%

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

      3. associate-+l+ 94.1%

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

      4. sqr-neg 94.1%

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

      5. distribute-rgt-out 94.1%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in m around 0 81.6%

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

   5. Taylor expanded in k around 0 41.9%

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

   if 9.6e8 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 960000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 10: 25.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1500000000000:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a_m \cdot k\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 1500000000000.0) a_m (* -10.0 (* a_m k)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1500000000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1500000000000.0d0) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (a_m * k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1500000000000.0) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (a_m * k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1500000000000.0:
		tmp = a_m
	else:
		tmp = -10.0 * (a_m * k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1500000000000.0)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(a_m * k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1500000000000.0)
		tmp = a_m;
	else
		tmp = -10.0 * (a_m * k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1500000000000.0], a$95$m, N[(-10.0 * N[(a$95$m * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.5e12

   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%

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

      2. sqr-neg 96.5%

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

      3. associate-+l+ 96.5%

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

      4. sqr-neg 96.5%

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

      5. distribute-rgt-out 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in m around 0 65.0%

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

   5. Taylor expanded in k around 0 26.6%

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

   if 1.5e12 < m

   1. Initial program 82.4%

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

      1. associate-/l* 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-+l+ 82.4%

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

      4. sqr-neg 82.4%

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

      5. distribute-rgt-out 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in m around 0 2.9%

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

   5. Taylor expanded in k around 0 5.3%

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

   6. Taylor expanded in k around inf 17.2%

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

      1. *-commutative 17.2%

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

   8. Simplified 17.2%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1500000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 19.8% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.5%

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

   1. associate-/l* 91.5%

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

   2. sqr-neg 91.5%

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

   3. associate-+l+ 91.5%

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

   4. sqr-neg 91.5%

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

   5. distribute-rgt-out 91.5%

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

3. Simplified 91.5%

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

4. Taylor expanded in m around 0 42.9%

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

5. Taylor expanded in k around 0 18.3%

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

6. Final simplification 18.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023333 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))