Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 99.6%

Time: 15.6s

Alternatives: 17

Speedup: 1.1×

• 23.2% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot \left(\sqrt{a_m} \cdot \sqrt{a_m}\right)\\ \mathbf{elif}\;m \leq 3.4:\\ \;\;\;\;\frac{{k}^{m}}{t_0} \cdot \frac{a_m}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a_m\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (*
    a_s
    (if (<= m -3.6e-9)
      (* (pow k m) (* (sqrt a_m) (sqrt a_m)))
      (if (<= m 3.4) (* (/ (pow k m) t_0) (/ a_m t_0)) (* (pow k m) 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 t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (m <= -3.6e-9) {
		tmp = pow(k, m) * (sqrt(a_m) * sqrt(a_m));
	} else if (m <= 3.4) {
		tmp = (pow(k, m) / t_0) * (a_m / t_0);
	} else {
		tmp = pow(k, m) * 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)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (m <= -3.6e-9)
		tmp = Float64((k ^ m) * Float64(sqrt(a_m) * sqrt(a_m)));
	elseif (m <= 3.4)
		tmp = Float64(Float64((k ^ m) / t_0) * Float64(a_m / t_0));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -3.6e-9], N[(N[Power[k, m], $MachinePrecision] * N[(N[Sqrt[a$95$m], $MachinePrecision] * N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.4], N[(N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(a$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -3.6e-9

   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. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 100.0%

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

      2. add-sqr-sqrt 47.7%

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

      3. associate-*l* 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. *-commutative 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-*l* 47.7%

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

   9. Simplified 47.7%

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

   10. Taylor expanded in k around 0 47.7%

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

   if -3.6e-9 < m < 3.39999999999999991

   1. Initial program 93.8%

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

      1. sqr-neg 93.8%

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

      2. cancel-sign-sub 93.8%

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

      3. distribute-lft-neg-in 93.8%

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

      4. neg-sub0 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 93.8%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 93.8%

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

      11. +-commutative 93.8%

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

      12. *-commutative 93.8%

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

      13. sqr-neg 93.8%

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

   3. Simplified 93.8%

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

      1. add-cube-cbrt 92.5%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 92.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 92.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 93.8%

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

      2. *-commutative 93.8%

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

      3. add-sqr-sqrt 93.8%

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

      4. times-frac 93.7%

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

      5. +-commutative 93.7%

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

      6. add-sqr-sqrt 93.7%

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

      7. hypot-def 93.7%

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

      8. +-commutative 93.7%

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

      9. fma-udef 93.7%

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

      10. +-commutative 93.7%

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

      11. add-sqr-sqrt 93.7%

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

      12. hypot-def 99.8%

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

      13. +-commutative 99.8%

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

      14. fma-udef 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 3.39999999999999991 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot \left(\sqrt{a} \cdot \sqrt{a}\right)\\ \mathbf{elif}\;m \leq 3.4:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot \left(\sqrt{a_m} \cdot \sqrt{a_m}\right)\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{a_m}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a_m\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (*
    a_s
    (if (<= m -3.4e-9)
      (* (pow k m) (* (sqrt a_m) (sqrt a_m)))
      (if (<= m 2.3e-14) (/ (/ a_m t_0) t_0) (* (pow k m) 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 t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (m <= -3.4e-9) {
		tmp = pow(k, m) * (sqrt(a_m) * sqrt(a_m));
	} else if (m <= 2.3e-14) {
		tmp = (a_m / t_0) / t_0;
	} else {
		tmp = pow(k, m) * 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)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (m <= -3.4e-9)
		tmp = Float64((k ^ m) * Float64(sqrt(a_m) * sqrt(a_m)));
	elseif (m <= 2.3e-14)
		tmp = Float64(Float64(a_m / t_0) / t_0);
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -3.4e-9], N[(N[Power[k, m], $MachinePrecision] * N[(N[Sqrt[a$95$m], $MachinePrecision] * N[Sqrt[a$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3e-14], N[(N[(a$95$m / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -3.3999999999999998e-9

   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. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   6. Step-by-step derivation

      1. rem-cube-cbrt 100.0%

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

      2. add-sqr-sqrt 47.7%

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

      3. associate-*l* 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. *-commutative 47.7%

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

      2. *-commutative 47.7%

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

      3. associate-*l* 47.7%

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

   9. Simplified 47.7%

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

   10. Taylor expanded in k around 0 47.7%

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

   if -3.3999999999999998e-9 < m < 2.29999999999999998e-14

   1. Initial program 93.5%

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

      1. sqr-neg 93.5%

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

      2. cancel-sign-sub 93.5%

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

      3. distribute-lft-neg-in 93.5%

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

      4. neg-sub0 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 93.5%

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

      11. +-commutative 93.5%

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

      12. *-commutative 93.5%

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

      13. sqr-neg 93.5%

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

   3. Simplified 93.5%

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

      1. add-cube-cbrt 92.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 92.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 93.5%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 93.5%

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

      2. *-lft-identity 93.5%

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

   8. Simplified 93.5%

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

      1. *-un-lft-identity 93.5%

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

      2. add-sqr-sqrt 93.5%

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

      3. times-frac 93.5%

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

      4. +-commutative 93.5%

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

      5. add-exp-log 93.5%

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

      6. log1p-udef 93.5%

         \[\leadsto \frac{1}{\sqrt{k \cdot k + e^{\color{blue}{\mathsf{log1p}\left(k \cdot 10\right)}}}} \cdot \frac{a}{\sqrt{\left(1 + k \cdot 10\right) + k \cdot k}} \]

      7. add-sqr-sqrt 93.5%

         \[\leadsto \frac{1}{\sqrt{k \cdot k + \color{blue}{\sqrt{e^{\mathsf{log1p}\left(k \cdot 10\right)}} \cdot \sqrt{e^{\mathsf{log1p}\left(k \cdot 10\right)}}}}} \cdot \frac{a}{\sqrt{\left(1 + k \cdot 10\right) + k \cdot k}} \]

      8. hypot-def 93.5%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(k, \sqrt{e^{\mathsf{log1p}\left(k \cdot 10\right)}}\right)}} \cdot \frac{a}{\sqrt{\left(1 + k \cdot 10\right) + k \cdot k}} \]

      9. log1p-udef 93.5%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(k, \sqrt{e^{\color{blue}{\log \left(1 + k \cdot 10\right)}}}\right)} \cdot \frac{a}{\sqrt{\left(1 + k \cdot 10\right) + k \cdot k}} \]

      10. add-exp-log 93.5%

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

      11. +-commutative 93.5%

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

      12. fma-def 93.5%

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

      13. +-commutative 93.5%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-commutative 99.8%

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

       3. associate-/r/ 99.8%

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

       4. /-rgt-identity 99.8%

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

   12. Simplified 99.8%

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

   if 2.29999999999999998e-14 < m

   1. Initial program 71.3%

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

      1. associate-*l/ 66.0%

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

      2. associate-+l+ 66.0%

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

      3. distribute-rgt-out 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 99.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.4 \cdot 10^{-9}:\\ \;\;\;\;{k}^{m} \cdot \left(\sqrt{a} \cdot \sqrt{a}\right)\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := {k}^{m} \cdot a_m\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{t_0}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a_m)))
   (* a_s (if (<= m 2.6) (/ t_0 (+ (* k k) (+ 1.0 (* k 10.0)))) t_0))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = pow(k, m) * a_m;
	double tmp;
	if (m <= 2.6) {
		tmp = t_0 / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a_m
    if (m <= 2.6d0) then
        tmp = t_0 / ((k * k) + (1.0d0 + (k * 10.0d0)))
    else
        tmp = t_0
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = Math.pow(k, m) * a_m;
	double tmp;
	if (m <= 2.6) {
		tmp = t_0 / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = math.pow(k, m) * a_m
	tmp = 0
	if m <= 2.6:
		tmp = t_0 / ((k * k) + (1.0 + (k * 10.0)))
	else:
		tmp = t_0
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64((k ^ m) * a_m)
	tmp = 0.0
	if (m <= 2.6)
		tmp = Float64(t_0 / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = (k ^ m) * a_m;
	tmp = 0.0;
	if (m <= 2.6)
		tmp = t_0 / ((k * k) + (1.0 + (k * 10.0)));
	else
		tmp = t_0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, 2.6], N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.60000000000000009

   1. Initial program 96.2%

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

   if 2.60000000000000009 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m 2.95)
    (* (pow k m) (/ a_m (+ 1.0 (* k (+ k 10.0)))))
    (* (pow k m) 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 <= 2.95) {
		tmp = pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = pow(k, m) * 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 <= 2.95d0) then
        tmp = (k ** m) * (a_m / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = (k ** m) * 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 <= 2.95) {
		tmp = Math.pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = Math.pow(k, m) * 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 <= 2.95:
		tmp = math.pow(k, m) * (a_m / (1.0 + (k * (k + 10.0))))
	else:
		tmp = math.pow(k, m) * 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 <= 2.95)
		tmp = Float64((k ^ m) * Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64((k ^ m) * 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 <= 2.95)
		tmp = (k ^ m) * (a_m / (1.0 + (k * (k + 10.0))));
	else
		tmp = (k ^ m) * a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 2.95], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 2.9500000000000002

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

   if 2.9500000000000002 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 97.5%

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

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := 1 + k \cdot 10\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-25}:\\ \;\;\;\;{k}^{m} \cdot \frac{a_m}{t_0}\\ \mathbf{elif}\;m \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{a_m}{k \cdot k + t_0}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a_m\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* k 10.0))))
   (*
    a_s
    (if (<= m -6.2e-25)
      (* (pow k m) (/ a_m t_0))
      (if (<= m 2.05e-14) (/ a_m (+ (* k k) t_0)) (* (pow k m) 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 t_0 = 1.0 + (k * 10.0);
	double tmp;
	if (m <= -6.2e-25) {
		tmp = pow(k, m) * (a_m / t_0);
	} else if (m <= 2.05e-14) {
		tmp = a_m / ((k * k) + t_0);
	} else {
		tmp = pow(k, m) * 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (k * 10.0d0)
    if (m <= (-6.2d-25)) then
        tmp = (k ** m) * (a_m / t_0)
    else if (m <= 2.05d-14) then
        tmp = a_m / ((k * k) + t_0)
    else
        tmp = (k ** m) * 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 t_0 = 1.0 + (k * 10.0);
	double tmp;
	if (m <= -6.2e-25) {
		tmp = Math.pow(k, m) * (a_m / t_0);
	} else if (m <= 2.05e-14) {
		tmp = a_m / ((k * k) + t_0);
	} else {
		tmp = Math.pow(k, m) * 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):
	t_0 = 1.0 + (k * 10.0)
	tmp = 0
	if m <= -6.2e-25:
		tmp = math.pow(k, m) * (a_m / t_0)
	elif m <= 2.05e-14:
		tmp = a_m / ((k * k) + t_0)
	else:
		tmp = math.pow(k, m) * 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)
	t_0 = Float64(1.0 + Float64(k * 10.0))
	tmp = 0.0
	if (m <= -6.2e-25)
		tmp = Float64((k ^ m) * Float64(a_m / t_0));
	elseif (m <= 2.05e-14)
		tmp = Float64(a_m / Float64(Float64(k * k) + t_0));
	else
		tmp = Float64((k ^ m) * 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)
	t_0 = 1.0 + (k * 10.0);
	tmp = 0.0;
	if (m <= -6.2e-25)
		tmp = (k ^ m) * (a_m / t_0);
	elseif (m <= 2.05e-14)
		tmp = a_m / ((k * k) + t_0);
	else
		tmp = (k ^ m) * 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_] := Block[{t$95$0 = N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -6.2e-25], N[(N[Power[k, m], $MachinePrecision] * N[(a$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.05e-14], N[(a$95$m / N[(N[(k * k), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -6.19999999999999989e-25

   1. Initial program 98.5%

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

      1. associate-*l/ 98.5%

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

      2. associate-+l+ 98.5%

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

      3. distribute-rgt-out 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in k around 0 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   if -6.19999999999999989e-25 < m < 2.0500000000000001e-14

   1. Initial program 94.4%

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

      1. sqr-neg 94.4%

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

      2. cancel-sign-sub 94.4%

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

      3. distribute-lft-neg-in 94.4%

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

      4. neg-sub0 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 94.4%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 94.4%

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

      11. +-commutative 94.4%

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

      12. *-commutative 94.4%

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

      13. sqr-neg 94.4%

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

   3. Simplified 94.4%

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

      1. add-cube-cbrt 93.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 93.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 93.1%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 94.4%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 94.4%

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

      2. *-lft-identity 94.4%

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

   8. Simplified 94.4%

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

   if 2.0500000000000001e-14 < m

   1. Initial program 71.3%

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

      1. associate-*l/ 66.0%

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

      2. associate-+l+ 66.0%

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

      3. distribute-rgt-out 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in k around 0 99.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.2 \cdot 10^{-25}:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 2.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 6: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-9} \lor \neg \left(m \leq 9 \cdot 10^{-15}\right):\\ \;\;\;\;{k}^{m} \cdot a_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -3.6e-9) (not (<= m 9e-15)))
    (* (pow k m) a_m)
    (/ a_m (+ (* k k) (+ 1.0 (* k 10.0)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -3.6e-9) || !(m <= 9e-15)) {
		tmp = pow(k, m) * a_m;
	} else {
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3.6d-9)) .or. (.not. (m <= 9d-15))) then
        tmp = (k ** m) * a_m
    else
        tmp = a_m / ((k * k) + (1.0d0 + (k * 10.0d0)))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -3.6e-9) || !(m <= 9e-15)) {
		tmp = Math.pow(k, m) * a_m;
	} else {
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if (m <= -3.6e-9) or not (m <= 9e-15):
		tmp = math.pow(k, m) * a_m
	else:
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((m <= -3.6e-9) || !(m <= 9e-15))
		tmp = Float64((k ^ m) * a_m);
	else
		tmp = Float64(a_m / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if ((m <= -3.6e-9) || ~((m <= 9e-15)))
		tmp = (k ^ m) * a_m;
	else
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[m, -3.6e-9], N[Not[LessEqual[m, 9e-15]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision], N[(a$95$m / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -3.6e-9 or 8.9999999999999995e-15 < m

   1. Initial program 83.0%

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

      1. associate-*l/ 79.9%

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

      2. associate-+l+ 79.9%

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

      3. distribute-rgt-out 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in k around 0 99.4%

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

   if -3.6e-9 < m < 8.9999999999999995e-15

   1. Initial program 93.5%

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

      1. sqr-neg 93.5%

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

      2. cancel-sign-sub 93.5%

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

      3. distribute-lft-neg-in 93.5%

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

      4. neg-sub0 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 93.5%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 93.5%

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

      11. +-commutative 93.5%

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

      12. *-commutative 93.5%

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

      13. sqr-neg 93.5%

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

   3. Simplified 93.5%

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

      1. add-cube-cbrt 92.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 92.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 92.3%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 93.5%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 93.5%

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

      2. *-lft-identity 93.5%

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

   8. Simplified 93.5%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-9} \lor \neg \left(m \leq 9 \cdot 10^{-15}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \end{array} \]

Alternative 7: 62.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -600:\\ \;\;\;\;\frac{a_m}{1 + \left(k \cdot \left(k + 10\right) + -1\right)}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a_m}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a_m\right)\\ \end{array} \end{array} \]

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 -600.0)
    (/ a_m (+ 1.0 (+ (* k (+ k 10.0)) -1.0)))
    (if (<= m 0.9)
      (/ a_m (+ (* k k) (+ 1.0 (* k 10.0))))
      (* -10.0 (* k a_m))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -600.0) {
		tmp = a_m / (1.0 + ((k * (k + 10.0)) + -1.0));
	} else if (m <= 0.9) {
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-600.0d0)) then
        tmp = a_m / (1.0d0 + ((k * (k + 10.0d0)) + (-1.0d0)))
    else if (m <= 0.9d0) then
        tmp = a_m / ((k * k) + (1.0d0 + (k * 10.0d0)))
    else
        tmp = (-10.0d0) * (k * a_m)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -600.0) {
		tmp = a_m / (1.0 + ((k * (k + 10.0)) + -1.0));
	} else if (m <= 0.9) {
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= -600.0:
		tmp = a_m / (1.0 + ((k * (k + 10.0)) + -1.0))
	elif m <= 0.9:
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)))
	else:
		tmp = -10.0 * (k * a_m)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -600.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(Float64(k * Float64(k + 10.0)) + -1.0)));
	elseif (m <= 0.9)
		tmp = Float64(a_m / Float64(Float64(k * k) + Float64(1.0 + Float64(k * 10.0))));
	else
		tmp = Float64(-10.0 * Float64(k * a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= -600.0)
		tmp = a_m / (1.0 + ((k * (k + 10.0)) + -1.0));
	elseif (m <= 0.9)
		tmp = a_m / ((k * k) + (1.0 + (k * 10.0)));
	else
		tmp = -10.0 * (k * a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -600.0], N[(a$95$m / N[(1.0 + N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.9], N[(a$95$m / N[(N[(k * k), $MachinePrecision] + N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -600

   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. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 31.1%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 31.1%

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

      2. *-lft-identity 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in k around inf 31.1%

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

       1. unpow2 31.1%

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

       2. distribute-rgt-in 31.1%

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

       3. +-commutative 31.1%

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

   11. Simplified 31.1%

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

       1. expm1-log1p-u 31.1%

          \[\leadsto \frac{a}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}} \]

       2. log1p-def 68.0%

          \[\leadsto \frac{a}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + k \cdot \left(k + 10\right)\right)}\right)} \]

       3. expm1-udef 68.0%

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

       4. add-exp-log 68.0%

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

       5. associate--l+ 68.0%

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

   13. Applied egg-rr 68.0%

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

   if -600 < m < 0.900000000000000022

   1. Initial program 93.9%

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

      1. sqr-neg 93.9%

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

      2. cancel-sign-sub 93.9%

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

      3. distribute-lft-neg-in 93.9%

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

      4. neg-sub0 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 93.9%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 93.9%

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

      11. +-commutative 93.9%

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

      12. *-commutative 93.9%

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

      13. sqr-neg 93.9%

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

   3. Simplified 93.9%

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

      1. add-cube-cbrt 92.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 92.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 92.6%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 92.0%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 92.0%

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

      2. *-lft-identity 92.0%

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

   8. Simplified 92.0%

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

   if 0.900000000000000022 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 12.6%

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

   6. Taylor expanded in k around inf 25.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -600:\\ \;\;\;\;\frac{a}{1 + \left(k \cdot \left(k + 10\right) + -1\right)}\\ \mathbf{elif}\;m \leq 0.9:\\ \;\;\;\;\frac{a}{k \cdot k + \left(1 + k \cdot 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8: 43.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a_m + -10 \cdot \left(k \cdot a_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a_m}{k + 10}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -5e-310)
    (* a_m (* k -10.0))
    (if (<= k 0.075)
      (+ a_m (* -10.0 (* k a_m)))
      (* (/ 1.0 k) (/ a_m (+ k 10.0)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = (1.0 / k) * (a_m / (k + 10.0));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5d-310)) then
        tmp = a_m * (k * (-10.0d0))
    else if (k <= 0.075d0) then
        tmp = a_m + ((-10.0d0) * (k * a_m))
    else
        tmp = (1.0d0 / k) * (a_m / (k + 10.0d0))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = (1.0 / k) * (a_m / (k + 10.0));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -5e-310:
		tmp = a_m * (k * -10.0)
	elif k <= 0.075:
		tmp = a_m + (-10.0 * (k * a_m))
	else:
		tmp = (1.0 / k) * (a_m / (k + 10.0))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -5e-310)
		tmp = Float64(a_m * Float64(k * -10.0));
	elseif (k <= 0.075)
		tmp = Float64(a_m + Float64(-10.0 * Float64(k * a_m)));
	else
		tmp = Float64(Float64(1.0 / k) * Float64(a_m / Float64(k + 10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = a_m * (k * -10.0);
	elseif (k <= 0.075)
		tmp = a_m + (-10.0 * (k * a_m));
	else
		tmp = (1.0 / k) * (a_m / (k + 10.0));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -5e-310], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a$95$m + N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / k), $MachinePrecision] * N[(a$95$m / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. associate-*l* 23.1%

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

      4. *-commutative 23.1%

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

   8. Simplified 23.1%

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

   if -4.999999999999985e-310 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 0.0749999999999999972 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

       1. *-un-lft-identity 61.6%

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

       2. times-frac 65.9%

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

   13. Applied egg-rr 65.9%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k + 10}\\ \end{array} \]

Alternative 9: 62.6% accurate, 8.7× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -600:\\ \;\;\;\;\frac{a_m}{1 + \left(t_0 + -1\right)}\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a_m}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (*
    a_s
    (if (<= m -600.0)
      (/ a_m (+ 1.0 (+ t_0 -1.0)))
      (if (<= m 1.25) (/ a_m (+ 1.0 t_0)) (* -10.0 (* k a_m)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -600.0) {
		tmp = a_m / (1.0 + (t_0 + -1.0));
	} else if (m <= 1.25) {
		tmp = a_m / (1.0 + t_0);
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-600.0d0)) then
        tmp = a_m / (1.0d0 + (t_0 + (-1.0d0)))
    else if (m <= 1.25d0) then
        tmp = a_m / (1.0d0 + t_0)
    else
        tmp = (-10.0d0) * (k * a_m)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -600.0) {
		tmp = a_m / (1.0 + (t_0 + -1.0));
	} else if (m <= 1.25) {
		tmp = a_m / (1.0 + t_0);
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -600.0:
		tmp = a_m / (1.0 + (t_0 + -1.0))
	elif m <= 1.25:
		tmp = a_m / (1.0 + t_0)
	else:
		tmp = -10.0 * (k * a_m)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -600.0)
		tmp = Float64(a_m / Float64(1.0 + Float64(t_0 + -1.0)));
	elseif (m <= 1.25)
		tmp = Float64(a_m / Float64(1.0 + t_0));
	else
		tmp = Float64(-10.0 * Float64(k * a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -600.0)
		tmp = a_m / (1.0 + (t_0 + -1.0));
	elseif (m <= 1.25)
		tmp = a_m / (1.0 + t_0);
	else
		tmp = -10.0 * (k * a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -600.0], N[(a$95$m / N[(1.0 + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.25], N[(a$95$m / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -600

   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. sqr-neg 100.0%

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

      2. cancel-sign-sub 100.0%

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

      3. distribute-lft-neg-in 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 100.0%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 100.0%

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

      11. +-commutative 100.0%

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

      12. *-commutative 100.0%

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

      13. sqr-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 100.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 31.1%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 31.1%

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

      2. *-lft-identity 31.1%

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

   8. Simplified 31.1%

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

   9. Taylor expanded in k around inf 31.1%

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

       1. unpow2 31.1%

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

       2. distribute-rgt-in 31.1%

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

       3. +-commutative 31.1%

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

   11. Simplified 31.1%

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

       1. expm1-log1p-u 31.1%

          \[\leadsto \frac{a}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}} \]

       2. log1p-def 68.0%

          \[\leadsto \frac{a}{\mathsf{expm1}\left(\color{blue}{\log \left(1 + k \cdot \left(k + 10\right)\right)}\right)} \]

       3. expm1-udef 68.0%

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

       4. add-exp-log 68.0%

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

       5. associate--l+ 68.0%

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

   13. Applied egg-rr 68.0%

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

   if -600 < m < 1.25

   1. Initial program 93.9%

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

      1. associate-*l/ 93.9%

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

      2. associate-+l+ 93.9%

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

      3. distribute-rgt-out 93.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in m around 0 92.0%

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

   if 1.25 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 12.6%

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

   6. Taylor expanded in k around inf 25.3%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -600:\\ \;\;\;\;\frac{a}{1 + \left(k \cdot \left(k + 10\right) + -1\right)}\\ \mathbf{elif}\;m \leq 1.25:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10: 29.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a_m + -10 \cdot \left(k \cdot a_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{k \cdot 10}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -5e-310)
    (* a_m (* k -10.0))
    (if (<= k 0.075) (+ a_m (* -10.0 (* k a_m))) (/ a_m (* k 10.0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = a_m / (k * 10.0);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5d-310)) then
        tmp = a_m * (k * (-10.0d0))
    else if (k <= 0.075d0) then
        tmp = a_m + ((-10.0d0) * (k * a_m))
    else
        tmp = a_m / (k * 10.0d0)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = a_m / (k * 10.0);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -5e-310:
		tmp = a_m * (k * -10.0)
	elif k <= 0.075:
		tmp = a_m + (-10.0 * (k * a_m))
	else:
		tmp = a_m / (k * 10.0)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -5e-310)
		tmp = Float64(a_m * Float64(k * -10.0));
	elseif (k <= 0.075)
		tmp = Float64(a_m + Float64(-10.0 * Float64(k * a_m)));
	else
		tmp = Float64(a_m / Float64(k * 10.0));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = a_m * (k * -10.0);
	elseif (k <= 0.075)
		tmp = a_m + (-10.0 * (k * a_m));
	else
		tmp = a_m / (k * 10.0);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -5e-310], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a$95$m + N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. associate-*l* 23.1%

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

      4. *-commutative 23.1%

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

   8. Simplified 23.1%

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

   if -4.999999999999985e-310 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 0.0749999999999999972 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in k around 0 21.2%

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

       1. *-commutative 21.2%

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

   14. Simplified 21.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 11: 42.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a_m + -10 \cdot \left(k \cdot a_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -5e-310)
    (* a_m (* k -10.0))
    (if (<= k 0.075) (+ a_m (* -10.0 (* k a_m))) (/ a_m (* k (+ k 10.0)))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = a_m / (k * (k + 10.0));
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5d-310)) then
        tmp = a_m * (k * (-10.0d0))
    else if (k <= 0.075d0) then
        tmp = a_m + ((-10.0d0) * (k * a_m))
    else
        tmp = a_m / (k * (k + 10.0d0))
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.075) {
		tmp = a_m + (-10.0 * (k * a_m));
	} else {
		tmp = a_m / (k * (k + 10.0));
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -5e-310:
		tmp = a_m * (k * -10.0)
	elif k <= 0.075:
		tmp = a_m + (-10.0 * (k * a_m))
	else:
		tmp = a_m / (k * (k + 10.0))
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -5e-310)
		tmp = Float64(a_m * Float64(k * -10.0));
	elseif (k <= 0.075)
		tmp = Float64(a_m + Float64(-10.0 * Float64(k * a_m)));
	else
		tmp = Float64(a_m / Float64(k * Float64(k + 10.0)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = a_m * (k * -10.0);
	elseif (k <= 0.075)
		tmp = a_m + (-10.0 * (k * a_m));
	else
		tmp = a_m / (k * (k + 10.0));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -5e-310], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a$95$m + N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a$95$m / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. associate-*l* 23.1%

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

      4. *-commutative 23.1%

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

   8. Simplified 23.1%

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

   if -4.999999999999985e-310 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 0.0749999999999999972 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 12: 50.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 1.4:\\ \;\;\;\;\frac{a_m}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a_m\right)\\ \end{array} \end{array} \]

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 1.4) (/ a_m (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* k a_m)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.4) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.4d0) then
        tmp = a_m / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = (-10.0d0) * (k * a_m)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 1.4) {
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 1.4:
		tmp = a_m / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (k * a_m)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 1.4)
		tmp = Float64(a_m / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(-10.0 * Float64(k * a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 1.4)
		tmp = a_m / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (k * a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 1.4], N[(a$95$m / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 1.3999999999999999

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in m around 0 68.9%

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

   if 1.3999999999999999 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 12.6%

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

   6. Taylor expanded in k around inf 25.3%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.4:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13: 29.3% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-309}:\\ \;\;\;\;-10 \cdot \left(k \cdot a_m\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 3.5e-309)
    (* -10.0 (* k a_m))
    (if (<= k 0.1) a_m (* 0.1 (/ a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 3.5e-309) {
		tmp = -10.0 * (k * a_m);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = 0.1 * (a_m / k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 3.5d-309) then
        tmp = (-10.0d0) * (k * a_m)
    else if (k <= 0.1d0) then
        tmp = a_m
    else
        tmp = 0.1d0 * (a_m / k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 3.5e-309) {
		tmp = -10.0 * (k * a_m);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = 0.1 * (a_m / k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= 3.5e-309:
		tmp = -10.0 * (k * a_m)
	elif k <= 0.1:
		tmp = a_m
	else:
		tmp = 0.1 * (a_m / k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= 3.5e-309)
		tmp = Float64(-10.0 * Float64(k * a_m));
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = Float64(0.1 * Float64(a_m / k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= 3.5e-309)
		tmp = -10.0 * (k * a_m);
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = 0.1 * (a_m / k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, 3.5e-309], N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], a$95$m, N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 3.4999999999999992e-309

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

   if 3.4999999999999992e-309 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.3%

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

   if 0.10000000000000001 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in k around 0 20.2%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{-309}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 14: 29.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a_m}{k}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k -5e-310)
    (* a_m (* k -10.0))
    (if (<= k 0.1) a_m (* 0.1 (/ a_m k))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = 0.1 * (a_m / k);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-5d-310)) then
        tmp = a_m * (k * (-10.0d0))
    else if (k <= 0.1d0) then
        tmp = a_m
    else
        tmp = 0.1d0 * (a_m / k)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= -5e-310) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = 0.1 * (a_m / k);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= -5e-310:
		tmp = a_m * (k * -10.0)
	elif k <= 0.1:
		tmp = a_m
	else:
		tmp = 0.1 * (a_m / k)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= -5e-310)
		tmp = Float64(a_m * Float64(k * -10.0));
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = Float64(0.1 * Float64(a_m / k));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= -5e-310)
		tmp = a_m * (k * -10.0);
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = 0.1 * (a_m / k);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, -5e-310], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], a$95$m, N[(0.1 * N[(a$95$m / k), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < -4.999999999999985e-310

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. associate-*l* 23.1%

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

      4. *-commutative 23.1%

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

   8. Simplified 23.1%

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

   if -4.999999999999985e-310 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.3%

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

   if 0.10000000000000001 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in k around 0 20.2%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]

Alternative 15: 29.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;a_m \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a_m}{k \cdot 10}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 2e-309)
    (* a_m (* k -10.0))
    (if (<= k 0.1) a_m (/ a_m (* k 10.0))))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 2e-309) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = a_m / (k * 10.0);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 2d-309) then
        tmp = a_m * (k * (-10.0d0))
    else if (k <= 0.1d0) then
        tmp = a_m
    else
        tmp = a_m / (k * 10.0d0)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 2e-309) {
		tmp = a_m * (k * -10.0);
	} else if (k <= 0.1) {
		tmp = a_m;
	} else {
		tmp = a_m / (k * 10.0);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= 2e-309:
		tmp = a_m * (k * -10.0)
	elif k <= 0.1:
		tmp = a_m
	else:
		tmp = a_m / (k * 10.0)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= 2e-309)
		tmp = Float64(a_m * Float64(k * -10.0));
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = Float64(a_m / Float64(k * 10.0));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= 2e-309)
		tmp = a_m * (k * -10.0);
	elseif (k <= 0.1)
		tmp = a_m;
	else
		tmp = a_m / (k * 10.0);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, 2e-309], N[(a$95$m * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], a$95$m, N[(a$95$m / N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if k < 1.9999999999999988e-309

   1. Initial program 86.8%

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

      1. associate-*l/ 82.9%

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

      2. associate-+l+ 82.9%

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

      3. distribute-rgt-out 82.9%

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

   3. Simplified 82.9%

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

   4. Taylor expanded in m around 0 13.6%

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

   5. Taylor expanded in k around 0 15.0%

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

   6. Taylor expanded in k around inf 23.1%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. associate-*l* 23.1%

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

      4. *-commutative 23.1%

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

   8. Simplified 23.1%

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

   if 1.9999999999999988e-309 < k < 0.10000000000000001

   1. Initial program 100.0%

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

      1. associate-*l/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.4%

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

   5. Taylor expanded in k around 0 56.3%

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

   if 0.10000000000000001 < k

   1. Initial program 76.2%

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

      1. sqr-neg 76.2%

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

      2. cancel-sign-sub 76.2%

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

      3. distribute-lft-neg-in 76.2%

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

      4. neg-sub0 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) - \color{blue}{\left(0 - \left(-k\right) \cdot \left(-k\right)\right)}} \]

      5. associate--r- 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right) + \left(-k\right) \cdot \left(-k\right)}} \]

      6. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(1 + \left(10 \cdot k - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      7. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(\left(1 + 10 \cdot k\right) - 0\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      8. +-commutative 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\left(\color{blue}{\left(10 \cdot k + 1\right)} - 0\right) + \left(-k\right) \cdot \left(-k\right)} \]

      9. associate--l+ 76.2%

         \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{\left(10 \cdot k + \left(1 - 0\right)\right)} + \left(-k\right) \cdot \left(-k\right)} \]

      10. metadata-eval 76.2%

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

      11. +-commutative 76.2%

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

      12. *-commutative 76.2%

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

      13. sqr-neg 76.2%

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

   3. Simplified 76.2%

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

      1. add-cube-cbrt 76.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{a \cdot {k}^{m}} \cdot \sqrt[3]{a \cdot {k}^{m}}\right) \cdot \sqrt[3]{a \cdot {k}^{m}}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

      2. pow3 76.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   5. Applied egg-rr 76.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{3}}}{\left(1 + k \cdot 10\right) + k \cdot k} \]

   6. Taylor expanded in m around 0 61.7%

      \[\leadsto \frac{\color{blue}{{1}^{0.3333333333333333} \cdot a}}{\left(1 + k \cdot 10\right) + k \cdot k} \]
   7. Step-by-step derivation

      1. pow-base-1 61.7%

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

      2. *-lft-identity 61.7%

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

   8. Simplified 61.7%

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

   9. Taylor expanded in k around inf 61.6%

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

       1. unpow2 61.6%

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

       2. distribute-rgt-in 61.6%

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

       3. +-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in k around 0 21.2%

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

       1. *-commutative 21.2%

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

   14. Simplified 21.2%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 16: 24.9% accurate, 16.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ a_s = \mathsf{copysign}\left(1, a\right) \\ a_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 0.195:\\ \;\;\;\;a_m\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
a_s = (copysign.f64 1 a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 0.195) a_m (* -10.0 (* k a_m)))))

C

a_m = fabs(a);
a_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 0.195) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Fortran

a_m = abs(a)
a_s = copysign(1.0d0, a)
real(8) function code(a_s, a_m, k, m)
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.195d0) then
        tmp = a_m
    else
        tmp = (-10.0d0) * (k * a_m)
    end if
    code = a_s * tmp
end function

Java

a_m = Math.abs(a);
a_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 0.195) {
		tmp = a_m;
	} else {
		tmp = -10.0 * (k * a_m);
	}
	return a_s * tmp;
}

Python

a_m = math.fabs(a)
a_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 0.195:
		tmp = a_m
	else:
		tmp = -10.0 * (k * a_m)
	return a_s * tmp

Julia

a_m = abs(a)
a_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 0.195)
		tmp = a_m;
	else
		tmp = Float64(-10.0 * Float64(k * a_m));
	end
	return Float64(a_s * tmp)
end

MATLAB

a_m = abs(a);
a_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 0.195)
		tmp = a_m;
	else
		tmp = -10.0 * (k * a_m);
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
a_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 0.195], a$95$m, N[(-10.0 * N[(k * a$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 0.19500000000000001

   1. Initial program 96.2%

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

      1. associate-*l/ 96.2%

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

      2. associate-+l+ 96.2%

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

      3. distribute-rgt-out 96.2%

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

   3. Simplified 96.2%

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

   4. Taylor expanded in m around 0 68.9%

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

   5. Taylor expanded in k around 0 29.8%

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

   if 0.19500000000000001 < m

   1. Initial program 70.0%

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

      1. associate-*l/ 64.4%

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

      2. associate-+l+ 64.4%

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

      3. distribute-rgt-out 64.4%

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

   3. Simplified 64.4%

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

   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 12.6%

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

   6. Taylor expanded in k around inf 25.3%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.195:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 17: 19.7% 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 87.0%

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

   1. associate-*l/ 85.0%

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

   2. associate-+l+ 85.0%

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

   3. distribute-rgt-out 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in m around 0 45.7%

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

5. Taylor expanded in k around 0 20.8%

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

6. Final simplification 20.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023336 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))