Falkner and Boettcher, Appendix A

Percentage Accurate: 89.8% → 99.8%

Time: 14.4s

Alternatives: 18

Speedup: 1.1×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ t_1 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;k \leq -2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)) (t_1 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= k -2.0) t_0 (/ (/ t_0 t_1) t_1))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double t_1 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (k <= -2.0) {
		tmp = t_0;
	} else {
		tmp = (t_0 / t_1) / t_1;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	t_1 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (k <= -2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_0 / t_1) / t_1);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, -2.0], t$95$0, N[(N[(t$95$0 / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < -2

   1. Initial program 95.1%

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

      1. associate-*r/ 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-+l+ 95.1%

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

      4. +-commutative 95.1%

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

      5. distribute-rgt-out 95.1%

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

      6. fma-def 95.1%

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

      7. +-commutative 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -2 < k

   1. Initial program 88.0%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

      3. associate-+l+ 88.0%

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

      4. +-commutative 88.0%

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

      5. distribute-rgt-out 88.0%

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

      6. fma-def 88.0%

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

      7. +-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. associate-*l/ 88.0%

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

      2. *-commutative 88.0%

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

      3. fma-udef 88.0%

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

      4. +-commutative 88.0%

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

      5. +-commutative 88.0%

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

      6. add-sqr-sqrt 88.0%

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

      7. associate-/r* 88.0%

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

      8. *-commutative 88.0%

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

      9. distribute-rgt-in 88.0%

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

      10. associate-+l+ 88.0%

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

      11. +-commutative 88.0%

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

      12. add-sqr-sqrt 88.0%

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

      13. hypot-def 88.0%

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

      14. +-commutative 88.0%

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

      15. *-commutative 88.0%

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

      16. fma-def 88.0%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{{k}^{m} \cdot a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2:\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{k}^{m} \cdot a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \end{array} \]

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+187}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 1e+187)
     (* (/ (pow k m) (hypot 1.0 k)) (/ a (hypot 1.0 k)))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+187) {
		tmp = (pow(k, m) / hypot(1.0, k)) * (a / hypot(1.0, k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+187) {
		tmp = (Math.pow(k, m) / Math.hypot(1.0, k)) * (a / Math.hypot(1.0, k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+187:
		tmp = (math.pow(k, m) / math.hypot(1.0, k)) * (a / math.hypot(1.0, k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 1e+187)
		tmp = Float64(Float64((k ^ m) / hypot(1.0, k)) * Float64(a / hypot(1.0, k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 1e+187)
		tmp = ((k ^ m) / hypot(1.0, k)) * (a / hypot(1.0, k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+187], N[(N[(N[Power[k, m], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 9.99999999999999907e186

   1. Initial program 97.0%

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

   2. Taylor expanded in k around 0 96.0%

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

      1. *-commutative 96.0%

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

      2. add-sqr-sqrt 96.0%

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

      3. times-frac 96.0%

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

      4. hypot-1-def 96.0%

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

      5. hypot-1-def 98.9%

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

   4. Applied egg-rr 98.9%

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

   if 9.99999999999999907e186 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 62.7%

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

      1. associate-*r/ 62.7%

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

      2. *-commutative 62.7%

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

      3. associate-+l+ 62.7%

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

      4. +-commutative 62.7%

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

      5. distribute-rgt-out 62.7%

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

      6. fma-def 62.7%

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

      7. +-commutative 62.7%

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

   3. Simplified 62.7%

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

   4. Taylor expanded in k around 0 99.4%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+187}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.6:\\ \;\;\;\;{k}^{m} \cdot \frac{-a}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.6)
   (* (pow k m) (/ (- a) (- -1.0 (* k (+ k 10.0)))))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.6) {
		tmp = pow(k, m) * (-a / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.6) {
		tmp = Math.pow(k, m) * (-a / (-1.0 - (k * (k + 10.0))));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 4.6:
		tmp = math.pow(k, m) * (-a / (-1.0 - (k * (k + 10.0))))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 4.6)
		tmp = Float64((k ^ m) * Float64(Float64(-a) / Float64(-1.0 - Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 4.6)
		tmp = (k ^ m) * (-a / (-1.0 - (k * (k + 10.0))));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 4.6], N[(N[Power[k, m], $MachinePrecision] * N[((-a) / N[(-1.0 - N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.5999999999999996

   1. Initial program 96.6%

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

      1. associate-*r/ 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-+l+ 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-rgt-out 96.6%

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

      6. fma-def 96.6%

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

      7. +-commutative 96.6%

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

   3. Simplified 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-commutative 96.6%

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

      3. fma-udef 96.6%

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

      4. +-commutative 96.6%

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

      5. +-commutative 96.6%

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

      6. frac-2neg 96.6%

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

      7. *-commutative 96.6%

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

      8. distribute-rgt-neg-in 96.6%

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

      9. neg-sub0 96.6%

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

      10. +-commutative 96.6%

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

      11. associate--r+ 96.6%

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

      12. metadata-eval 96.6%

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

   5. Applied egg-rr 96.6%

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

      1. *-commutative 96.6%

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

      2. associate-*l/ 96.7%

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

      3. *-commutative 96.7%

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

   7. Simplified 96.7%

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

   if 4.5999999999999996 < m

   1. Initial program 73.5%

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

      1. associate-*r/ 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-+l+ 73.5%

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

      4. +-commutative 73.5%

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

      5. distribute-rgt-out 73.5%

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

      6. fma-def 73.5%

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

      7. +-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 4.6:\\ \;\;\;\;{k}^{m} \cdot \frac{-a}{-1 - k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.9) (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m))) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.9d0) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.9:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.9)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.9], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. associate-+l+ 96.6%

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

      3. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   if 2.89999999999999991 < m

   1. Initial program 73.5%

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

      1. associate-*r/ 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-+l+ 73.5%

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

      4. +-commutative 73.5%

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

      5. distribute-rgt-out 73.5%

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

      6. fma-def 73.5%

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

      7. +-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot \left(k + 10\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 5: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;\frac{t_0}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a))) (if (<= m 2.5) (/ t_0 (+ 1.0 (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= 2.5) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a
    if (m <= 2.5d0) then
        tmp = t_0 / (1.0d0 + (k * k))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (m <= 2.5) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if m <= 2.5:
		tmp = t_0 / (1.0 + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= 2.5)
		tmp = Float64(t_0 / Float64(1.0 + Float64(k * k)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (m <= 2.5)
		tmp = t_0 / (1.0 + (k * k));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, 2.5], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5

   1. Initial program 96.6%

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

   2. Taylor expanded in k around 0 95.3%

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

   if 2.5 < m

   1. Initial program 73.5%

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

      1. associate-*r/ 73.5%

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

      2. *-commutative 73.5%

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

      3. associate-+l+ 73.5%

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

      4. +-commutative 73.5%

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

      5. distribute-rgt-out 73.5%

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

      6. fma-def 73.5%

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

      7. +-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 6: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.2 \cdot 10^{-7} \lor \neg \left(m \leq 8.2 \cdot 10^{-8}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.2e-7) (not (<= m 8.2e-8)))
   (* (pow k m) a)
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.2e-7) || !(m <= 8.2e-8)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3.2d-7)) .or. (.not. (m <= 8.2d-8))) then
        tmp = (k ** m) * a
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.2e-7) || !(m <= 8.2e-8)) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -3.2e-7) or not (m <= 8.2e-8):
		tmp = math.pow(k, m) * a
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3.2e-7) || !(m <= 8.2e-8))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -3.2e-7) || ~((m <= 8.2e-8)))
		tmp = (k ^ m) * a;
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3.2e-7], N[Not[LessEqual[m, 8.2e-8]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.2000000000000001e-7 or 8.20000000000000063e-8 < m

   1. Initial program 86.4%

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

      1. associate-*r/ 86.4%

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

      2. *-commutative 86.4%

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

      3. associate-+l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. distribute-rgt-out 86.4%

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

      6. fma-def 86.4%

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

      7. +-commutative 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -3.2000000000000001e-7 < m < 8.20000000000000063e-8

   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. associate-*r/ 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-+l+ 93.8%

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

      4. +-commutative 93.8%

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

      5. distribute-rgt-out 93.8%

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

      6. fma-def 93.8%

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

      7. +-commutative 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in m around 0 91.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.2 \cdot 10^{-7} \lor \neg \left(m \leq 8.2 \cdot 10^{-8}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7: 46.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{if}\;k \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -9 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-309} \lor \neg \left(k \leq 0.0042\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 k) (/ a k))))
   (if (<= k -1.18e+111)
     t_0
     (if (<= k -9e-132)
       (* a (* k -10.0))
       (if (or (<= k 7e-309) (not (<= k 0.0042))) t_0 a)))))

C

double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.18e+111) {
		tmp = t_0;
	} else if (k <= -9e-132) {
		tmp = a * (k * -10.0);
	} else if ((k <= 7e-309) || !(k <= 0.0042)) {
		tmp = t_0;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / k) * (a / k)
    if (k <= (-1.18d+111)) then
        tmp = t_0
    else if (k <= (-9d-132)) then
        tmp = a * (k * (-10.0d0))
    else if ((k <= 7d-309) .or. (.not. (k <= 0.0042d0))) then
        tmp = t_0
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.18e+111) {
		tmp = t_0;
	} else if (k <= -9e-132) {
		tmp = a * (k * -10.0);
	} else if ((k <= 7e-309) || !(k <= 0.0042)) {
		tmp = t_0;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (1.0 / k) * (a / k)
	tmp = 0
	if k <= -1.18e+111:
		tmp = t_0
	elif k <= -9e-132:
		tmp = a * (k * -10.0)
	elif (k <= 7e-309) or not (k <= 0.0042):
		tmp = t_0
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(1.0 / k) * Float64(a / k))
	tmp = 0.0
	if (k <= -1.18e+111)
		tmp = t_0;
	elseif (k <= -9e-132)
		tmp = Float64(a * Float64(k * -10.0));
	elseif ((k <= 7e-309) || !(k <= 0.0042))
		tmp = t_0;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (1.0 / k) * (a / k);
	tmp = 0.0;
	if (k <= -1.18e+111)
		tmp = t_0;
	elseif (k <= -9e-132)
		tmp = a * (k * -10.0);
	elseif ((k <= 7e-309) || ~((k <= 0.0042)))
		tmp = t_0;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.18e+111], t$95$0, If[LessEqual[k, -9e-132], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k, 7e-309], N[Not[LessEqual[k, 0.0042]], $MachinePrecision]], t$95$0, a]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.1799999999999999e111 or -8.9999999999999999e-132 < k < 6.9999999999999984e-309 or 0.00419999999999999974 < k

   1. Initial program 79.7%

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

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-+l+ 79.7%

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

      4. +-commutative 79.7%

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

      5. distribute-rgt-out 79.7%

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

      6. fma-def 79.7%

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

      7. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in m around 0 51.4%

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

   5. Taylor expanded in k around inf 56.9%

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

      1. *-un-lft-identity 56.9%

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

      2. unpow2 56.9%

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

      3. times-frac 57.0%

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

   7. Applied egg-rr 57.0%

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

   if -1.1799999999999999e111 < k < -8.9999999999999999e-132

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if 6.9999999999999984e-309 < k < 0.00419999999999999974

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 55.4%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.18 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq -9 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-309} \lor \neg \left(k \leq 0.0042\right):\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 8: 46.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{if}\;k \leq -1.2 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -2.85 \cdot 10^{-139}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310} \lor \neg \left(k \leq 0.0036\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 k) (/ a k))))
   (if (<= k -1.2e+111)
     t_0
     (if (<= k -2.85e-139)
       (* a (* k -10.0))
       (if (or (<= k -1e-310) (not (<= k 0.0036)))
         t_0
         (+ a (* -10.0 (* k a))))))))

C

double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.2e+111) {
		tmp = t_0;
	} else if (k <= -2.85e-139) {
		tmp = a * (k * -10.0);
	} else if ((k <= -1e-310) || !(k <= 0.0036)) {
		tmp = t_0;
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / k) * (a / k)
    if (k <= (-1.2d+111)) then
        tmp = t_0
    else if (k <= (-2.85d-139)) then
        tmp = a * (k * (-10.0d0))
    else if ((k <= (-1d-310)) .or. (.not. (k <= 0.0036d0))) then
        tmp = t_0
    else
        tmp = a + ((-10.0d0) * (k * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.2e+111) {
		tmp = t_0;
	} else if (k <= -2.85e-139) {
		tmp = a * (k * -10.0);
	} else if ((k <= -1e-310) || !(k <= 0.0036)) {
		tmp = t_0;
	} else {
		tmp = a + (-10.0 * (k * a));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (1.0 / k) * (a / k)
	tmp = 0
	if k <= -1.2e+111:
		tmp = t_0
	elif k <= -2.85e-139:
		tmp = a * (k * -10.0)
	elif (k <= -1e-310) or not (k <= 0.0036):
		tmp = t_0
	else:
		tmp = a + (-10.0 * (k * a))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(1.0 / k) * Float64(a / k))
	tmp = 0.0
	if (k <= -1.2e+111)
		tmp = t_0;
	elseif (k <= -2.85e-139)
		tmp = Float64(a * Float64(k * -10.0));
	elseif ((k <= -1e-310) || !(k <= 0.0036))
		tmp = t_0;
	else
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (1.0 / k) * (a / k);
	tmp = 0.0;
	if (k <= -1.2e+111)
		tmp = t_0;
	elseif (k <= -2.85e-139)
		tmp = a * (k * -10.0);
	elseif ((k <= -1e-310) || ~((k <= 0.0036)))
		tmp = t_0;
	else
		tmp = a + (-10.0 * (k * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.2e+111], t$95$0, If[LessEqual[k, -2.85e-139], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[k, -1e-310], N[Not[LessEqual[k, 0.0036]], $MachinePrecision]], t$95$0, N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.20000000000000003e111 or -2.8499999999999999e-139 < k < -9.999999999999969e-311 or 0.0035999999999999999 < k

   1. Initial program 79.7%

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

      1. associate-*r/ 79.7%

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

      2. *-commutative 79.7%

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

      3. associate-+l+ 79.7%

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

      4. +-commutative 79.7%

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

      5. distribute-rgt-out 79.7%

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

      6. fma-def 79.7%

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

      7. +-commutative 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in m around 0 51.4%

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

   5. Taylor expanded in k around inf 56.9%

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

      1. *-un-lft-identity 56.9%

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

      2. unpow2 56.9%

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

      3. times-frac 57.0%

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

   7. Applied egg-rr 57.0%

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

   if -1.20000000000000003e111 < k < -2.8499999999999999e-139

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if -9.999999999999969e-311 < k < 0.0035999999999999999

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 56.4%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq -2.85 \cdot 10^{-139}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310} \lor \neg \left(k \leq 0.0036\right):\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9: 46.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -8.2 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 k) (/ a k))))
   (if (<= k -1.35e+111)
     t_0
     (if (<= k -8.2e-132)
       (* a (* k -10.0))
       (if (<= k -1e-310)
         t_0
         (if (<= k 0.0042) (+ a (* -10.0 (* k a))) (/ 1.0 (* k (/ k a)))))))))

C

double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.35e+111) {
		tmp = t_0;
	} else if (k <= -8.2e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= -1e-310) {
		tmp = t_0;
	} else if (k <= 0.0042) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / k) * (a / k)
    if (k <= (-1.35d+111)) then
        tmp = t_0
    else if (k <= (-8.2d-132)) then
        tmp = a * (k * (-10.0d0))
    else if (k <= (-1d-310)) then
        tmp = t_0
    else if (k <= 0.0042d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = (1.0 / k) * (a / k);
	double tmp;
	if (k <= -1.35e+111) {
		tmp = t_0;
	} else if (k <= -8.2e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= -1e-310) {
		tmp = t_0;
	} else if (k <= 0.0042) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (1.0 / k) * (a / k)
	tmp = 0
	if k <= -1.35e+111:
		tmp = t_0
	elif k <= -8.2e-132:
		tmp = a * (k * -10.0)
	elif k <= -1e-310:
		tmp = t_0
	elif k <= 0.0042:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(1.0 / k) * Float64(a / k))
	tmp = 0.0
	if (k <= -1.35e+111)
		tmp = t_0;
	elseif (k <= -8.2e-132)
		tmp = Float64(a * Float64(k * -10.0));
	elseif (k <= -1e-310)
		tmp = t_0;
	elseif (k <= 0.0042)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (1.0 / k) * (a / k);
	tmp = 0.0;
	if (k <= -1.35e+111)
		tmp = t_0;
	elseif (k <= -8.2e-132)
		tmp = a * (k * -10.0);
	elseif (k <= -1e-310)
		tmp = t_0;
	elseif (k <= 0.0042)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.35e+111], t$95$0, If[LessEqual[k, -8.2e-132], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-310], t$95$0, If[LessEqual[k, 0.0042], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -1.3499999999999999e111 or -8.20000000000000013e-132 < k < -9.999999999999969e-311

   1. Initial program 94.9%

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

      1. associate-*r/ 94.9%

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

      2. *-commutative 94.9%

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

      3. associate-+l+ 94.9%

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

      4. +-commutative 94.9%

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

      5. distribute-rgt-out 94.9%

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

      6. fma-def 94.9%

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

      7. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in m around 0 43.8%

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

   5. Taylor expanded in k around inf 65.2%

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

      1. *-un-lft-identity 65.2%

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

      2. unpow2 65.2%

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

      3. times-frac 58.1%

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

   7. Applied egg-rr 58.1%

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

   if -1.3499999999999999e111 < k < -8.20000000000000013e-132

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if -9.999999999999969e-311 < k < 0.00419999999999999974

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 0.00419999999999999974 < k

   1. Initial program 73.7%

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

      1. associate-*r/ 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-+l+ 73.6%

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

      4. +-commutative 73.6%

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

      5. distribute-rgt-out 73.6%

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

      6. fma-def 73.6%

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

      7. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in m around 0 54.4%

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

   5. Taylor expanded in k around inf 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. unpow2 53.6%

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

      3. times-frac 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. associate-*l/ 56.6%

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

      2. associate-/l* 56.6%

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

      3. associate-/l* 53.6%

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

      4. unpow2 53.6%

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

   9. Simplified 53.6%

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

       1. div-inv 53.6%

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

       2. unpow2 53.6%

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

       3. associate-*l* 56.5%

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

       4. div-inv 56.6%

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

   11. Applied egg-rr 56.6%

       \[\leadsto \frac{1}{\color{blue}{k \cdot \frac{k}{a}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq -8.2 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 10: 47.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.35e+111)
   (/ a (* k (+ k 10.0)))
   (if (<= k -4.4e-132)
     (* a (* k -10.0))
     (if (<= k -1e-310)
       (* (/ 1.0 k) (/ a k))
       (if (<= k 0.0042) (+ a (* -10.0 (* k a))) (/ 1.0 (* k (/ k a))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.35e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -4.4e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= -1e-310) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.35d+111)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= (-4.4d-132)) then
        tmp = a * (k * (-10.0d0))
    else if (k <= (-1d-310)) then
        tmp = (1.0d0 / k) * (a / k)
    else if (k <= 0.0042d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.35e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -4.4e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= -1e-310) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.35e+111:
		tmp = a / (k * (k + 10.0))
	elif k <= -4.4e-132:
		tmp = a * (k * -10.0)
	elif k <= -1e-310:
		tmp = (1.0 / k) * (a / k)
	elif k <= 0.0042:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.35e+111)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= -4.4e-132)
		tmp = Float64(a * Float64(k * -10.0));
	elseif (k <= -1e-310)
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	elseif (k <= 0.0042)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.35e+111)
		tmp = a / (k * (k + 10.0));
	elseif (k <= -4.4e-132)
		tmp = a * (k * -10.0);
	elseif (k <= -1e-310)
		tmp = (1.0 / k) * (a / k);
	elseif (k <= 0.0042)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.35e+111], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -4.4e-132], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1e-310], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0042], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -1.3499999999999999e111

   1. Initial program 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-+l+ 92.0%

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

      4. +-commutative 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. fma-def 92.0%

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

      7. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in m around 0 65.4%

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

   5. Taylor expanded in k around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. distribute-rgt-in 65.4%

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

   7. Simplified 65.4%

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

   if -1.3499999999999999e111 < k < -4.39999999999999981e-132

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if -4.39999999999999981e-132 < k < -9.999999999999969e-311

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 5.4%

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

   5. Taylor expanded in k around inf 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. unpow2 64.8%

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

      3. times-frac 64.9%

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

   7. Applied egg-rr 64.9%

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

   if -9.999999999999969e-311 < k < 0.00419999999999999974

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 56.4%

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

   if 0.00419999999999999974 < k

   1. Initial program 73.7%

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

      1. associate-*r/ 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-+l+ 73.6%

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

      4. +-commutative 73.6%

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

      5. distribute-rgt-out 73.6%

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

      6. fma-def 73.6%

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

      7. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in m around 0 54.4%

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

   5. Taylor expanded in k around inf 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. unpow2 53.6%

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

      3. times-frac 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. associate-*l/ 56.6%

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

      2. associate-/l* 56.6%

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

      3. associate-/l* 53.6%

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

      4. unpow2 53.6%

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

   9. Simplified 53.6%

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

       1. div-inv 53.6%

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

       2. unpow2 53.6%

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

       3. associate-*l* 56.5%

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

       4. div-inv 56.6%

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

   11. Applied egg-rr 56.6%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -4.4 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 11: 47.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.12 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -3.6 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.12e+111)
   (/ a (* k (+ k 10.0)))
   (if (<= k -3.6e-132)
     (* a (* k -10.0))
     (if (<= k 2e-309)
       (* (/ 1.0 k) (/ a k))
       (if (<= k 0.0042) (/ a (+ 1.0 (* k 10.0))) (/ 1.0 (* k (/ k a))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.12e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -3.6e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= 2e-309) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.12d+111)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= (-3.6d-132)) then
        tmp = a * (k * (-10.0d0))
    else if (k <= 2d-309) then
        tmp = (1.0d0 / k) * (a / k)
    else if (k <= 0.0042d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = 1.0d0 / (k * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.12e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -3.6e-132) {
		tmp = a * (k * -10.0);
	} else if (k <= 2e-309) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = 1.0 / (k * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.12e+111:
		tmp = a / (k * (k + 10.0))
	elif k <= -3.6e-132:
		tmp = a * (k * -10.0)
	elif k <= 2e-309:
		tmp = (1.0 / k) * (a / k)
	elif k <= 0.0042:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = 1.0 / (k * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.12e+111)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= -3.6e-132)
		tmp = Float64(a * Float64(k * -10.0));
	elseif (k <= 2e-309)
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	elseif (k <= 0.0042)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.12e+111)
		tmp = a / (k * (k + 10.0));
	elseif (k <= -3.6e-132)
		tmp = a * (k * -10.0);
	elseif (k <= 2e-309)
		tmp = (1.0 / k) * (a / k);
	elseif (k <= 0.0042)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = 1.0 / (k * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.12e+111], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.6e-132], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2e-309], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0042], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -1.11999999999999995e111

   1. Initial program 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-+l+ 92.0%

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

      4. +-commutative 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. fma-def 92.0%

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

      7. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in m around 0 65.4%

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

   5. Taylor expanded in k around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. distribute-rgt-in 65.4%

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

   7. Simplified 65.4%

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

   if -1.11999999999999995e111 < k < -3.60000000000000007e-132

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if -3.60000000000000007e-132 < k < 1.9999999999999988e-309

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 5.4%

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

   5. Taylor expanded in k around inf 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. unpow2 64.8%

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

      3. times-frac 64.9%

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

   7. Applied egg-rr 64.9%

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

   if 1.9999999999999988e-309 < k < 0.00419999999999999974

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

   if 0.00419999999999999974 < k

   1. Initial program 73.7%

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

      1. associate-*r/ 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-+l+ 73.6%

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

      4. +-commutative 73.6%

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

      5. distribute-rgt-out 73.6%

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

      6. fma-def 73.6%

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

      7. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in m around 0 54.4%

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

   5. Taylor expanded in k around inf 53.6%

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

      1. *-un-lft-identity 53.6%

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

      2. unpow2 53.6%

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

      3. times-frac 56.6%

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

   7. Applied egg-rr 56.6%

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

      1. associate-*l/ 56.6%

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

      2. associate-/l* 56.6%

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

      3. associate-/l* 53.6%

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

      4. unpow2 53.6%

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

   9. Simplified 53.6%

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

       1. div-inv 53.6%

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

       2. unpow2 53.6%

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

       3. associate-*l* 56.5%

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

       4. div-inv 56.6%

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

   11. Applied egg-rr 56.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.12 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -3.6 \cdot 10^{-132}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 12: 47.7% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.85 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-134}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -1.85e+111)
   (/ a (* k (+ k 10.0)))
   (if (<= k -3.2e-134)
     (* a (* k -10.0))
     (if (<= k 4e-309)
       (* (/ 1.0 k) (/ a k))
       (if (<= k 0.0042) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.85e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -3.2e-134) {
		tmp = a * (k * -10.0);
	} else if (k <= 4e-309) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-1.85d+111)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= (-3.2d-134)) then
        tmp = a * (k * (-10.0d0))
    else if (k <= 4d-309) then
        tmp = (1.0d0 / k) * (a / k)
    else if (k <= 0.0042d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / (k + 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -1.85e+111) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= -3.2e-134) {
		tmp = a * (k * -10.0);
	} else if (k <= 4e-309) {
		tmp = (1.0 / k) * (a / k);
	} else if (k <= 0.0042) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / (k + 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -1.85e+111:
		tmp = a / (k * (k + 10.0))
	elif k <= -3.2e-134:
		tmp = a * (k * -10.0)
	elif k <= 4e-309:
		tmp = (1.0 / k) * (a / k)
	elif k <= 0.0042:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / (k + 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -1.85e+111)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= -3.2e-134)
		tmp = Float64(a * Float64(k * -10.0));
	elseif (k <= 4e-309)
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	elseif (k <= 0.0042)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / Float64(k + 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -1.85e+111)
		tmp = a / (k * (k + 10.0));
	elseif (k <= -3.2e-134)
		tmp = a * (k * -10.0);
	elseif (k <= 4e-309)
		tmp = (1.0 / k) * (a / k);
	elseif (k <= 0.0042)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / (k + 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -1.85e+111], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.2e-134], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4e-309], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.0042], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -1.8500000000000001e111

   1. Initial program 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-+l+ 92.0%

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

      4. +-commutative 92.0%

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

      5. distribute-rgt-out 92.0%

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

      6. fma-def 92.0%

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

      7. +-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in m around 0 65.4%

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

   5. Taylor expanded in k around inf 65.4%

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

      1. +-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. distribute-rgt-in 65.4%

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

   7. Simplified 65.4%

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

   if -1.8500000000000001e111 < k < -3.2000000000000001e-134

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.5%

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

   5. Taylor expanded in k around 0 9.7%

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

   6. Taylor expanded in k around inf 15.3%

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

      1. *-commutative 15.3%

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

      2. *-commutative 15.3%

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

      3. *-commutative 15.3%

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

      4. associate-*r* 15.4%

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

   8. Simplified 15.4%

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

   if -3.2000000000000001e-134 < k < 3.9999999999999977e-309

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 5.4%

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

   5. Taylor expanded in k around inf 64.8%

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

      1. *-un-lft-identity 64.8%

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

      2. unpow2 64.8%

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

      3. times-frac 64.9%

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

   7. Applied egg-rr 64.9%

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

   if 3.9999999999999977e-309 < k < 0.00419999999999999974

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 56.8%

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

   5. Taylor expanded in k around 0 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

   if 0.00419999999999999974 < k

   1. Initial program 73.7%

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

      1. associate-*r/ 73.6%

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

      2. *-commutative 73.6%

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

      3. associate-+l+ 73.6%

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

      4. +-commutative 73.6%

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

      5. distribute-rgt-out 73.6%

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

      6. fma-def 73.6%

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

      7. +-commutative 73.6%

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

   3. Simplified 73.6%

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

   4. Taylor expanded in m around 0 54.4%

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

   5. Taylor expanded in k around inf 54.3%

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

      1. +-commutative 54.3%

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

      2. unpow2 54.3%

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

      3. distribute-rgt-in 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in a around 0 54.3%

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

      1. +-commutative 54.3%

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

      2. associate-/r* 57.4%

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

   10. Simplified 57.4%

       \[\leadsto \color{blue}{\frac{\frac{a}{k}}{k + 10}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.85 \cdot 10^{+111}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-134}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.0042:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k + 10}\\ \end{array} \]

Alternative 13: 54.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.2e+28)
   (* (/ 1.0 k) (/ a k))
   (if (<= m 2.6e+24) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.2e+28) {
		tmp = (1.0 / k) * (a / k);
	} else if (m <= 2.6e+24) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.2d+28)) then
        tmp = (1.0d0 / k) * (a / k)
    else if (m <= 2.6d+24) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.2e+28) {
		tmp = (1.0 / k) * (a / k);
	} else if (m <= 2.6e+24) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.2e+28:
		tmp = (1.0 / k) * (a / k)
	elif m <= 2.6e+24:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.2e+28)
		tmp = Float64(Float64(1.0 / k) * Float64(a / k));
	elseif (m <= 2.6e+24)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.2e+28)
		tmp = (1.0 / k) * (a / k);
	elseif (m <= 2.6e+24)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.2e+28], N[(N[(1.0 / k), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.6e+24], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.19999999999999991e28

   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-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 39.4%

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

   5. Taylor expanded in k around inf 65.0%

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

      1. *-un-lft-identity 65.0%

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

      2. unpow2 65.0%

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

      3. times-frac 57.4%

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

   7. Applied egg-rr 57.4%

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

   if -1.19999999999999991e28 < m < 2.5999999999999998e24

   1. Initial program 93.6%

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

      1. associate-*r/ 93.5%

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

      2. *-commutative 93.5%

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

      3. associate-+l+ 93.5%

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

      4. +-commutative 93.5%

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

      5. distribute-rgt-out 93.5%

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

      6. fma-def 93.5%

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

      7. +-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in m around 0 84.9%

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

   if 2.5999999999999998e24 < m

   1. Initial program 73.4%

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

      1. associate-*r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-+l+ 73.4%

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

      4. +-commutative 73.4%

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

      5. distribute-rgt-out 73.4%

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

      6. fma-def 73.4%

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

      7. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 5.4%

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

   6. Taylor expanded in k around inf 12.6%

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

      1. *-commutative 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r* 12.6%

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

   8. Simplified 12.6%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{k} \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 14: 31.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;a \cdot \frac{0.1}{k}\\ \mathbf{elif}\;m \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.19) (* a (/ 0.1 k)) (if (<= m 8.8e+27) a (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = a * (0.1 / k);
	} else if (m <= 8.8e+27) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.19d0)) then
        tmp = a * (0.1d0 / k)
    else if (m <= 8.8d+27) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = a * (0.1 / k);
	} else if (m <= 8.8e+27) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.19:
		tmp = a * (0.1 / k)
	elif m <= 8.8e+27:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.19)
		tmp = Float64(a * Float64(0.1 / k));
	elseif (m <= 8.8e+27)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.19)
		tmp = a * (0.1 / k);
	elseif (m <= 8.8e+27)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.19], N[(a * N[(0.1 / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 8.8e+27], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.19

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.3%

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

   5. Taylor expanded in k around 0 17.6%

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

      1. *-commutative 17.6%

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

   7. Simplified 17.6%

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

   8. Taylor expanded in k around inf 23.4%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   9. Step-by-step derivation

      1. *-commutative 23.4%

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

   10. Simplified 23.4%

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

   11. Taylor expanded in a around 0 23.4%

       \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   12. Step-by-step derivation

       1. associate-*r/ 23.4%

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

       2. associate-*l/ 23.4%

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

       3. *-commutative 23.4%

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

   13. Simplified 23.4%

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

   if -0.19 < m < 8.7999999999999995e27

   1. Initial program 93.2%

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

      1. associate-*r/ 93.1%

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

      2. *-commutative 93.1%

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

      3. associate-+l+ 93.1%

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

      4. +-commutative 93.1%

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

      5. distribute-rgt-out 93.1%

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

      6. fma-def 93.1%

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

      7. +-commutative 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in m around 0 86.9%

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

   5. Taylor expanded in k around 0 48.7%

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

   if 8.7999999999999995e27 < m

   1. Initial program 73.4%

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

      1. associate-*r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-+l+ 73.4%

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

      4. +-commutative 73.4%

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

      5. distribute-rgt-out 73.4%

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

      6. fma-def 73.4%

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

      7. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 5.4%

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

   6. Taylor expanded in k around inf 12.6%

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

      1. *-commutative 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r* 12.6%

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

   8. Simplified 12.6%

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

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;a \cdot \frac{0.1}{k}\\ \mathbf{elif}\;m \leq 8.8 \cdot 10^{+27}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 15: 31.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.19) (/ 0.1 (/ k a)) (if (<= m 2.6e+24) a (* a (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = 0.1 / (k / a);
	} else if (m <= 2.6e+24) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.19d0)) then
        tmp = 0.1d0 / (k / a)
    else if (m <= 2.6d+24) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.19) {
		tmp = 0.1 / (k / a);
	} else if (m <= 2.6e+24) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.19:
		tmp = 0.1 / (k / a)
	elif m <= 2.6e+24:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.19)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (m <= 2.6e+24)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.19)
		tmp = 0.1 / (k / a);
	elseif (m <= 2.6e+24)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.19], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.6e+24], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.19

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l+ 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-rgt-out 100.0%

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

      6. fma-def 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 40.3%

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

   5. Taylor expanded in k around 0 17.6%

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

      1. *-commutative 17.6%

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

   7. Simplified 17.6%

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

   8. Taylor expanded in k around inf 23.4%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   9. Step-by-step derivation

      1. *-commutative 23.4%

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

   10. Simplified 23.4%

       \[\leadsto \color{blue}{\frac{a}{k} \cdot 0.1} \]
   11. Step-by-step derivation

       1. *-commutative 23.4%

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

       2. clear-num 23.8%

          \[\leadsto 0.1 \cdot \color{blue}{\frac{1}{\frac{k}{a}}} \]

       3. un-div-inv 23.8%

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

   12. Applied egg-rr 23.8%

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

   if -0.19 < m < 2.5999999999999998e24

   1. Initial program 93.2%

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

      1. associate-*r/ 93.1%

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

      2. *-commutative 93.1%

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

      3. associate-+l+ 93.1%

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

      4. +-commutative 93.1%

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

      5. distribute-rgt-out 93.1%

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

      6. fma-def 93.1%

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

      7. +-commutative 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in m around 0 86.9%

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

   5. Taylor expanded in k around 0 48.7%

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

   if 2.5999999999999998e24 < m

   1. Initial program 73.4%

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

      1. associate-*r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-+l+ 73.4%

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

      4. +-commutative 73.4%

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

      5. distribute-rgt-out 73.4%

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

      6. fma-def 73.4%

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

      7. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 5.4%

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

   6. Taylor expanded in k around inf 12.6%

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

      1. *-commutative 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r* 12.6%

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

   8. Simplified 12.6%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.19:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 16: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 2.6e+24) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.6e+24) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.6d+24) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.6e+24) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.6e+24:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.6e+24)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.6e+24)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.6e+24], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5999999999999998e24

   1. Initial program 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-+l+ 96.1%

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

      4. +-commutative 96.1%

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

      5. distribute-rgt-out 96.1%

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

      6. fma-def 96.1%

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

      7. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 66.6%

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

   5. Taylor expanded in k around 0 29.2%

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

   if 2.5999999999999998e24 < m

   1. Initial program 73.4%

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

      1. associate-*r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-+l+ 73.4%

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

      4. +-commutative 73.4%

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

      5. distribute-rgt-out 73.4%

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

      6. fma-def 73.4%

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

      7. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 5.4%

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

   6. Taylor expanded in k around inf 12.6%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.6 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 17: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 8e+24) a (* a (* k -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8e+24) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8d+24) then
        tmp = a
    else
        tmp = a * (k * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8e+24) {
		tmp = a;
	} else {
		tmp = a * (k * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8e+24:
		tmp = a
	else:
		tmp = a * (k * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8e+24)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8e+24)
		tmp = a;
	else
		tmp = a * (k * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8e+24], a, N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 7.9999999999999999e24

   1. Initial program 96.1%

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

      1. associate-*r/ 96.1%

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

      2. *-commutative 96.1%

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

      3. associate-+l+ 96.1%

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

      4. +-commutative 96.1%

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

      5. distribute-rgt-out 96.1%

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

      6. fma-def 96.1%

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

      7. +-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in m around 0 66.6%

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

   5. Taylor expanded in k around 0 29.2%

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

   if 7.9999999999999999e24 < m

   1. Initial program 73.4%

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

      1. associate-*r/ 73.4%

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

      2. *-commutative 73.4%

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

      3. associate-+l+ 73.4%

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

      4. +-commutative 73.4%

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

      5. distribute-rgt-out 73.4%

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

      6. fma-def 73.4%

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

      7. +-commutative 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in m around 0 3.1%

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

   5. Taylor expanded in k around 0 5.4%

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

   6. Taylor expanded in k around inf 12.6%

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

      1. *-commutative 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

      4. associate-*r* 12.6%

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

   8. Simplified 12.6%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{+24}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot -10\right)\\ \end{array} \]

Alternative 18: 20.7% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

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
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.1%

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

   1. associate-*r/ 89.1%

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

   2. *-commutative 89.1%

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

   3. associate-+l+ 89.1%

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

   4. +-commutative 89.1%

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

   5. distribute-rgt-out 89.1%

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

   6. fma-def 89.1%

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

   7. +-commutative 89.1%

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

3. Simplified 89.1%

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

4. Taylor expanded in m around 0 47.0%

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

5. Taylor expanded in k around 0 21.5%

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

6. Final simplification 21.5%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023305 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))