Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 97.0%

Time: 13.6s

Alternatives: 14

Speedup: 1.1×

• 22.9% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 1e+105) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 1d+105) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 1e+105) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 1e+105:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 1e+105)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 1e+105)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+105], t$95$1, 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))) < 9.9999999999999994e104

   1. Initial program 97.9%

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

   if 9.9999999999999994e104 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 64.7%

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

   2. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 10^{+105}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 3.1:\\ \;\;\;\;\frac{t_0}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= m 3.1) (/ t_0 (+ 1.0 (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 3.1) {
		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 = a * (k ** m)
    if (m <= 3.1d0) 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 = a * Math.pow(k, m);
	double tmp;
	if (m <= 3.1) {
		tmp = t_0 / (1.0 + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 3.1:
		tmp = t_0 / (1.0 + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 3.1)
		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 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 3.1)
		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[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 3.1], 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 < 3.10000000000000009

   1. Initial program 97.5%

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

   2. Taylor expanded in k around 0 96.0%

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

   if 3.10000000000000009 < m

   1. Initial program 78.3%

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

   2. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.1:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 3: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00022 \lor \neg \left(m \leq 7.2 \cdot 10^{-35}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.00022) (not (<= m 7.2e-35)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00022) || !(m <= 7.2e-35)) {
		tmp = a * pow(k, m);
	} 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 <= (-0.00022d0)) .or. (.not. (m <= 7.2d-35))) then
        tmp = a * (k ** m)
    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 <= -0.00022) || !(m <= 7.2e-35)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.00022) or not (m <= 7.2e-35):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.00022) || !(m <= 7.2e-35))
		tmp = Float64(a * (k ^ m));
	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 <= -0.00022) || ~((m <= 7.2e-35)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.00022], N[Not[LessEqual[m, 7.2e-35]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $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 < -2.20000000000000008e-4 or 7.20000000000000038e-35 < m

   1. Initial program 89.3%

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

   2. Taylor expanded in k around 0 99.4%

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

   if -2.20000000000000008e-4 < m < 7.20000000000000038e-35

   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. clear-num 94.9%

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

      2. inv-pow 94.9%

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

      3. div-inv 94.9%

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

      4. metadata-eval 94.9%

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

      5. unpow-prod-down 94.9%

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

      6. metadata-eval 94.9%

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

      7. inv-pow 94.9%

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

      8. associate-+l+ 94.9%

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

      9. +-commutative 94.9%

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

      10. distribute-rgt-out 94.9%

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

      11. fma-def 94.9%

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

      12. +-commutative 94.9%

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

      13. *-commutative 94.9%

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

      14. associate-/r* 94.9%

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

      15. pow-flip 94.9%

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

      16. metadata-eval 94.9%

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

   3. Applied egg-rr 94.9%

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

      1. unpow-1 94.9%

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

   5. Simplified 94.9%

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

   6. Taylor expanded in m around 0 94.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00022 \lor \neg \left(m \leq 7.2 \cdot 10^{-35}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.6e-20) (* a (pow k m)) (/ a (pow k (- 2.0 m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.6e-20) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / pow(k, (2.0 - m));
	}
	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 <= 2.6d-20) then
        tmp = a * (k ** m)
    else
        tmp = a / (k ** (2.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.6e-20) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / Math.pow(k, (2.0 - m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.6e-20:
		tmp = a * math.pow(k, m)
	else:
		tmp = a / math.pow(k, (2.0 - m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.6e-20)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / (k ^ Float64(2.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.6e-20)
		tmp = a * (k ^ m);
	else
		tmp = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.6e-20], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, N[(2.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.59999999999999995e-20

   1. Initial program 97.5%

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

   2. Taylor expanded in k around 0 100.0%

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

   if 2.59999999999999995e-20 < k

   1. Initial program 80.7%

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

      1. clear-num 80.7%

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

      2. inv-pow 80.7%

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

      3. div-inv 80.7%

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

      4. metadata-eval 80.7%

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

      5. unpow-prod-down 80.7%

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

      6. metadata-eval 80.7%

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

      7. inv-pow 80.7%

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

      8. associate-+l+ 80.7%

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

      9. +-commutative 80.7%

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

      10. distribute-rgt-out 80.7%

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

      11. fma-def 80.7%

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

      12. +-commutative 80.7%

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

      13. *-commutative 80.7%

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

      14. associate-/r* 80.7%

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

      15. pow-flip 80.7%

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

      16. metadata-eval 80.7%

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

   3. Applied egg-rr 80.7%

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

      1. unpow-1 80.7%

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

   5. Simplified 80.7%

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

   6. Taylor expanded in k around inf 78.0%

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

      1. associate-/r* 75.9%

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

      2. unpow2 75.9%

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

   8. Simplified 75.9%

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

      1. frac-2neg 75.9%

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

      2. neg-sub0 75.9%

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

      3. div-sub 63.3%

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

      4. inv-pow 63.3%

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

      5. metadata-eval 63.3%

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

      6. pow-pow 63.3%

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

      7. metadata-eval 63.3%

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

      8. clear-num 63.2%

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

      9. div-inv 63.2%

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

      10. clear-num 63.3%

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

      11. neg-mul-1 63.3%

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

      12. metadata-eval 63.3%

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

      13. times-frac 63.3%

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

      14. metadata-eval 63.3%

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

      15. metadata-eval 63.3%

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

      16. associate-/l/ 63.3%

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

      17. *-commutative 63.3%

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

      18. pow2 63.3%

          \[\leadsto \frac{0}{-{k}^{\left(-1 \cdot m\right)}} - -1 \cdot \frac{a}{\color{blue}{{k}^{2}} \cdot {\left(\frac{1}{k}\right)}^{m}} \]

   10. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{\frac{0}{-{k}^{\left(-1 \cdot m\right)}} - -1 \cdot \frac{a}{{k}^{\left(2 + -1 \cdot m\right)}}} \]
   11. Step-by-step derivation

       1. div0 92.6%

          \[\leadsto \color{blue}{0} - -1 \cdot \frac{a}{{k}^{\left(2 + -1 \cdot m\right)}} \]

       2. neg-sub0 92.6%

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

       3. distribute-lft-neg-in 92.6%

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

       4. metadata-eval 92.6%

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

       5. *-lft-identity 92.6%

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

       6. neg-mul-1 92.6%

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

   12. Simplified 92.6%

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

   13. Taylor expanded in a around 0 91.8%

       \[\leadsto \color{blue}{\frac{a}{e^{\log k \cdot \left(2 - m\right)}}} \]
   14. Step-by-step derivation

       1. exp-to-pow 92.6%

          \[\leadsto \frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}} \]

   15. Simplified 92.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(2 - m\right)}}\\ \end{array} \]

Alternative 5: 55.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.05:\\ \;\;\;\;k \cdot \frac{\frac{-a}{k}}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \frac{--1}{k}}{a \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.05)
   (* k (/ (/ (- a) k) (* k (- k))))
   (if (<= m 4.2e+82)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ (* (* a a) (/ (- -1.0) k)) (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.05) {
		tmp = k * ((-a / k) / (k * -k));
	} else if (m <= 4.2e+82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	}
	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 <= (-0.05d0)) then
        tmp = k * ((-a / k) / (k * -k))
    else if (m <= 4.2d+82) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = ((a * a) * (-(-1.0d0) / k)) / (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.05) {
		tmp = k * ((-a / k) / (k * -k));
	} else if (m <= 4.2e+82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.05:
		tmp = k * ((-a / k) / (k * -k))
	elif m <= 4.2e+82:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.05)
		tmp = Float64(k * Float64(Float64(Float64(-a) / k) / Float64(k * Float64(-k))));
	elseif (m <= 4.2e+82)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(-(-1.0)) / k)) / Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.05)
		tmp = k * ((-a / k) / (k * -k));
	elseif (m <= 4.2e+82)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.05], N[(k * N[(N[((-a) / k), $MachinePrecision] / N[(k * (-k)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.2e+82], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[((--1.0) / k), $MachinePrecision]), $MachinePrecision] / N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.050000000000000003

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in a around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-/r* 43.7%

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

   8. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. neg-sub0 43.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{0 - k}} \]

      3. flip-- 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{\frac{0 \cdot 0 - k \cdot k}{0 + k}}} \]

      4. metadata-eval 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{0} - k \cdot k}{0 + k}} \]

      5. neg-sub0 61.9%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{-k \cdot k}}{0 + k}} \]

      6. distribute-rgt-neg-out 61.9%

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

      7. +-lft-identity 61.9%

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

      8. associate-/r/ 68.2%

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

      9. distribute-neg-frac 68.2%

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

      10. distribute-rgt-neg-out 68.2%

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

   10. Applied egg-rr 68.2%

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

   if -0.050000000000000003 < m < 4.2e82

   1. Initial program 92.3%

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

      1. clear-num 92.1%

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

      2. inv-pow 92.1%

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

      3. div-inv 92.1%

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

      4. metadata-eval 92.1%

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

      5. unpow-prod-down 92.1%

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

      6. metadata-eval 92.1%

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

      7. inv-pow 92.1%

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

      8. associate-+l+ 92.1%

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

      9. +-commutative 92.1%

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

      10. distribute-rgt-out 92.1%

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

      11. fma-def 92.1%

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

      12. +-commutative 92.1%

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

      13. *-commutative 92.1%

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

      14. associate-/r* 92.1%

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

      15. pow-flip 92.2%

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

      16. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

   5. Simplified 92.2%

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

   6. Taylor expanded in m around 0 79.3%

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

   if 4.2e82 < m

   1. Initial program 78.8%

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

   2. Taylor expanded in m around 0 3.5%

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

   3. Taylor expanded in k around inf 2.2%

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

      1. unpow2 2.2%

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

   5. Simplified 2.2%

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

      1. frac-2neg 2.2%

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

      2. div-inv 2.2%

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

      3. distribute-rgt-neg-in 2.2%

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

   7. Applied egg-rr 2.2%

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

      1. associate-*r/ 2.2%

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

      2. times-frac 2.3%

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

      3. frac-2neg 2.3%

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

      4. remove-double-neg 2.3%

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

      5. distribute-neg-frac 2.3%

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

      6. associate-*l/ 2.3%

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

      7. *-commutative 2.3%

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

      8. associate-*l/ 2.3%

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

      9. neg-sub0 2.3%

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

      10. flip-- 17.9%

          \[\leadsto \frac{-\frac{1}{k}}{k} \cdot \color{blue}{\frac{0 \cdot 0 - a \cdot a}{0 + a}} \]

      11. +-lft-identity 17.9%

          \[\leadsto \frac{-\frac{1}{k}}{k} \cdot \frac{0 \cdot 0 - a \cdot a}{\color{blue}{a}} \]

      12. frac-times 10.3%

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

      13. distribute-neg-frac 10.3%

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

      14. metadata-eval 10.3%

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

      15. metadata-eval 10.3%

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

      16. sub0-neg 10.3%

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

      17. *-commutative 10.3%

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

   9. Applied egg-rr 10.3%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.05:\\ \;\;\;\;k \cdot \frac{\frac{-a}{k}}{k \cdot \left(-k\right)}\\ \mathbf{elif}\;m \leq 4.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \frac{--1}{k}}{a \cdot k}\\ \end{array} \]

Alternative 6: 57.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.85:\\ \;\;\;\;k \cdot \left(k \cdot \frac{\frac{\frac{\frac{a}{k}}{k}}{k}}{k}\right)\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \frac{--1}{k}}{a \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.85)
   (* k (* k (/ (/ (/ (/ a k) k) k) k)))
   (if (<= m 2.3e+82)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ (* (* a a) (/ (- -1.0) k)) (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.85) {
		tmp = k * (k * ((((a / k) / k) / k) / k));
	} else if (m <= 2.3e+82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	}
	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 <= (-0.85d0)) then
        tmp = k * (k * ((((a / k) / k) / k) / k))
    else if (m <= 2.3d+82) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = ((a * a) * (-(-1.0d0) / k)) / (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.85) {
		tmp = k * (k * ((((a / k) / k) / k) / k));
	} else if (m <= 2.3e+82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.85:
		tmp = k * (k * ((((a / k) / k) / k) / k))
	elif m <= 2.3e+82:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.85)
		tmp = Float64(k * Float64(k * Float64(Float64(Float64(Float64(a / k) / k) / k) / k)));
	elseif (m <= 2.3e+82)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(Float64(Float64(a * a) * Float64(Float64(-(-1.0)) / k)) / Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.85)
		tmp = k * (k * ((((a / k) / k) / k) / k));
	elseif (m <= 2.3e+82)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = ((a * a) * (-(-1.0) / k)) / (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.85], N[(k * N[(k * N[(N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3e+82], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * a), $MachinePrecision] * N[((--1.0) / k), $MachinePrecision]), $MachinePrecision] / N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.849999999999999978

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in a around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-/r* 43.7%

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

   8. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. neg-sub0 43.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{0 - k}} \]

      3. flip3-- 24.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{\frac{{0}^{3} - {k}^{3}}{0 \cdot 0 + \left(k \cdot k + 0 \cdot k\right)}}} \]

      4. metadata-eval 24.7%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{{0}^{3} - {k}^{3}}{\color{blue}{0} + \left(k \cdot k + 0 \cdot k\right)}} \]

      5. +-lft-identity 24.7%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{{0}^{3} - {k}^{3}}{\color{blue}{k \cdot k + 0 \cdot k}}} \]

      6. associate-/r/ 28.7%

         \[\leadsto \color{blue}{\frac{-\frac{a}{k}}{{0}^{3} - {k}^{3}} \cdot \left(k \cdot k + 0 \cdot k\right)} \]

      7. distribute-neg-frac 28.7%

         \[\leadsto \frac{\color{blue}{\frac{-a}{k}}}{{0}^{3} - {k}^{3}} \cdot \left(k \cdot k + 0 \cdot k\right) \]

      8. metadata-eval 28.7%

         \[\leadsto \frac{\frac{-a}{k}}{\color{blue}{0} - {k}^{3}} \cdot \left(k \cdot k + 0 \cdot k\right) \]

      9. sub0-neg 33.5%

         \[\leadsto \frac{\frac{-a}{k}}{\color{blue}{-{k}^{3}}} \cdot \left(k \cdot k + 0 \cdot k\right) \]

      10. distribute-rgt-out 33.5%

          \[\leadsto \frac{\frac{-a}{k}}{-{k}^{3}} \cdot \color{blue}{\left(k \cdot \left(k + 0\right)\right)} \]

      11. +-commutative 33.5%

          \[\leadsto \frac{\frac{-a}{k}}{-{k}^{3}} \cdot \left(k \cdot \color{blue}{\left(0 + k\right)}\right) \]

      12. +-lft-identity 33.5%

          \[\leadsto \frac{\frac{-a}{k}}{-{k}^{3}} \cdot \left(k \cdot \color{blue}{k}\right) \]

   10. Applied egg-rr 33.5%

       \[\leadsto \color{blue}{\frac{\frac{-a}{k}}{-{k}^{3}} \cdot \left(k \cdot k\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 81.1%

          \[\leadsto \color{blue}{\left(\frac{\frac{-a}{k}}{-{k}^{3}} \cdot k\right) \cdot k} \]

       2. *-commutative 81.1%

          \[\leadsto \color{blue}{k \cdot \left(\frac{\frac{-a}{k}}{-{k}^{3}} \cdot k\right)} \]

       3. *-commutative 81.1%

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \frac{\frac{-a}{k}}{-{k}^{3}}\right)} \]

       4. unpow3 81.1%

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

       5. distribute-lft-neg-out 81.1%

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

       6. associate-/l/ 83.5%

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

       7. associate-/r* 81.1%

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

       8. associate-/r* 77.6%

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

       9. neg-mul-1 77.6%

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

       10. neg-mul-1 77.6%

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

       11. times-frac 77.6%

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

       12. metadata-eval 77.6%

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

       13. associate-/r* 77.6%

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

       14. *-lft-identity 77.6%

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

   12. Simplified 77.6%

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

   if -0.849999999999999978 < m < 2.29999999999999988e82

   1. Initial program 92.3%

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

      1. clear-num 92.1%

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

      2. inv-pow 92.1%

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

      3. div-inv 92.1%

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

      4. metadata-eval 92.1%

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

      5. unpow-prod-down 92.1%

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

      6. metadata-eval 92.1%

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

      7. inv-pow 92.1%

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

      8. associate-+l+ 92.1%

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

      9. +-commutative 92.1%

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

      10. distribute-rgt-out 92.1%

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

      11. fma-def 92.1%

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

      12. +-commutative 92.1%

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

      13. *-commutative 92.1%

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

      14. associate-/r* 92.1%

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

      15. pow-flip 92.2%

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

      16. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

   5. Simplified 92.2%

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

   6. Taylor expanded in m around 0 79.3%

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

   if 2.29999999999999988e82 < m

   1. Initial program 78.8%

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

   2. Taylor expanded in m around 0 3.5%

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

   3. Taylor expanded in k around inf 2.2%

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

      1. unpow2 2.2%

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

   5. Simplified 2.2%

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

      1. frac-2neg 2.2%

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

      2. div-inv 2.2%

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

      3. distribute-rgt-neg-in 2.2%

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

   7. Applied egg-rr 2.2%

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

      1. associate-*r/ 2.2%

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

      2. times-frac 2.3%

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

      3. frac-2neg 2.3%

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

      4. remove-double-neg 2.3%

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

      5. distribute-neg-frac 2.3%

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

      6. associate-*l/ 2.3%

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

      7. *-commutative 2.3%

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

      8. associate-*l/ 2.3%

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

      9. neg-sub0 2.3%

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

      10. flip-- 17.9%

          \[\leadsto \frac{-\frac{1}{k}}{k} \cdot \color{blue}{\frac{0 \cdot 0 - a \cdot a}{0 + a}} \]

      11. +-lft-identity 17.9%

          \[\leadsto \frac{-\frac{1}{k}}{k} \cdot \frac{0 \cdot 0 - a \cdot a}{\color{blue}{a}} \]

      12. frac-times 10.3%

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

      13. distribute-neg-frac 10.3%

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

      14. metadata-eval 10.3%

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

      15. metadata-eval 10.3%

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

      16. sub0-neg 10.3%

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

      17. *-commutative 10.3%

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

   9. Applied egg-rr 10.3%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.85:\\ \;\;\;\;k \cdot \left(k \cdot \frac{\frac{\frac{\frac{a}{k}}{k}}{k}}{k}\right)\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot a\right) \cdot \frac{--1}{k}}{a \cdot k}\\ \end{array} \]

Alternative 7: 54.5% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.17:\\ \;\;\;\;k \cdot \frac{\frac{-a}{k}}{k \cdot \left(-k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.17)
   (* k (/ (/ (- a) k) (* k (- k))))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.17) {
		tmp = k * ((-a / k) / (k * -k));
	} 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 <= (-0.17d0)) then
        tmp = k * ((-a / k) / (k * -k))
    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 <= -0.17) {
		tmp = k * ((-a / k) / (k * -k));
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.17:
		tmp = k * ((-a / k) / (k * -k))
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.17)
		tmp = Float64(k * Float64(Float64(Float64(-a) / k) / Float64(k * Float64(-k))));
	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 <= -0.17)
		tmp = k * ((-a / k) / (k * -k));
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.17], N[(k * N[(N[((-a) / k), $MachinePrecision] / N[(k * (-k)), $MachinePrecision]), $MachinePrecision]), $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 < -0.170000000000000012

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in a around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-/r* 43.7%

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

   8. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. neg-sub0 43.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{0 - k}} \]

      3. flip-- 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{\frac{0 \cdot 0 - k \cdot k}{0 + k}}} \]

      4. metadata-eval 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{0} - k \cdot k}{0 + k}} \]

      5. neg-sub0 61.9%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{-k \cdot k}}{0 + k}} \]

      6. distribute-rgt-neg-out 61.9%

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

      7. +-lft-identity 61.9%

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

      8. associate-/r/ 68.2%

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

      9. distribute-neg-frac 68.2%

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

      10. distribute-rgt-neg-out 68.2%

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

   10. Applied egg-rr 68.2%

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

   if -0.170000000000000012 < m

   1. Initial program 87.2%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. div-inv 87.0%

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

      4. metadata-eval 87.0%

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

      5. unpow-prod-down 87.1%

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

      6. metadata-eval 87.1%

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

      7. inv-pow 87.1%

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

      8. associate-+l+ 87.1%

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

      9. +-commutative 87.1%

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

      10. distribute-rgt-out 87.1%

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

      11. fma-def 87.1%

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

      12. +-commutative 87.1%

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

      13. *-commutative 87.1%

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

      14. associate-/r* 87.1%

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

      15. pow-flip 87.1%

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

      16. metadata-eval 87.1%

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

   3. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in m around 0 50.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.17:\\ \;\;\;\;k \cdot \frac{\frac{-a}{k}}{k \cdot \left(-k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 45.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-264} \lor \neg \left(k \leq 1.38 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 2.6e-264) (not (<= k 1.38e-5)))
   (/ a (* k k))
   (+ a (* (* a k) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.6e-264) || !(k <= 1.38e-5)) {
		tmp = a / (k * k);
	} else {
		tmp = a + ((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 ((k <= 2.6d-264) .or. (.not. (k <= 1.38d-5))) then
        tmp = a / (k * k)
    else
        tmp = a + ((a * k) * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.6e-264) || !(k <= 1.38e-5)) {
		tmp = a / (k * k);
	} else {
		tmp = a + ((a * k) * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 2.6e-264) or not (k <= 1.38e-5):
		tmp = a / (k * k)
	else:
		tmp = a + ((a * k) * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 2.6e-264) || !(k <= 1.38e-5))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a + Float64(Float64(a * k) * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 2.6e-264) || ~((k <= 1.38e-5)))
		tmp = a / (k * k);
	else
		tmp = a + ((a * k) * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 2.6e-264], N[Not[LessEqual[k, 1.38e-5]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a + N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.6000000000000002e-264 or 1.38e-5 < k

   1. Initial program 86.8%

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

   2. Taylor expanded in m around 0 44.6%

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

   3. Taylor expanded in k around inf 46.4%

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

      1. unpow2 46.4%

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

   5. Simplified 46.4%

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

   if 2.6000000000000002e-264 < k < 1.38e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 51.6%

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

   3. Taylor expanded in k around 0 51.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.6 \cdot 10^{-264} \lor \neg \left(k \leq 1.38 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(a \cdot k\right) \cdot -10\\ \end{array} \]

Alternative 9: 45.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;a + \left(a \cdot k\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7.2e-264)
   (/ 1.0 (/ (* k k) a))
   (if (<= k 1.38e-5) (+ a (* (* a k) -10.0)) (/ a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.2e-264) {
		tmp = 1.0 / ((k * k) / a);
	} else if (k <= 1.38e-5) {
		tmp = a + ((a * k) * -10.0);
	} else {
		tmp = a / (k * k);
	}
	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 <= 7.2d-264) then
        tmp = 1.0d0 / ((k * k) / a)
    else if (k <= 1.38d-5) then
        tmp = a + ((a * k) * (-10.0d0))
    else
        tmp = a / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7.2e-264) {
		tmp = 1.0 / ((k * k) / a);
	} else if (k <= 1.38e-5) {
		tmp = a + ((a * k) * -10.0);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7.2e-264:
		tmp = 1.0 / ((k * k) / a)
	elif k <= 1.38e-5:
		tmp = a + ((a * k) * -10.0)
	else:
		tmp = a / (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7.2e-264)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (k <= 1.38e-5)
		tmp = Float64(a + Float64(Float64(a * k) * -10.0));
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7.2e-264)
		tmp = 1.0 / ((k * k) / a);
	elseif (k <= 1.38e-5)
		tmp = a + ((a * k) * -10.0);
	else
		tmp = a / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7.2e-264], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.38e-5], N[(a + N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.2000000000000004e-264

   1. Initial program 94.8%

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

   2. Taylor expanded in m around 0 27.8%

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

   3. Taylor expanded in k around inf 35.0%

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

      1. unpow2 35.0%

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

   5. Simplified 35.0%

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

      1. clear-num 35.8%

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

      2. inv-pow 35.8%

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

      3. frac-2neg 35.8%

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

      4. distribute-lft-neg-in 35.8%

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

      5. neg-mul-1 35.8%

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

      6. metadata-eval 35.8%

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

      7. times-frac 28.6%

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

      8. frac-2neg 28.6%

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

      9. /-rgt-identity 28.6%

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

   7. Applied egg-rr 28.6%

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

      1. unpow-1 28.6%

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

      2. associate-*r/ 35.8%

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

   9. Simplified 35.8%

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

   if 7.2000000000000004e-264 < k < 1.38e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 51.6%

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

   3. Taylor expanded in k around 0 51.6%

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

   if 1.38e-5 < k

   1. Initial program 80.1%

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

   2. Taylor expanded in m around 0 58.7%

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

   3. Taylor expanded in k around inf 55.9%

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

      1. unpow2 55.9%

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

   5. Simplified 55.9%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{-264}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;k \leq 1.38 \cdot 10^{-5}:\\ \;\;\;\;a + \left(a \cdot k\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \]

Alternative 10: 53.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.068:\\ \;\;\;\;k \cdot \frac{\frac{\frac{a}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.068) (* k (/ (/ (/ a k) k) k)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.068) {
		tmp = k * (((a / k) / k) / k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	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 <= (-0.068d0)) then
        tmp = k * (((a / k) / k) / k)
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.068) {
		tmp = k * (((a / k) / k) / k);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.068:
		tmp = k * (((a / k) / k) / k)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.068)
		tmp = Float64(k * Float64(Float64(Float64(a / k) / k) / k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.068)
		tmp = k * (((a / k) / k) / k);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.068], N[(k * N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.068000000000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in a around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-/r* 43.7%

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

   8. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. neg-sub0 43.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{0 - k}} \]

      3. flip-- 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{\frac{0 \cdot 0 - k \cdot k}{0 + k}}} \]

      4. metadata-eval 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{0} - k \cdot k}{0 + k}} \]

      5. neg-sub0 61.9%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{-k \cdot k}}{0 + k}} \]

      6. distribute-rgt-neg-out 61.9%

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

      7. +-lft-identity 61.9%

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

      8. associate-/r/ 68.2%

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

      9. distribute-neg-frac 68.2%

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

      10. distribute-rgt-neg-out 68.2%

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

   10. Applied egg-rr 68.2%

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

       1. *-commutative 68.2%

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

       2. associate-/l/ 78.8%

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

       3. associate-/r* 68.2%

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

       4. neg-mul-1 68.2%

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

       5. neg-mul-1 68.2%

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

       6. times-frac 68.2%

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

       7. metadata-eval 68.2%

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

       8. associate-/r* 65.9%

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

       9. *-lft-identity 65.9%

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

   12. Simplified 65.9%

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

   if -0.068000000000000005 < m

   1. Initial program 87.2%

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

   2. Taylor expanded in m around 0 50.5%

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

   3. Taylor expanded in k around 0 49.0%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.068:\\ \;\;\;\;k \cdot \frac{\frac{\frac{a}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 11: 53.8% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.048:\\ \;\;\;\;k \cdot \frac{\frac{\frac{a}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.048) (* k (/ (/ (/ a k) k) k)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.048) {
		tmp = k * (((a / k) / k) / k);
	} 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 <= (-0.048d0)) then
        tmp = k * (((a / k) / k) / k)
    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 <= -0.048) {
		tmp = k * (((a / k) / k) / k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.048:
		tmp = k * (((a / k) / k) / k)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.048)
		tmp = Float64(k * Float64(Float64(Float64(a / k) / k) / k));
	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 <= -0.048)
		tmp = k * (((a / k) / k) / k);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.048], N[(k * N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $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 < -0.048000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

   6. Taylor expanded in a around 0 61.9%

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

      1. unpow2 61.9%

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

      2. associate-/r* 43.7%

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

   8. Simplified 43.7%

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

      1. frac-2neg 43.7%

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

      2. neg-sub0 43.7%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{0 - k}} \]

      3. flip-- 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\color{blue}{\frac{0 \cdot 0 - k \cdot k}{0 + k}}} \]

      4. metadata-eval 44.8%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{0} - k \cdot k}{0 + k}} \]

      5. neg-sub0 61.9%

         \[\leadsto \frac{-\frac{a}{k}}{\frac{\color{blue}{-k \cdot k}}{0 + k}} \]

      6. distribute-rgt-neg-out 61.9%

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

      7. +-lft-identity 61.9%

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

      8. associate-/r/ 68.2%

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

      9. distribute-neg-frac 68.2%

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

      10. distribute-rgt-neg-out 68.2%

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

   10. Applied egg-rr 68.2%

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

       1. *-commutative 68.2%

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

       2. associate-/l/ 78.8%

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

       3. associate-/r* 68.2%

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

       4. neg-mul-1 68.2%

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

       5. neg-mul-1 68.2%

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

       6. times-frac 68.2%

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

       7. metadata-eval 68.2%

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

       8. associate-/r* 65.9%

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

       9. *-lft-identity 65.9%

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

   12. Simplified 65.9%

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

   if -0.048000000000000001 < m

   1. Initial program 87.2%

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

      1. clear-num 87.1%

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

      2. inv-pow 87.1%

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

      3. div-inv 87.0%

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

      4. metadata-eval 87.0%

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

      5. unpow-prod-down 87.1%

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

      6. metadata-eval 87.1%

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

      7. inv-pow 87.1%

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

      8. associate-+l+ 87.1%

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

      9. +-commutative 87.1%

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

      10. distribute-rgt-out 87.1%

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

      11. fma-def 87.1%

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

      12. +-commutative 87.1%

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

      13. *-commutative 87.1%

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

      14. associate-/r* 87.1%

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

      15. pow-flip 87.1%

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

      16. metadata-eval 87.1%

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

   3. Applied egg-rr 87.1%

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

      1. unpow-1 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in m around 0 50.5%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.048:\\ \;\;\;\;k \cdot \frac{\frac{\frac{a}{k}}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 12: 45.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-264} \lor \neg \left(k \leq 1.38 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 2.7e-264) (not (<= k 1.38e-5))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.7e-264) || !(k <= 1.38e-5)) {
		tmp = a / (k * k);
	} 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) :: tmp
    if ((k <= 2.7d-264) .or. (.not. (k <= 1.38d-5))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 2.7e-264) || !(k <= 1.38e-5)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 2.7e-264) or not (k <= 1.38e-5):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 2.7e-264) || !(k <= 1.38e-5))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 2.7e-264) || ~((k <= 1.38e-5)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 2.7e-264], N[Not[LessEqual[k, 1.38e-5]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 2.69999999999999994e-264 or 1.38e-5 < k

   1. Initial program 86.8%

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

   2. Taylor expanded in m around 0 44.6%

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

   3. Taylor expanded in k around inf 46.4%

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

      1. unpow2 46.4%

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

   5. Simplified 46.4%

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

   if 2.69999999999999994e-264 < k < 1.38e-5

   1. Initial program 100.0%

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

   2. Taylor expanded in k around 0 100.0%

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

   3. Taylor expanded in m around 0 51.6%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-264} \lor \neg \left(k \leq 1.38 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 13: 51.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.36:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.36) (/ 1.0 (/ (* k k) a)) (/ a (+ 1.0 (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.36) {
		tmp = 1.0 / ((k * k) / a);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	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 <= (-0.36d0)) then
        tmp = 1.0d0 / ((k * k) / a)
    else
        tmp = a / (1.0d0 + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.36) {
		tmp = 1.0 / ((k * k) / a);
	} else {
		tmp = a / (1.0 + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.36:
		tmp = 1.0 / ((k * k) / a)
	else:
		tmp = a / (1.0 + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.36)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.36)
		tmp = 1.0 / ((k * k) / a);
	else
		tmp = a / (1.0 + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.36], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.35999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.6%

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

   3. Taylor expanded in k around inf 61.9%

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

      1. unpow2 61.9%

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

   5. Simplified 61.9%

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

      1. clear-num 62.6%

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

      2. inv-pow 62.6%

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

      3. frac-2neg 62.6%

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

      4. distribute-lft-neg-in 62.6%

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

      5. neg-mul-1 62.6%

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

      6. metadata-eval 62.6%

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

      7. times-frac 44.4%

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

      8. frac-2neg 44.4%

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

      9. /-rgt-identity 44.4%

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

   7. Applied egg-rr 44.4%

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

      1. unpow-1 44.4%

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

      2. associate-*r/ 62.6%

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

   9. Simplified 62.6%

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

   if -0.35999999999999999 < m

   1. Initial program 87.2%

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

   2. Taylor expanded in m around 0 50.5%

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

   3. Taylor expanded in k around 0 49.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.36:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \end{array} \]

Alternative 14: 20.1% 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 91.3%

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

2. Taylor expanded in k around 0 84.1%

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

3. Taylor expanded in m around 0 21.8%

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

4. Final simplification 21.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023297 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))