Falkner and Boettcher, Appendix A

Percentage Accurate: 90.0% → 98.7%

Time: 10.8s

Alternatives: 16

Speedup: 1.1×

• 23.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% 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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;m \leq 0.0033:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(1, k\right)}}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5.4e-22)
   (/ (pow k m) (/ (+ 1.0 (* k (+ k 10.0))) a))
   (if (<= m 0.0033) (/ (/ a (hypot 1.0 k)) (hypot 1.0 k)) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-22) {
		tmp = pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else if (m <= 0.0033) {
		tmp = (a / hypot(1.0, k)) / hypot(1.0, k);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -5.4e-22) {
		tmp = Math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a);
	} else if (m <= 0.0033) {
		tmp = (a / Math.hypot(1.0, k)) / Math.hypot(1.0, k);
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -5.4e-22:
		tmp = math.pow(k, m) / ((1.0 + (k * (k + 10.0))) / a)
	elif m <= 0.0033:
		tmp = (a / math.hypot(1.0, k)) / math.hypot(1.0, k)
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5.4e-22)
		tmp = Float64((k ^ m) / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / a));
	elseif (m <= 0.0033)
		tmp = Float64(Float64(a / hypot(1.0, k)) / hypot(1.0, k));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -5.4e-22)
		tmp = (k ^ m) / ((1.0 + (k * (k + 10.0))) / a);
	elseif (m <= 0.0033)
		tmp = (a / hypot(1.0, k)) / hypot(1.0, k);
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5.4e-22], N[(N[Power[k, m], $MachinePrecision] / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.0033], N[(N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.4000000000000004e-22

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. fma-udef 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -5.4000000000000004e-22 < m < 0.0033

   1. Initial program 91.5%

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

      1. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in k around 0 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. add-sqr-sqrt 89.4%

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

      3. times-frac 89.4%

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

      4. hypot-1-def 89.4%

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

      5. hypot-1-def 97.7%

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

   6. Applied egg-rr 97.7%

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

      1. associate-*l/ 97.8%

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

      2. *-lft-identity 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in m around 0 96.8%

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

   if 0.0033 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{{k}^{m}}{\frac{1 + k \cdot \left(k + 10\right)}{a}}\\ \mathbf{elif}\;m \leq 0.0033:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(1, k\right)}}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := N[(N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

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

   1. *-commutative 89.9%

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

3. Simplified 89.9%

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

4. Taylor expanded in k around 0 88.4%

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

   1. *-un-lft-identity 88.4%

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

   2. add-sqr-sqrt 88.4%

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

   3. times-frac 88.4%

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

   4. hypot-1-def 88.4%

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

   5. hypot-1-def 98.4%

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

6. Applied egg-rr 98.4%

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

   1. associate-*l/ 98.4%

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

   2. *-lft-identity 98.4%

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

8. Simplified 98.4%

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

9. Final simplification 98.4%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;\frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\frac{\frac{a}{{k}^{\left(-m\right)}}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) 2e+244)
     (/ (/ a (pow k (- m))) (+ 1.0 (* k (+ k 10.0))))
     t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+244) {
		tmp = (a / pow(k, -m)) / (1.0 + (k * (k + 10.0)));
	} 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 ((t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))) <= 2d+244) then
        tmp = (a / (k ** -m)) / (1.0d0 + (k * (k + 10.0d0)))
    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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+244) {
		tmp = (a / Math.pow(k, -m)) / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if (t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+244:
		tmp = (a / math.pow(k, -m)) / (1.0 + (k * (k + 10.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k))) <= 2e+244)
		tmp = Float64(Float64(a / (k ^ Float64(-m))) / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 ((t_0 / ((1.0 + (k * 10.0)) + (k * k))) <= 2e+244)
		tmp = (a / (k ^ -m)) / (1.0 + (k * (k + 10.0)));
	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[N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+244], N[(N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 2.00000000000000015e244

   1. Initial program 96.4%

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

      1. associate-/l* 96.4%

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

      2. sqr-neg 96.4%

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

      3. associate-+l+ 96.4%

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

      4. sqr-neg 96.4%

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

      5. distribute-rgt-out 96.4%

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

   3. Simplified 96.4%

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

      1. *-un-lft-identity 96.4%

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

      2. div-inv 96.4%

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

      3. distribute-lft-in 96.4%

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

      4. associate-+l+ 96.4%

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

      5. times-frac 96.4%

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

      6. associate-+l+ 96.4%

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

      7. distribute-lft-in 96.4%

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

      8. +-commutative 96.4%

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

      9. +-commutative 96.4%

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

      10. fma-udef 96.4%

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

      11. pow-flip 96.4%

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

   5. Applied egg-rr 96.4%

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

      1. associate-*l/ 96.4%

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

      2. *-lft-identity 96.4%

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

   7. Simplified 96.4%

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

      1. fma-udef 94.6%

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

   9. Applied egg-rr 96.4%

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

   if 2.00000000000000015e244 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 56.1%

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

      1. associate-/l* 56.1%

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

      2. sqr-neg 56.1%

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

      3. associate-+l+ 56.1%

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

      4. sqr-neg 56.1%

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

      5. distribute-rgt-out 56.1%

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

   3. Simplified 56.1%

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

   4. Taylor expanded in k around 0 97.6%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+244}:\\ \;\;\;\;\frac{\frac{a}{{k}^{\left(-m\right)}}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.2) (/ a (/ (+ 1.0 (* k (+ k 10.0))) (pow k m))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / pow(k, m));
	} else {
		tmp = a * pow(k, 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 (m <= 2.2d0) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) / (k ** m))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.2) {
		tmp = a / ((1.0 + (k * (k + 10.0))) / Math.pow(k, m));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.2:
		tmp = a / ((1.0 + (k * (k + 10.0))) / math.pow(k, m))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.2)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) / (k ^ m)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.2)
		tmp = a / ((1.0 + (k * (k + 10.0))) / (k ^ m));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.2], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 95.7%

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

      1. associate-/l* 95.6%

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

      2. sqr-neg 95.6%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   if 2.2000000000000002 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.0%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7e-16)
   (/ a (/ (+ 1.0 (* k 10.0)) (pow k m)))
   (if (<= m 0.000225) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7e-16) {
		tmp = a / ((1.0 + (k * 10.0)) / pow(k, m));
	} else if (m <= 0.000225) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * pow(k, 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 (m <= (-7d-16)) then
        tmp = a / ((1.0d0 + (k * 10.0d0)) / (k ** m))
    else if (m <= 0.000225d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -7e-16) {
		tmp = a / ((1.0 + (k * 10.0)) / Math.pow(k, m));
	} else if (m <= 0.000225) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -7e-16:
		tmp = a / ((1.0 + (k * 10.0)) / math.pow(k, m))
	elif m <= 0.000225:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7e-16)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) / (k ^ m)));
	elseif (m <= 0.000225)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -7e-16)
		tmp = a / ((1.0 + (k * 10.0)) / (k ^ m));
	elseif (m <= 0.000225)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7e-16], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.000225], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.00000000000000035e-16

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 98.0%

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

      1. *-commutative 19.7%

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

   6. Simplified 98.0%

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

   if -7.00000000000000035e-16 < m < 2.2499999999999999e-4

   1. Initial program 91.5%

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

      1. associate-/l* 91.5%

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

      2. sqr-neg 91.5%

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

      3. associate-+l+ 91.5%

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

      4. sqr-neg 91.5%

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

      5. distribute-rgt-out 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in m around 0 90.5%

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

   if 2.2499999999999999e-4 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 0.000225:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 6: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{t_0}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= m 2.2) (/ 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 <= 2.2) {
		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 <= 2.2d0) 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 <= 2.2) {
		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 <= 2.2:
		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 <= 2.2)
		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 <= 2.2)
		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, 2.2], N[(t$95$0 / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 2.2000000000000002

   1. Initial program 95.7%

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

      1. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in k around 0 93.4%

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

   if 2.2000000000000002 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.2:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 7: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.25 \cdot 10^{-5} \lor \neg \left(m \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.25e-5) (not (<= m 2.6e-6)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.25e-5) || !(m <= 2.6e-6)) {
		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 <= (-1.25d-5)) .or. (.not. (m <= 2.6d-6))) 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 <= -1.25e-5) || !(m <= 2.6e-6)) {
		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 <= -1.25e-5) or not (m <= 2.6e-6):
		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 <= -1.25e-5) || !(m <= 2.6e-6))
		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 <= -1.25e-5) || ~((m <= 2.6e-6)))
		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, -1.25e-5], N[Not[LessEqual[m, 2.6e-6]], $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 < -1.25000000000000006e-5 or 2.60000000000000009e-6 < m

   1. Initial program 89.1%

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

      1. associate-/l* 89.1%

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

      2. sqr-neg 89.1%

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

      3. associate-+l+ 89.1%

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

      4. sqr-neg 89.1%

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

      5. distribute-rgt-out 89.1%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in k around 0 98.3%

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

   if -1.25000000000000006e-5 < m < 2.60000000000000009e-6

   1. Initial program 91.5%

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

      1. associate-/l* 91.5%

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

      2. sqr-neg 91.5%

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

      3. associate-+l+ 91.5%

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

      4. sqr-neg 91.5%

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

      5. distribute-rgt-out 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in m around 0 90.5%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.25 \cdot 10^{-5} \lor \neg \left(m \leq 2.6 \cdot 10^{-6}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 30.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;a - a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.011)
   (* 0.1 (/ a k))
   (if (<= m 7.5e+79) (- a (* a (* k -10.0))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 0.1 * (a / k);
	} else if (m <= 7.5e+79) {
		tmp = a - (a * (k * -10.0));
	} else {
		tmp = -10.0 * (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 <= (-0.011d0)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 7.5d+79) then
        tmp = a - (a * (k * (-10.0d0)))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 0.1 * (a / k);
	} else if (m <= 7.5e+79) {
		tmp = a - (a * (k * -10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.011:
		tmp = 0.1 * (a / k)
	elif m <= 7.5e+79:
		tmp = a - (a * (k * -10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.011)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 7.5e+79)
		tmp = Float64(a - Float64(a * Float64(k * -10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.011)
		tmp = 0.1 * (a / k);
	elseif (m <= 7.5e+79)
		tmp = a - (a * (k * -10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.011], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.5e+79], N[(a - N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.010999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 98.0%

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

   5. Taylor expanded in m around 0 21.6%

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

   6. Taylor expanded in k around inf 21.7%

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

   if -0.010999999999999999 < m < 7.49999999999999967e79

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. sqr-neg 85.2%

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

      3. associate-+l+ 85.2%

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

      4. sqr-neg 85.2%

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

      5. distribute-rgt-out 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in m around 0 73.5%

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

   5. Taylor expanded in k around 0 40.9%

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

   6. Taylor expanded in a around 0 40.9%

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

      1. *-commutative 40.9%

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

   8. Simplified 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. add-sqr-sqrt 23.1%

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

      3. sqrt-unprod 18.3%

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

      4. sqr-neg 18.3%

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

      5. sqrt-unprod 1.1%

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

      6. add-sqr-sqrt 4.5%

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

      7. *-un-lft-identity 4.5%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.9%

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

      10. swap-sqr 1.9%

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

      11. metadata-eval 1.9%

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

      12. metadata-eval 1.9%

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

      13. swap-sqr 1.9%

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

      14. sqrt-unprod 1.9%

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

      15. add-sqr-sqrt 2.3%

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

      16. distribute-rgt-out 2.3%

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

      17. *-commutative 2.3%

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

      18. *-un-lft-identity 2.3%

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

      19. distribute-lft-neg-out 2.3%

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

      20. unsub-neg 2.3%

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

   10. Applied egg-rr 42.7%

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

   if 7.49999999999999967e79 < m

   1. Initial program 83.9%

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

      1. associate-/l* 83.9%

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

      2. sqr-neg 83.9%

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

      3. associate-+l+ 83.9%

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

      4. sqr-neg 83.9%

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

      5. distribute-rgt-out 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in m around 0 3.6%

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

   5. Taylor expanded in k around 0 8.8%

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

   6. Taylor expanded in k around inf 25.9%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 7.5 \cdot 10^{+79}:\\ \;\;\;\;a - a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 9: 31.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+79}:\\ \;\;\;\;a - a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.011)
   (/ 1.0 (* 10.0 (/ k a)))
   (if (<= m 4e+79) (- a (* a (* k -10.0))) (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 1.0 / (10.0 * (k / a));
	} else if (m <= 4e+79) {
		tmp = a - (a * (k * -10.0));
	} else {
		tmp = -10.0 * (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 <= (-0.011d0)) then
        tmp = 1.0d0 / (10.0d0 * (k / a))
    else if (m <= 4d+79) then
        tmp = a - (a * (k * (-10.0d0)))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 1.0 / (10.0 * (k / a));
	} else if (m <= 4e+79) {
		tmp = a - (a * (k * -10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.011:
		tmp = 1.0 / (10.0 * (k / a))
	elif m <= 4e+79:
		tmp = a - (a * (k * -10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.011)
		tmp = Float64(1.0 / Float64(10.0 * Float64(k / a)));
	elseif (m <= 4e+79)
		tmp = Float64(a - Float64(a * Float64(k * -10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.011)
		tmp = 1.0 / (10.0 * (k / a));
	elseif (m <= 4e+79)
		tmp = a - (a * (k * -10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.011], N[(1.0 / N[(10.0 * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4e+79], N[(a - N[(a * N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.010999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 98.0%

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

   5. Taylor expanded in m around 0 21.6%

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

   6. Taylor expanded in k around inf 23.6%

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

   if -0.010999999999999999 < m < 3.99999999999999987e79

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. sqr-neg 85.2%

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

      3. associate-+l+ 85.2%

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

      4. sqr-neg 85.2%

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

      5. distribute-rgt-out 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in m around 0 73.5%

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

   5. Taylor expanded in k around 0 40.9%

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

   6. Taylor expanded in a around 0 40.9%

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

      1. *-commutative 40.9%

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

   8. Simplified 40.9%

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

      1. *-un-lft-identity 40.9%

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

      2. add-sqr-sqrt 23.1%

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

      3. sqrt-unprod 18.3%

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

      4. sqr-neg 18.3%

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

      5. sqrt-unprod 1.1%

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

      6. add-sqr-sqrt 4.5%

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

      7. *-un-lft-identity 4.5%

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

      8. add-sqr-sqrt 0.0%

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

      9. sqrt-unprod 1.9%

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

      10. swap-sqr 1.9%

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

      11. metadata-eval 1.9%

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

      12. metadata-eval 1.9%

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

      13. swap-sqr 1.9%

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

      14. sqrt-unprod 1.9%

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

      15. add-sqr-sqrt 2.3%

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

      16. distribute-rgt-out 2.3%

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

      17. *-commutative 2.3%

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

      18. *-un-lft-identity 2.3%

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

      19. distribute-lft-neg-out 2.3%

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

      20. unsub-neg 2.3%

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

   10. Applied egg-rr 42.7%

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

   if 3.99999999999999987e79 < m

   1. Initial program 83.9%

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

      1. associate-/l* 83.9%

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

      2. sqr-neg 83.9%

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

      3. associate-+l+ 83.9%

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

      4. sqr-neg 83.9%

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

      5. distribute-rgt-out 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in m around 0 3.6%

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

   5. Taylor expanded in k around 0 8.8%

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

   6. Taylor expanded in k around inf 25.9%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;\frac{1}{10 \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{+79}:\\ \;\;\;\;a - a \cdot \left(k \cdot -10\right)\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 10: 52.4% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.88:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-a}{a \cdot \left(-a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.88) (/ a (+ 1.0 (* k k))) (/ 1.0 (/ (- a) (* a (- a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.88) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = 1.0 / (-a / (a * -a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.88d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = 1.0d0 / (-a / (a * -a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.88) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = 1.0 / (-a / (a * -a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.88:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = 1.0 / (-a / (a * -a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.88)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(1.0 / Float64(Float64(-a) / Float64(a * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.88)
		tmp = a / (1.0 + (k * k));
	else
		tmp = 1.0 / (-a / (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.88], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((-a) / N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.880000000000000004

   1. Initial program 95.7%

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

      1. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in k around 0 93.4%

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

   5. Taylor expanded in m around 0 63.6%

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

   if 0.880000000000000004 < m

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/l* 68.8%

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

      3. sqr-neg 68.8%

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

      4. associate-+l+ 68.8%

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

      5. +-commutative 68.8%

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

      6. sqr-neg 68.8%

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

      7. distribute-rgt-out 68.8%

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

      8. fma-def 68.8%

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

      9. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in k around 0 70.1%

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

   5. Taylor expanded in m around 0 3.1%

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

      1. +-commutative 3.1%

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

      2. frac-2neg 3.1%

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

      3. metadata-eval 3.1%

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

      4. associate-*r/ 3.1%

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

      5. *-commutative 3.1%

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

      6. frac-add 21.0%

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

      7. neg-mul-1 21.0%

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

   7. Applied egg-rr 21.0%

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

   8. Taylor expanded in k around 0 30.5%

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

      1. mul-1-neg 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.88:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-a}{a \cdot \left(-a\right)}}\\ \end{array} \]

Alternative 11: 53.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.76:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-a}{a \cdot \left(-a\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.76) (/ a (+ 1.0 (* k (+ k 10.0)))) (/ 1.0 (/ (- a) (* a (- a))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.76) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = 1.0 / (-a / (a * -a));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.76d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = 1.0d0 / (-a / (a * -a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.76) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = 1.0 / (-a / (a * -a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.76:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = 1.0 / (-a / (a * -a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.76)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(-a) / Float64(a * Float64(-a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.76)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = 1.0 / (-a / (a * -a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.76], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[((-a) / N[(a * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.76000000000000001

   1. Initial program 95.7%

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

      1. associate-/l* 95.6%

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

      2. sqr-neg 95.6%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 64.7%

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

   if 0.76000000000000001 < m

   1. Initial program 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-/l* 68.8%

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

      3. sqr-neg 68.8%

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

      4. associate-+l+ 68.8%

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

      5. +-commutative 68.8%

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

      6. sqr-neg 68.8%

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

      7. distribute-rgt-out 68.8%

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

      8. fma-def 68.8%

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

      9. +-commutative 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in k around 0 70.1%

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

   5. Taylor expanded in m around 0 3.1%

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

      1. +-commutative 3.1%

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

      2. frac-2neg 3.1%

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

      3. metadata-eval 3.1%

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

      4. associate-*r/ 3.1%

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

      5. *-commutative 3.1%

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

      6. frac-add 21.0%

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

      7. neg-mul-1 21.0%

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

   7. Applied egg-rr 21.0%

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

   8. Taylor expanded in k around 0 30.5%

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

      1. mul-1-neg 30.5%

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

   10. Simplified 30.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.76:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-a}{a \cdot \left(-a\right)}}\\ \end{array} \]

Alternative 12: 32.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.55:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.011) (* 0.1 (/ a k)) (if (<= m 0.55) a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.55) {
		tmp = a;
	} else {
		tmp = -10.0 * (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 <= (-0.011d0)) then
        tmp = 0.1d0 * (a / k)
    else if (m <= 0.55d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.011) {
		tmp = 0.1 * (a / k);
	} else if (m <= 0.55) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.011:
		tmp = 0.1 * (a / k)
	elif m <= 0.55:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.011)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (m <= 0.55)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.011)
		tmp = 0.1 * (a / k);
	elseif (m <= 0.55)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.011], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.55], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.010999999999999999

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 98.0%

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

   5. Taylor expanded in m around 0 21.6%

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

   6. Taylor expanded in k around inf 21.7%

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

   if -0.010999999999999999 < m < 0.55000000000000004

   1. Initial program 91.6%

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

      1. associate-/l* 91.5%

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

      2. sqr-neg 91.5%

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

      3. associate-+l+ 91.6%

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

      4. sqr-neg 91.6%

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

      5. distribute-rgt-out 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in m around 0 89.6%

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

   5. Taylor expanded in k around 0 49.6%

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

   6. Taylor expanded in k around 0 50.3%

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

   if 0.55000000000000004 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 7.2%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.011:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;m \leq 0.55:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 13: 34.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.72:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.72) (/ a (+ 1.0 (* k 10.0))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.72) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (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 <= 0.72d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.72) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.72:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.72)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.72)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.72], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.71999999999999997

   1. Initial program 95.7%

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

      1. associate-/l* 95.6%

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

      2. sqr-neg 95.6%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 64.7%

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

   5. Taylor expanded in k around 0 41.4%

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

      1. *-commutative 41.4%

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

   7. Simplified 41.4%

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

   if 0.71999999999999997 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 7.2%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.72:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 14: 50.5% accurate, 12.6× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.4) (/ a (+ 1.0 (* k k))) (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.4) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (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 <= 1.4d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

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

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.4:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.4], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3999999999999999

   1. Initial program 95.7%

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

      1. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in k around 0 93.4%

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

   5. Taylor expanded in m around 0 63.6%

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

   if 1.3999999999999999 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 7.2%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 50.8%

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

Alternative 15: 25.7% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.21) a (* -10.0 (* a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = -10.0 * (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 <= 0.21d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (a * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.21) {
		tmp = a;
	} else {
		tmp = -10.0 * (a * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.21:
		tmp = a
	else:
		tmp = -10.0 * (a * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.21)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(a * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.21)
		tmp = a;
	else
		tmp = -10.0 * (a * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.21], a, N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.209999999999999992

   1. Initial program 95.7%

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

      1. associate-/l* 95.6%

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

      2. sqr-neg 95.6%

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

      3. associate-+l+ 95.7%

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

      4. sqr-neg 95.7%

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

      5. distribute-rgt-out 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 64.7%

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

   5. Taylor expanded in k around 0 26.9%

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

   6. Taylor expanded in k around 0 27.7%

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

   if 0.209999999999999992 < m

   1. Initial program 76.6%

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

      1. associate-/l* 76.6%

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

      2. sqr-neg 76.6%

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

      3. associate-+l+ 76.6%

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

      4. sqr-neg 76.6%

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

      5. distribute-rgt-out 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in m around 0 3.3%

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

   5. Taylor expanded in k around 0 7.2%

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

   6. Taylor expanded in k around inf 21.1%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.21:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 16: 20.2% accurate, 114.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. sqr-neg 89.9%

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

   3. associate-+l+ 89.9%

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

   4. sqr-neg 89.9%

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

   5. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

4. Taylor expanded in m around 0 46.2%

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

5. Taylor expanded in k around 0 21.0%

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

6. Taylor expanded in k around 0 20.6%

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

7. Final simplification 20.6%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023339 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))