Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 98.9%

Time: 9.9s

Alternatives: 15

Speedup: 1.1×

• 23.4% 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: 90.5% 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: 98.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. *-commutative 88.0%

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

3. Simplified 88.0%

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

4. Taylor expanded in k around 0 87.2%

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

   1. *-un-lft-identity 87.2%

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

   2. add-sqr-sqrt 87.2%

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

   3. times-frac 87.2%

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

   4. hypot-1-def 87.2%

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

   5. hypot-1-def 99.1%

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

6. Applied egg-rr 99.1%

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

7. Final simplification 99.1%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.0023)
   (* (/ (pow k m) (hypot 1.0 k)) (/ a (hypot 1.0 k)))
   (/ a (pow k (- m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.0023) {
		tmp = (pow(k, m) / hypot(1.0, k)) * (a / 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 <= 0.0023) {
		tmp = (Math.pow(k, m) / Math.hypot(1.0, k)) * (a / Math.hypot(1.0, k));
	} else {
		tmp = a / Math.pow(k, -m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.0023:
		tmp = (math.pow(k, m) / math.hypot(1.0, k)) * (a / 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 <= 0.0023)
		tmp = Float64(Float64((k ^ m) / hypot(1.0, k)) * Float64(a / hypot(1.0, k)));
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.0023)
		tmp = ((k ^ m) / hypot(1.0, k)) * (a / hypot(1.0, k));
	else
		tmp = a / (k ^ -m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.0023], N[(N[(N[Power[k, m], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.0023

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in k around 0 93.1%

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

      1. *-commutative 93.1%

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

      2. add-sqr-sqrt 93.1%

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

      3. times-frac 93.1%

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

      4. hypot-1-def 93.1%

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

      5. hypot-1-def 98.7%

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

   6. Applied egg-rr 98.7%

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

   if 0.0023 < m

   1. Initial program 71.8%

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

      1. associate-/l* 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-+l+ 71.8%

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

      4. sqr-neg 71.8%

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

      5. distribute-rgt-out 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in k around 0 100.0%

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

   5. Taylor expanded in k around inf 53.5%

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

      1. rec-exp 53.5%

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

      2. mul-1-neg 53.5%

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

      3. remove-double-neg 53.5%

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

      4. *-commutative 53.5%

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

      5. log-rec 53.5%

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

      6. distribute-lft-neg-in 53.5%

         \[\leadsto \frac{a}{e^{\color{blue}{-\log k \cdot m}}} \]

      7. distribute-rgt-neg-out 53.5%

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

      8. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(1, k\right)} \cdot \frac{a}{\mathsf{hypot}\left(1, k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.0023)
   (* (pow k m) (/ a (+ 1.0 (* k (+ k 10.0)))))
   (/ a (pow k (- m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.0023) {
		tmp = pow(k, m) * (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 <= 0.0023d0) then
        tmp = (k ** m) * (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 <= 0.0023) {
		tmp = Math.pow(k, m) * (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 <= 0.0023:
		tmp = math.pow(k, m) * (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 <= 0.0023)
		tmp = Float64((k ^ m) * Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.0023)
		tmp = (k ^ m) * (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, 0.0023], N[(N[Power[k, m], $MachinePrecision] * N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.0023

   1. Initial program 94.2%

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

      1. associate-*l/ 94.2%

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

      2. *-commutative 94.2%

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

      3. sqr-neg 94.2%

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

      4. associate-+l+ 94.2%

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

      5. +-commutative 94.2%

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

      6. sqr-neg 94.2%

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

      7. distribute-rgt-out 94.2%

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

      8. fma-def 94.2%

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

      9. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in a around 0 94.2%

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

   if 0.0023 < m

   1. Initial program 71.8%

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

      1. associate-/l* 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-+l+ 71.8%

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

      4. sqr-neg 71.8%

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

      5. distribute-rgt-out 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in k around 0 100.0%

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

   5. Taylor expanded in k around inf 53.5%

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

      1. rec-exp 53.5%

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

      2. mul-1-neg 53.5%

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

      3. remove-double-neg 53.5%

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

      4. *-commutative 53.5%

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

      5. log-rec 53.5%

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

      6. distribute-lft-neg-in 53.5%

         \[\leadsto \frac{a}{e^{\color{blue}{-\log k \cdot m}}} \]

      7. distribute-rgt-neg-out 53.5%

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

      8. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;{k}^{m} \cdot \frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 4: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8e-17)
   (/ a (/ (+ 1.0 (* k 10.0)) (pow k m)))
   (if (<= m 2.4e-18) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (pow k m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8e-17) {
		tmp = a / ((1.0 + (k * 10.0)) / pow(k, m));
	} else if (m <= 2.4e-18) {
		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 <= (-8d-17)) then
        tmp = a / ((1.0d0 + (k * 10.0d0)) / (k ** m))
    else if (m <= 2.4d-18) 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 <= -8e-17) {
		tmp = a / ((1.0 + (k * 10.0)) / Math.pow(k, m));
	} else if (m <= 2.4e-18) {
		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 <= -8e-17:
		tmp = a / ((1.0 + (k * 10.0)) / math.pow(k, m))
	elif m <= 2.4e-18:
		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 <= -8e-17)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) / (k ^ m)));
	elseif (m <= 2.4e-18)
		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 <= -8e-17)
		tmp = a / ((1.0 + (k * 10.0)) / (k ^ m));
	elseif (m <= 2.4e-18)
		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, -8e-17], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.4e-18], 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 < -8.00000000000000057e-17

   1. Initial program 99.1%

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

      1. associate-/l* 99.1%

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

      2. sqr-neg 99.1%

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

      3. associate-+l+ 99.1%

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

      4. sqr-neg 99.1%

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

      5. distribute-rgt-out 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in k around 0 97.9%

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

      1. *-commutative 18.3%

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

   6. Simplified 97.9%

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

   if -8.00000000000000057e-17 < m < 2.39999999999999994e-18

   1. Initial program 90.2%

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

      1. associate-/l* 90.2%

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

      2. sqr-neg 90.2%

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

      3. associate-+l+ 90.2%

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

      4. sqr-neg 90.2%

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

      5. distribute-rgt-out 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in m around 0 90.2%

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

   if 2.39999999999999994e-18 < m

   1. Initial program 72.1%

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

      1. associate-*l/ 69.4%

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

      2. *-commutative 69.4%

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

      3. sqr-neg 69.4%

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

      4. associate-+l+ 69.4%

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

      5. +-commutative 69.4%

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

      6. sqr-neg 69.4%

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

      7. distribute-rgt-out 69.4%

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

      8. fma-def 69.4%

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

      9. +-commutative 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in k around 0 98.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{\frac{1 + k \cdot 10}{{k}^{m}}}\\ \mathbf{elif}\;m \leq 2.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 5: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.0023) (/ (* a (pow k m)) (+ 1.0 (* k k))) (/ a (pow k (- m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.0023) {
		tmp = (a * pow(k, m)) / (1.0 + (k * k));
	} 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 <= 0.0023d0) then
        tmp = (a * (k ** m)) / (1.0d0 + (k * k))
    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 <= 0.0023) {
		tmp = (a * Math.pow(k, m)) / (1.0 + (k * k));
	} else {
		tmp = a / Math.pow(k, -m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.0023:
		tmp = (a * math.pow(k, m)) / (1.0 + (k * k))
	else:
		tmp = a / math.pow(k, -m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.0023)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a / (k ^ Float64(-m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.0023)
		tmp = (a * (k ^ m)) / (1.0 + (k * k));
	else
		tmp = a / (k ^ -m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.0023], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, (-m)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.0023

   1. Initial program 94.2%

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

      1. *-commutative 94.2%

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

   3. Simplified 94.2%

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

   4. Taylor expanded in k around 0 93.1%

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

   if 0.0023 < m

   1. Initial program 71.8%

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

      1. associate-/l* 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-+l+ 71.8%

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

      4. sqr-neg 71.8%

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

      5. distribute-rgt-out 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in k around 0 100.0%

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

   5. Taylor expanded in k around inf 53.5%

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

      1. rec-exp 53.5%

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

      2. mul-1-neg 53.5%

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

      3. remove-double-neg 53.5%

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

      4. *-commutative 53.5%

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

      5. log-rec 53.5%

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

      6. distribute-lft-neg-in 53.5%

         \[\leadsto \frac{a}{e^{\color{blue}{-\log k \cdot m}}} \]

      7. distribute-rgt-neg-out 53.5%

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

      8. exp-to-pow 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.0023:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)}}\\ \end{array} \]

Alternative 6: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -47000 \lor \neg \left(m \leq 2.4 \cdot 10^{-18}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -47000.0) (not (<= m 2.4e-18)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -47000.0) || !(m <= 2.4e-18)) {
		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 <= (-47000.0d0)) .or. (.not. (m <= 2.4d-18))) 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 <= -47000.0) || !(m <= 2.4e-18)) {
		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 <= -47000.0) or not (m <= 2.4e-18):
		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 <= -47000.0) || !(m <= 2.4e-18))
		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 <= -47000.0) || ~((m <= 2.4e-18)))
		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, -47000.0], N[Not[LessEqual[m, 2.4e-18]], $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 < -47000 or 2.39999999999999994e-18 < m

   1. Initial program 86.7%

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

      1. associate-*l/ 85.5%

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

      2. *-commutative 85.5%

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

      3. sqr-neg 85.5%

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

      4. associate-+l+ 85.5%

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

      5. +-commutative 85.5%

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

      6. sqr-neg 85.5%

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

      7. distribute-rgt-out 85.5%

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

      8. fma-def 85.5%

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

      9. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in k around 0 99.1%

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

   if -47000 < m < 2.39999999999999994e-18

   1. Initial program 90.0%

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

      1. associate-/l* 90.0%

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

      2. sqr-neg 90.0%

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

      3. associate-+l+ 90.0%

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

      4. sqr-neg 90.0%

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

      5. distribute-rgt-out 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in m around 0 88.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -47000 \lor \neg \left(m \leq 2.4 \cdot 10^{-18}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7: 50.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -1.04 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{elif}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot -0.01 + \left(0.001 \cdot \left(k \cdot a\right) + 0.1 \cdot \frac{a}{k}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -1.04e+18)
     (/ a t_0)
     (if (<= m 2.1)
       (/ a (+ 1.0 t_0))
       (+ (* a -0.01) (+ (* 0.001 (* k a)) (* 0.1 (/ a k))))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -1.04e+18) {
		tmp = a / t_0;
	} else if (m <= 2.1) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = (a * -0.01) + ((0.001 * (k * a)) + (0.1 * (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) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-1.04d+18)) then
        tmp = a / t_0
    else if (m <= 2.1d0) then
        tmp = a / (1.0d0 + t_0)
    else
        tmp = (a * (-0.01d0)) + ((0.001d0 * (k * a)) + (0.1d0 * (a / k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -1.04e+18) {
		tmp = a / t_0;
	} else if (m <= 2.1) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = (a * -0.01) + ((0.001 * (k * a)) + (0.1 * (a / k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -1.04e+18:
		tmp = a / t_0
	elif m <= 2.1:
		tmp = a / (1.0 + t_0)
	else:
		tmp = (a * -0.01) + ((0.001 * (k * a)) + (0.1 * (a / k)))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -1.04e+18)
		tmp = Float64(a / t_0);
	elseif (m <= 2.1)
		tmp = Float64(a / Float64(1.0 + t_0));
	else
		tmp = Float64(Float64(a * -0.01) + Float64(Float64(0.001 * Float64(k * a)) + Float64(0.1 * Float64(a / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -1.04e+18)
		tmp = a / t_0;
	elseif (m <= 2.1)
		tmp = a / (1.0 + t_0);
	else
		tmp = (a * -0.01) + ((0.001 * (k * a)) + (0.1 * (a / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.04e+18], N[(a / t$95$0), $MachinePrecision], If[LessEqual[m, 2.1], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a * -0.01), $MachinePrecision] + N[(N[(0.001 * N[(k * a), $MachinePrecision]), $MachinePrecision] + N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.04e18

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 29.3%

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

      1. clear-num 29.3%

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

      2. inv-pow 29.3%

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

      3. +-commutative 29.3%

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

      4. +-commutative 29.3%

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

      5. fma-udef 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. unpow-1 29.3%

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

   8. Simplified 29.3%

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

   9. Taylor expanded in k around inf 42.9%

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

       1. +-commutative 42.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 42.9%

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

       3. distribute-rgt-in 42.9%

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

   11. Simplified 42.9%

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

   12. Taylor expanded in a around 0 42.9%

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

       1. associate-/r* 39.5%

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

       2. +-commutative 39.5%

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

       3. associate-/r* 42.9%

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

   14. Simplified 42.9%

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

   if -1.04e18 < m < 2.10000000000000009

   1. Initial program 89.8%

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

      1. associate-/l* 89.8%

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

      2. sqr-neg 89.8%

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

      3. associate-+l+ 89.8%

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

      4. sqr-neg 89.8%

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

      5. distribute-rgt-out 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in m around 0 84.8%

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

   if 2.10000000000000009 < m

   1. Initial program 71.4%

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

      1. associate-/l* 71.4%

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

      2. sqr-neg 71.4%

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

      3. associate-+l+ 71.4%

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

      4. sqr-neg 71.4%

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

      5. distribute-rgt-out 71.4%

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

   3. Simplified 71.4%

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

   4. Taylor expanded in m around 0 3.0%

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

      1. clear-num 3.0%

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

      2. inv-pow 3.0%

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

      3. +-commutative 3.0%

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

      4. +-commutative 3.0%

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

      5. fma-udef 3.0%

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

   6. Applied egg-rr 3.0%

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

      1. unpow-1 3.0%

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

   8. Simplified 3.0%

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

   9. Taylor expanded in k around inf 2.3%

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

       1. +-commutative 2.3%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 2.3%

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

       3. distribute-rgt-in 2.3%

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

   11. Simplified 2.3%

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

   12. Taylor expanded in k around 0 9.8%

       \[\leadsto \color{blue}{-0.01 \cdot a + \left(0.001 \cdot \left(a \cdot k\right) + 0.1 \cdot \frac{a}{k}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.04 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.1:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot -0.01 + \left(0.001 \cdot \left(k \cdot a\right) + 0.1 \cdot \frac{a}{k}\right)\\ \end{array} \]

Alternative 8: 46.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -0.245:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 3.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k + 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -0.245)
   (/ a (* k (+ k 10.0)))
   (if (<= k 3.5) (/ a (+ 1.0 (* k 10.0))) (* (/ a k) (/ 1.0 (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -0.245) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 3.5) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) * (1.0 / (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-0.245d0)) then
        tmp = a / (k * (k + 10.0d0))
    else if (k <= 3.5d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) * (1.0d0 / (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -0.245) {
		tmp = a / (k * (k + 10.0));
	} else if (k <= 3.5) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) * (1.0 / (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -0.245:
		tmp = a / (k * (k + 10.0))
	elif k <= 3.5:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) * (1.0 / (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -0.245)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (k <= 3.5)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -0.245)
		tmp = a / (k * (k + 10.0));
	elseif (k <= 3.5)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) * (1.0 / (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -0.245], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.5], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(1.0 / N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -0.245

   1. Initial program 78.9%

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

      1. associate-/l* 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-+l+ 78.9%

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

      4. sqr-neg 78.9%

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

      5. distribute-rgt-out 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in m around 0 41.4%

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

      1. clear-num 41.4%

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

      2. inv-pow 41.4%

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

      3. +-commutative 41.4%

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

      4. +-commutative 41.4%

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

      5. fma-udef 41.4%

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

   6. Applied egg-rr 41.4%

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

      1. unpow-1 41.4%

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

   8. Simplified 41.4%

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

   9. Taylor expanded in k around inf 41.4%

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

       1. +-commutative 41.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 41.4%

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

       3. distribute-rgt-in 41.4%

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

   11. Simplified 41.4%

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

   12. Taylor expanded in a around 0 41.4%

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

       1. associate-/r* 36.4%

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

       2. +-commutative 36.4%

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

       3. associate-/r* 41.4%

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

   14. Simplified 41.4%

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

   if -0.245 < k < 3.5

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.5%

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

   5. Taylor expanded in k around 0 36.8%

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

      1. *-commutative 36.8%

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

   7. Simplified 36.8%

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

   if 3.5 < k

   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 55.4%

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

      1. clear-num 54.0%

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

      2. inv-pow 54.0%

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

      3. +-commutative 54.0%

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

      4. +-commutative 54.0%

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

      5. fma-udef 54.0%

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

   6. Applied egg-rr 54.0%

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

      1. unpow-1 54.0%

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

   8. Simplified 54.0%

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

   9. Taylor expanded in k around inf 54.0%

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

       1. +-commutative 54.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 54.0%

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

       3. distribute-rgt-in 54.0%

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

   11. Simplified 54.0%

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

       1. associate-/l* 55.4%

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

       2. *-commutative 55.4%

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

       3. times-frac 63.6%

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

   13. Applied egg-rr 63.6%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -0.245:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;k \leq 3.5:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k + 10}\\ \end{array} \]

Alternative 9: 28.8% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -33000000000000 \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -33000000000000.0) (not (<= k 0.075)))
   (* 0.1 (/ a k))
   (+ a (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -33000000000000.0) || !(k <= 0.075)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a + ((k * a) * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-33000000000000.0d0)) .or. (.not. (k <= 0.075d0))) then
        tmp = 0.1d0 * (a / k)
    else
        tmp = a + ((k * a) * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -33000000000000.0) || !(k <= 0.075)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a + ((k * a) * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -33000000000000.0) or not (k <= 0.075):
		tmp = 0.1 * (a / k)
	else:
		tmp = a + ((k * a) * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -33000000000000.0) || !(k <= 0.075))
		tmp = Float64(0.1 * Float64(a / k));
	else
		tmp = Float64(a + Float64(Float64(k * a) * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -33000000000000.0) || ~((k <= 0.075)))
		tmp = 0.1 * (a / k);
	else
		tmp = a + ((k * a) * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -33000000000000.0], N[Not[LessEqual[k, 0.075]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], N[(a + N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -3.3e13 or 0.0749999999999999972 < k

   1. Initial program 77.3%

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

      1. associate-/l* 77.3%

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

      2. sqr-neg 77.3%

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

      3. associate-+l+ 77.3%

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

      4. sqr-neg 77.3%

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

      5. distribute-rgt-out 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 51.4%

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

      1. clear-num 50.4%

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

      2. inv-pow 50.4%

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

      3. +-commutative 50.4%

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

      4. +-commutative 50.4%

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

      5. fma-udef 50.4%

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

   6. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in k around inf 50.4%

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

       1. +-commutative 50.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 50.4%

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

       3. distribute-rgt-in 50.4%

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

   11. Simplified 50.4%

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

   12. Taylor expanded in k around 0 21.2%

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

   if -3.3e13 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.5%

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

   5. Taylor expanded in k around 0 36.7%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -33000000000000 \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \end{array} \]

Alternative 10: 45.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -5000000 \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -5000000.0) (not (<= k 0.075)))
   (/ a (* k (+ k 10.0)))
   (+ a (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -5000000.0) || !(k <= 0.075)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a + ((k * a) * -10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-5000000.0d0)) .or. (.not. (k <= 0.075d0))) then
        tmp = a / (k * (k + 10.0d0))
    else
        tmp = a + ((k * a) * (-10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -5000000.0) || !(k <= 0.075)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a + ((k * a) * -10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -5000000.0) or not (k <= 0.075):
		tmp = a / (k * (k + 10.0))
	else:
		tmp = a + ((k * a) * -10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -5000000.0) || !(k <= 0.075))
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(a + Float64(Float64(k * a) * -10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -5000000.0) || ~((k <= 0.075)))
		tmp = a / (k * (k + 10.0));
	else
		tmp = a + ((k * a) * -10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -5000000.0], N[Not[LessEqual[k, 0.075]], $MachinePrecision]], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5e6 or 0.0749999999999999972 < k

   1. Initial program 77.3%

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

      1. associate-/l* 77.3%

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

      2. sqr-neg 77.3%

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

      3. associate-+l+ 77.3%

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

      4. sqr-neg 77.3%

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

      5. distribute-rgt-out 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 51.4%

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

      1. clear-num 50.4%

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

      2. inv-pow 50.4%

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

      3. +-commutative 50.4%

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

      4. +-commutative 50.4%

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

      5. fma-udef 50.4%

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

   6. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in k around inf 50.4%

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

       1. +-commutative 50.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 50.4%

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

       3. distribute-rgt-in 50.4%

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

   11. Simplified 50.4%

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

   12. Taylor expanded in a around 0 51.4%

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

       1. associate-/r* 56.0%

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

       2. +-commutative 56.0%

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

       3. associate-/r* 51.4%

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

   14. Simplified 51.4%

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

   if -5e6 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.5%

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

   5. Taylor expanded in k around 0 36.7%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5000000 \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \end{array} \]

Alternative 11: 46.0% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -0.14 \lor \neg \left(k \leq 3.5\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -0.14) (not (<= k 3.5)))
   (/ a (* k (+ k 10.0)))
   (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -0.14) || !(k <= 3.5)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-0.14d0)) .or. (.not. (k <= 3.5d0))) then
        tmp = a / (k * (k + 10.0d0))
    else
        tmp = a / (1.0d0 + (k * 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -0.14) || !(k <= 3.5)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -0.14) or not (k <= 3.5):
		tmp = a / (k * (k + 10.0))
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -0.14) || !(k <= 3.5))
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -0.14) || ~((k <= 3.5)))
		tmp = a / (k * (k + 10.0));
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -0.14], N[Not[LessEqual[k, 3.5]], $MachinePrecision]], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -0.14000000000000001 or 3.5 < k

   1. Initial program 77.3%

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

      1. associate-/l* 77.3%

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

      2. sqr-neg 77.3%

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

      3. associate-+l+ 77.3%

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

      4. sqr-neg 77.3%

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

      5. distribute-rgt-out 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 51.4%

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

      1. clear-num 50.4%

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

      2. inv-pow 50.4%

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

      3. +-commutative 50.4%

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

      4. +-commutative 50.4%

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

      5. fma-udef 50.4%

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

   6. Applied egg-rr 50.4%

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

      1. unpow-1 50.4%

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

   8. Simplified 50.4%

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

   9. Taylor expanded in k around inf 50.4%

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

       1. +-commutative 50.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 50.4%

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

       3. distribute-rgt-in 50.4%

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

   11. Simplified 50.4%

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

   12. Taylor expanded in a around 0 51.4%

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

       1. associate-/r* 56.0%

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

       2. +-commutative 56.0%

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

       3. associate-/r* 51.4%

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

   14. Simplified 51.4%

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

   if -0.14000000000000001 < k < 3.5

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.5%

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

   5. Taylor expanded in k around 0 36.8%

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

      1. *-commutative 36.8%

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

   7. Simplified 36.8%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -0.14 \lor \neg \left(k \leq 3.5\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]

Alternative 12: 29.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -6.4 \cdot 10^{+15}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot 10}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -6.4e+15)
   (* 0.1 (/ a k))
   (if (<= k 0.075) (+ a (* (* k a) -10.0)) (/ 1.0 (/ (* k 10.0) a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.4e+15) {
		tmp = 0.1 * (a / k);
	} else if (k <= 0.075) {
		tmp = a + ((k * a) * -10.0);
	} else {
		tmp = 1.0 / ((k * 10.0) / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= (-6.4d+15)) then
        tmp = 0.1d0 * (a / k)
    else if (k <= 0.075d0) then
        tmp = a + ((k * a) * (-10.0d0))
    else
        tmp = 1.0d0 / ((k * 10.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.4e+15) {
		tmp = 0.1 * (a / k);
	} else if (k <= 0.075) {
		tmp = a + ((k * a) * -10.0);
	} else {
		tmp = 1.0 / ((k * 10.0) / a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -6.4e+15:
		tmp = 0.1 * (a / k)
	elif k <= 0.075:
		tmp = a + ((k * a) * -10.0)
	else:
		tmp = 1.0 / ((k * 10.0) / a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -6.4e+15)
		tmp = Float64(0.1 * Float64(a / k));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(Float64(k * a) * -10.0));
	else
		tmp = Float64(1.0 / Float64(Float64(k * 10.0) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -6.4e+15)
		tmp = 0.1 * (a / k);
	elseif (k <= 0.075)
		tmp = a + ((k * a) * -10.0);
	else
		tmp = 1.0 / ((k * 10.0) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -6.4e+15], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k * 10.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < -6.4e15

   1. Initial program 78.4%

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

      1. associate-/l* 78.4%

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

      2. sqr-neg 78.4%

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

      3. associate-+l+ 78.4%

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

      4. sqr-neg 78.4%

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

      5. distribute-rgt-out 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in m around 0 42.4%

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

      1. clear-num 42.4%

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

      2. inv-pow 42.4%

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

      3. +-commutative 42.4%

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

      4. +-commutative 42.4%

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

      5. fma-udef 42.4%

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

   6. Applied egg-rr 42.4%

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

      1. unpow-1 42.4%

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

   8. Simplified 42.4%

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

   9. Taylor expanded in k around inf 42.4%

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

       1. +-commutative 42.4%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 42.4%

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

       3. distribute-rgt-in 42.4%

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

   11. Simplified 42.4%

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

   12. Taylor expanded in k around 0 19.4%

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

   if -6.4e15 < k < 0.0749999999999999972

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.5%

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

   5. Taylor expanded in k around 0 36.7%

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

   if 0.0749999999999999972 < k

   1. Initial program 76.9%

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

      1. associate-/l* 76.9%

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

      2. sqr-neg 76.9%

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

      3. associate-+l+ 76.9%

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

      4. sqr-neg 76.9%

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

      5. distribute-rgt-out 76.9%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in m around 0 54.8%

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

      1. clear-num 53.5%

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

      2. inv-pow 53.5%

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

      3. +-commutative 53.5%

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

      4. +-commutative 53.5%

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

      5. fma-udef 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. unpow-1 53.5%

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

   8. Simplified 53.5%

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

   9. Taylor expanded in k around inf 53.5%

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

       1. +-commutative 53.5%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 53.5%

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

       3. distribute-rgt-in 53.5%

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

   11. Simplified 53.5%

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

   12. Taylor expanded in k around 0 23.2%

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

       1. associate-*r/ 23.2%

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

       2. *-commutative 23.2%

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

   14. Simplified 23.2%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.4 \cdot 10^{+15}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + \left(k \cdot a\right) \cdot -10\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k \cdot 10}{a}}\\ \end{array} \]

Alternative 13: 48.3% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -5.2e+19) (/ a t_0) (/ a (+ 1.0 t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -5.2e+19) {
		tmp = a / t_0;
	} else {
		tmp = a / (1.0 + t_0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-5.2d+19)) then
        tmp = a / t_0
    else
        tmp = a / (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -5.2e+19) {
		tmp = a / t_0;
	} else {
		tmp = a / (1.0 + t_0);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -5.2e+19:
		tmp = a / t_0
	else:
		tmp = a / (1.0 + t_0)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -5.2e+19)
		tmp = Float64(a / t_0);
	else
		tmp = Float64(a / Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -5.2e+19)
		tmp = a / t_0;
	else
		tmp = a / (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.2e+19], N[(a / t$95$0), $MachinePrecision], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.2e19

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 29.3%

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

      1. clear-num 29.3%

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

      2. inv-pow 29.3%

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

      3. +-commutative 29.3%

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

      4. +-commutative 29.3%

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

      5. fma-udef 29.3%

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

   6. Applied egg-rr 29.3%

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

      1. unpow-1 29.3%

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

   8. Simplified 29.3%

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

   9. Taylor expanded in k around inf 42.9%

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

       1. +-commutative 42.9%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 42.9%

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

       3. distribute-rgt-in 42.9%

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

   11. Simplified 42.9%

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

   12. Taylor expanded in a around 0 42.9%

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

       1. associate-/r* 39.5%

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

       2. +-commutative 39.5%

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

       3. associate-/r* 42.9%

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

   14. Simplified 42.9%

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

   if -5.2e19 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. sqr-neg 82.5%

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

      3. associate-+l+ 82.5%

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

      4. sqr-neg 82.5%

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

      5. distribute-rgt-out 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 52.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 14: 28.6% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -1.45 \cdot 10^{+16} \lor \neg \left(k \leq 3.5\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -1.45e+16) (not (<= k 3.5))) (* 0.1 (/ a k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.45e+16) || !(k <= 3.5)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-1.45d+16)) .or. (.not. (k <= 3.5d0))) then
        tmp = 0.1d0 * (a / k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -1.45e+16) || !(k <= 3.5)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -1.45e+16) or not (k <= 3.5):
		tmp = 0.1 * (a / k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -1.45e+16) || !(k <= 3.5))
		tmp = Float64(0.1 * Float64(a / k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -1.45e+16) || ~((k <= 3.5)))
		tmp = 0.1 * (a / k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -1.45e+16], N[Not[LessEqual[k, 3.5]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -1.45e16 or 3.5 < k

   1. Initial program 77.1%

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

      1. associate-/l* 77.1%

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

      2. sqr-neg 77.1%

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

      3. associate-+l+ 77.1%

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

      4. sqr-neg 77.1%

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

      5. distribute-rgt-out 77.1%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in m around 0 51.8%

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

      1. clear-num 50.8%

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

      2. inv-pow 50.8%

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

      3. +-commutative 50.8%

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

      4. +-commutative 50.8%

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

      5. fma-udef 50.8%

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

   6. Applied egg-rr 50.8%

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

      1. unpow-1 50.8%

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

   8. Simplified 50.8%

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

   9. Taylor expanded in k around inf 50.8%

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

       1. +-commutative 50.8%

          \[\leadsto \frac{1}{\frac{\color{blue}{{k}^{2} + 10 \cdot k}}{a}} \]

       2. unpow2 50.8%

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

       3. distribute-rgt-in 50.8%

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

   11. Simplified 50.8%

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

   12. Taylor expanded in k around 0 21.4%

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

   if -1.45e16 < k < 3.5

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. associate-+l+ 100.0%

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

      4. sqr-neg 100.0%

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

      5. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 37.2%

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

   5. Taylor expanded in k around 0 36.1%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.45 \cdot 10^{+16} \lor \neg \left(k \leq 3.5\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 15: 20.4% 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 88.0%

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

   1. associate-/l* 88.0%

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

   2. sqr-neg 88.0%

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

   3. associate-+l+ 88.0%

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

   4. sqr-neg 88.0%

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

   5. distribute-rgt-out 88.0%

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

3. Simplified 88.0%

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

4. Taylor expanded in m around 0 44.8%

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

5. Taylor expanded in k around 0 19.3%

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

6. Final simplification 19.3%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023322 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))