Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.6%

Time: 15.0s

Alternatives: 13

Speedup: 1.1×

• 23.6% 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: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 2e+162) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 2e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 2d+162) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 2e+162) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 2e+162:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 2e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 2e+162)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+162], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 1.9999999999999999e162

   1. Initial program 98.6%

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

   if 1.9999999999999999e162 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 67.9%

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

   2. Taylor expanded in k around 0 50.0%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+162}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.225:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.225) (* a (pow k m)) (/ (* (pow k m) (/ a k)) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.225) {
		tmp = a * pow(k, m);
	} else {
		tmp = (pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.225d0) then
        tmp = a * (k ** m)
    else
        tmp = ((k ** m) * (a / k)) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.225) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (Math.pow(k, m) * (a / k)) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.225:
		tmp = a * math.pow(k, m)
	else:
		tmp = (math.pow(k, m) * (a / k)) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.225)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64((k ^ m) * Float64(a / k)) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.225)
		tmp = a * (k ^ m);
	else
		tmp = ((k ^ m) * (a / k)) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.225], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.225000000000000006

   1. Initial program 94.3%

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

   2. Taylor expanded in k around 0 48.8%

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

      1. exp-to-pow 99.4%

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

      2. *-commutative 99.4%

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

   4. Simplified 99.4%

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

   if 0.225000000000000006 < k

   1. Initial program 86.6%

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

      1. div-inv 86.5%

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

      2. add-cube-cbrt 86.3%

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

      3. associate-*l* 86.3%

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

      4. pow2 86.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      5. associate-+l+ 86.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      6. +-commutative 86.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

      7. distribute-rgt-out 86.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right) \]

      8. fma-def 86.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right) \]

   3. Applied egg-rr 86.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)} \]

   4. Taylor expanded in k around inf 84.6%

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

      1. unpow2 84.6%

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

      2. times-frac 93.0%

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

      3. mul-1-neg 93.0%

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

      4. *-commutative 93.0%

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

      5. distribute-rgt-neg-in 93.0%

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

      6. log-rec 93.0%

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

   6. Simplified 93.0%

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

      1. associate-*r/ 93.0%

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

      2. remove-double-neg 93.0%

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

      3. *-commutative 93.0%

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

      4. pow-to-exp 93.0%

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

   8. Applied egg-rr 93.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.225:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{a}{k}}{k}\\ \end{array} \]

Alternative 3: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{-17} \lor \neg \left(m \leq 9.5 \cdot 10^{-14}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.2e-17) (not (<= m 9.5e-14)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.2e-17) || !(m <= 9.5e-14)) {
		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 <= (-5.2d-17)) .or. (.not. (m <= 9.5d-14))) 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 <= -5.2e-17) || !(m <= 9.5e-14)) {
		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 <= -5.2e-17) or not (m <= 9.5e-14):
		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 <= -5.2e-17) || !(m <= 9.5e-14))
		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 <= -5.2e-17) || ~((m <= 9.5e-14)))
		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, -5.2e-17], N[Not[LessEqual[m, 9.5e-14]], $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 < -5.20000000000000006e-17 or 9.4999999999999999e-14 < m

   1. Initial program 89.6%

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

   2. Taylor expanded in k around 0 48.8%

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

      1. exp-to-pow 100.0%

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

      2. *-commutative 100.0%

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

   4. Simplified 100.0%

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

   if -5.20000000000000006e-17 < m < 9.4999999999999999e-14

   1. Initial program 96.6%

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

      1. div-inv 96.6%

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

      2. add-cube-cbrt 95.2%

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

      3. associate-*l* 95.2%

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

      4. pow2 95.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      5. associate-+l+ 95.2%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      6. +-commutative 95.2%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

      7. distribute-rgt-out 95.2%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right) \]

      8. fma-def 95.2%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right) \]

   3. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)} \]

   4. Taylor expanded in m around 0 96.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{-17} \lor \neg \left(m \leq 9.5 \cdot 10^{-14}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 62.5% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10 + k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 1.95)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (+ a (* a (+ (* k -10.0) (* k (* k 100.0))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * -10.0) + (k * (k * 100.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 1.95d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + (a * ((k * (-10.0d0)) + (k * (k * 100.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 1.95) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (a * ((k * -10.0) + (k * (k * 100.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 1.95:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (a * ((k * -10.0) + (k * (k * 100.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.95)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(a * Float64(Float64(k * -10.0) + Float64(k * Float64(k * 100.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 1.95)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (a * ((k * -10.0) + (k * (k * 100.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.95], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(N[(k * -10.0), $MachinePrecision] + N[(k * N[(k * 100.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 1.94999999999999996

   1. Initial program 96.7%

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

      1. div-inv 96.7%

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

      2. add-cube-cbrt 95.3%

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

      3. associate-*l* 95.3%

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

      4. pow2 95.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      5. associate-+l+ 95.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      6. +-commutative 95.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

      7. distribute-rgt-out 95.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right) \]

      8. fma-def 95.3%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right) \]

   3. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)} \]

   4. Taylor expanded in m around 0 94.7%

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

   if 1.94999999999999996 < m

   1. Initial program 79.5%

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

   2. Taylor expanded in k around 0 77.3%

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

   3. Taylor expanded in m around 0 2.7%

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

   4. Taylor expanded in k around 0 23.2%

      \[\leadsto \color{blue}{a + \left(-10 \cdot \left(k \cdot a\right) + 100 \cdot \left({k}^{2} \cdot a\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 23.2%

         \[\leadsto a + \left(\color{blue}{\left(-10 \cdot k\right) \cdot a} + 100 \cdot \left({k}^{2} \cdot a\right)\right) \]

      2. associate-*r* 23.2%

         \[\leadsto a + \left(\left(-10 \cdot k\right) \cdot a + \color{blue}{\left(100 \cdot {k}^{2}\right) \cdot a}\right) \]

      3. distribute-rgt-out 30.0%

         \[\leadsto a + \color{blue}{a \cdot \left(-10 \cdot k + 100 \cdot {k}^{2}\right)} \]

      4. *-commutative 30.0%

         \[\leadsto a + a \cdot \left(\color{blue}{k \cdot -10} + 100 \cdot {k}^{2}\right) \]

      5. *-commutative 30.0%

         \[\leadsto a + a \cdot \left(k \cdot -10 + \color{blue}{{k}^{2} \cdot 100}\right) \]

      6. unpow2 30.0%

         \[\leadsto a + a \cdot \left(k \cdot -10 + \color{blue}{\left(k \cdot k\right)} \cdot 100\right) \]

      7. associate-*l* 30.0%

         \[\leadsto a + a \cdot \left(k \cdot -10 + \color{blue}{k \cdot \left(k \cdot 100\right)}\right) \]

   6. Simplified 30.0%

      \[\leadsto \color{blue}{a + a \cdot \left(k \cdot -10 + k \cdot \left(k \cdot 100\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.95:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot -10 + k \cdot \left(k \cdot 100\right)\right)\\ \end{array} \]

Alternative 5: 56.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{+50}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10 + 10 \cdot \frac{1}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 1.32e+50)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ a (+ (* k 10.0) (* 10.0 (/ 1.0 k)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 1.32e+50) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / ((k * 10.0) + (10.0 * (1.0 / k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 1.32d+50) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a / ((k * 10.0d0) + (10.0d0 * (1.0d0 / k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 1.32e+50) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / ((k * 10.0) + (10.0 * (1.0 / k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 1.32e+50:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a / ((k * 10.0) + (10.0 * (1.0 / k)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.32e+50)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a / Float64(Float64(k * 10.0) + Float64(10.0 * Float64(1.0 / k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 1.32e+50)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a / ((k * 10.0) + (10.0 * (1.0 / k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.32e+50], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * 10.0), $MachinePrecision] + N[(10.0 * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 1.3199999999999999e50

   1. Initial program 93.8%

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

      1. div-inv 93.7%

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

      2. add-cube-cbrt 92.4%

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

      3. associate-*l* 92.4%

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

      4. pow2 92.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      5. associate-+l+ 92.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      6. +-commutative 92.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

      7. distribute-rgt-out 92.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right) \]

      8. fma-def 92.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right) \]

   3. Applied egg-rr 92.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)} \]

   4. Taylor expanded in m around 0 87.7%

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

   if 1.3199999999999999e50 < m

   1. Initial program 81.5%

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

   2. Taylor expanded in m around 0 2.9%

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

      1. associate-+l+ 2.9%

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

      2. flip-+ 2.5%

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

      3. metadata-eval 2.5%

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

      4. distribute-rgt-out 2.5%

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

      5. distribute-rgt-out 2.5%

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

      6. distribute-rgt-out 2.5%

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

   4. Applied egg-rr 2.5%

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

      1. associate-*r* 2.5%

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

      2. *-commutative 2.5%

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

      3. +-commutative 2.5%

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

      4. *-commutative 2.5%

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

      5. associate-*l* 2.5%

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

      6. +-commutative 2.5%

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

      7. +-commutative 2.5%

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

   6. Simplified 2.5%

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

   7. Taylor expanded in k around 0 2.5%

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

      1. *-commutative 2.5%

         \[\leadsto \frac{a}{\frac{1 - \left(k + 10\right) \cdot \color{blue}{\left({k}^{2} \cdot 10\right)}}{1 - k \cdot \left(k + 10\right)}} \]

      2. unpow2 2.5%

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

      3. associate-*l* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in k around inf 14.1%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.32 \cdot 10^{+50}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10 + 10 \cdot \frac{1}{k}}\\ \end{array} \]

Alternative 6: 54.8% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 5.5e+15) (/ a (+ 1.0 (* k (+ k 10.0)))) (+ a (* -10.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 5.5e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 5.5d+15) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 5.5e+15) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 5.5e+15:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 5.5e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 5.5e+15)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.5e+15], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 5.5e15

   1. Initial program 96.8%

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

      1. div-inv 96.7%

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

      2. add-cube-cbrt 95.4%

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

      3. associate-*l* 95.4%

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

      4. pow2 95.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}\right) \]

      5. associate-+l+ 95.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{1 + \left(10 \cdot k + k \cdot k\right)}}\right) \]

      6. +-commutative 95.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\left(10 \cdot k + k \cdot k\right) + 1}}\right) \]

      7. distribute-rgt-out 95.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{k \cdot \left(10 + k\right)} + 1}\right) \]

      8. fma-def 95.4%

         \[\leadsto {\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(k, 10 + k, 1\right)}}\right) \]

   3. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2} \cdot \left(\sqrt[3]{a \cdot {k}^{m}} \cdot \frac{1}{\mathsf{fma}\left(k, 10 + k, 1\right)}\right)} \]

   4. Taylor expanded in m around 0 93.6%

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

   if 5.5e15 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in m around 0 2.9%

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

   3. Taylor expanded in k around 0 7.6%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 7: 46.9% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-294} \lor \neg \left(k \leq 0.23\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 6.5e-294) (not (<= k 0.23)))
   (/ a (* k k))
   (+ a (* -10.0 (* a k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 6.5e-294) || !(k <= 0.23)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 6.5d-294) .or. (.not. (k <= 0.23d0))) then
        tmp = a / (k * k)
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 6.5e-294) || !(k <= 0.23)) {
		tmp = a / (k * k);
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 6.5e-294) or not (k <= 0.23):
		tmp = a / (k * k)
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 6.5e-294) || !(k <= 0.23))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 6.5e-294) || ~((k <= 0.23)))
		tmp = a / (k * k);
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 6.5e-294], N[Not[LessEqual[k, 0.23]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.4999999999999995e-294 or 0.23000000000000001 < k

   1. Initial program 88.1%

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

   2. Taylor expanded in m around 0 38.7%

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

   3. Taylor expanded in k around inf 41.5%

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

      1. unpow2 41.5%

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

   5. Simplified 41.5%

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

   if 6.4999999999999995e-294 < k < 0.23000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 56.0%

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

   3. Taylor expanded in k around 0 55.5%

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

4. Final simplification 45.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-294} \lor \neg \left(k \leq 0.23\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 8: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-294}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.25e-294)
   (/ a (* k k))
   (if (<= k 0.075) (+ a (* -10.0 (* a k))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.25e-294) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.25d-294) then
        tmp = a / (k * k)
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.25e-294) {
		tmp = a / (k * k);
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.25e-294:
		tmp = a / (k * k)
	elif k <= 0.075:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.25e-294)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.25e-294)
		tmp = a / (k * k);
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.25e-294], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.2500000000000001e-294

   1. Initial program 89.6%

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

   2. Taylor expanded in m around 0 15.4%

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

   3. Taylor expanded in k around inf 22.1%

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

      1. unpow2 22.1%

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

   5. Simplified 22.1%

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

   if 1.2500000000000001e-294 < k < 0.0749999999999999972

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 56.6%

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

   3. Taylor expanded in k around 0 56.1%

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

   if 0.0749999999999999972 < k

   1. Initial program 86.6%

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

   2. Taylor expanded in m around 0 66.3%

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

   3. Taylor expanded in k around inf 65.0%

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

      1. unpow2 65.0%

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

      2. distribute-rgt-in 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-294}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 9: 47.2% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5e-295)
   (/ a (* k k))
   (if (<= k 1.0) (/ a (+ 1.0 (* k 10.0))) (/ a (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-295) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 5d-295) then
        tmp = a / (k * k)
    else if (k <= 1.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-295) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 5e-295:
		tmp = a / (k * k)
	elif k <= 1.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5e-295)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 5e-295)
		tmp = a / (k * k);
	elseif (k <= 1.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5e-295], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.00000000000000008e-295

   1. Initial program 89.6%

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

   2. Taylor expanded in m around 0 15.4%

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

   3. Taylor expanded in k around inf 22.1%

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

      1. unpow2 22.1%

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

   5. Simplified 22.1%

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

   if 5.00000000000000008e-295 < k < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in k around 0 99.6%

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

   3. Taylor expanded in m around 0 55.6%

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

   if 1 < k

   1. Initial program 86.4%

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

   2. Taylor expanded in m around 0 67.0%

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

   3. Taylor expanded in k around inf 65.8%

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

      1. unpow2 65.8%

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

      2. distribute-rgt-in 65.8%

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

   5. Simplified 65.8%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-295}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10: 54.1% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.94)
   (/ a (* k k))
   (if (<= m 6.4e+15) (/ a (+ 1.0 (* k k))) (+ a (* -10.0 (* a k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 6.4e+15) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.94d0)) then
        tmp = a / (k * k)
    else if (m <= 6.4d+15) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a + ((-10.0d0) * (a * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.94) {
		tmp = a / (k * k);
	} else if (m <= 6.4e+15) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a + (-10.0 * (a * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.94:
		tmp = a / (k * k)
	elif m <= 6.4e+15:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a + (-10.0 * (a * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.94)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 6.4e+15)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.94)
		tmp = a / (k * k);
	elseif (m <= 6.4e+15)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a + (-10.0 * (a * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.94], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.4e+15], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.93999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 35.5%

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

   3. Taylor expanded in k around inf 59.2%

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

      1. unpow2 59.2%

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

   5. Simplified 59.2%

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

   if -0.93999999999999995 < m < 6.4e15

   1. Initial program 96.8%

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

   2. Taylor expanded in m around 0 93.6%

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

   3. Taylor expanded in k around 0 90.7%

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

   if 6.4e15 < m

   1. Initial program 79.3%

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

   2. Taylor expanded in m around 0 2.9%

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

   3. Taylor expanded in k around 0 7.6%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.94:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \end{array} \]

Alternative 11: 46.8% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-295} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 4e-295) (not (<= k 1.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 4e-295) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= 4d-295) .or. (.not. (k <= 1.0d0))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= 4e-295) || !(k <= 1.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 4e-295) or not (k <= 1.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 4e-295) || !(k <= 1.0))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= 4e-295) || ~((k <= 1.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 4e-295], N[Not[LessEqual[k, 1.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 4.00000000000000024e-295 or 1 < k

   1. Initial program 88.1%

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

   2. Taylor expanded in m around 0 38.7%

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

   3. Taylor expanded in k around inf 41.5%

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

      1. unpow2 41.5%

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

   5. Simplified 41.5%

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

   if 4.00000000000000024e-295 < k < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0 56.0%

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

   3. Taylor expanded in k around 0 54.8%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{-295} \lor \neg \left(k \leq 1\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 12: 26.4% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 94.3%

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

   2. Taylor expanded in m around 0 34.1%

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

   3. Taylor expanded in k around 0 28.1%

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

   if 0.10000000000000001 < k

   1. Initial program 86.6%

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

   2. Taylor expanded in k around 0 68.1%

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

   3. Taylor expanded in m around 0 25.1%

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

   4. Taylor expanded in k around inf 25.1%

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

      1. *-commutative 25.1%

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

   6. Simplified 25.1%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 13: 20.2% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.9%

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

2. Taylor expanded in m around 0 44.2%

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

3. Taylor expanded in k around 0 20.8%

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

4. Final simplification 20.8%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))