Falkner and Boettcher, Appendix A

Percentage Accurate: 90.9% → 97.7%

Time: 10.6s

Alternatives: 10

Speedup: 1.0×

• 21.4% 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: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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+180) 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+180) {
		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+180) 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+180) {
		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+180:
		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+180)
		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+180)
		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+180], 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))) < 2e180

   1. Initial program 97.3%

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

   if 2e180 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 53.5%

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

      1. associate-/l* 53.5%

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

      2. remove-double-neg 53.5%

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

      3. distribute-frac-neg2 53.5%

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

      4. distribute-neg-frac2 53.5%

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

      5. remove-double-neg 53.5%

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

      6. sqr-neg 53.5%

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

      7. associate-+l+ 53.5%

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

      8. sqr-neg 53.5%

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

      9. distribute-rgt-out 53.5%

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

   3. Simplified 53.5%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 2 \cdot 10^{+180}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-28}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2e-28)
   (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0)))))
   (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2e-28) {
		tmp = a * (pow(k, m) / (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 <= 2d-28) then
        tmp = a * ((k ** m) / (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 <= 2e-28) {
		tmp = a * (Math.pow(k, m) / (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 <= 2e-28:
		tmp = a * (math.pow(k, m) / (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 <= 2e-28)
		tmp = Float64(a * Float64((k ^ m) / 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 <= 2e-28)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2e-28], N[(a * N[(N[Power[k, m], $MachinePrecision] / 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 < 1.99999999999999994e-28

   1. Initial program 96.6%

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

      1. associate-/l* 96.6%

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

      2. remove-double-neg 96.6%

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

      3. distribute-frac-neg2 96.6%

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

      4. distribute-neg-frac2 96.6%

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

      5. remove-double-neg 96.6%

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

      6. sqr-neg 96.6%

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

      7. associate-+l+ 96.6%

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

      8. sqr-neg 96.6%

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

      9. distribute-rgt-out 96.6%

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

   3. Simplified 96.6%

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

   if 1.99999999999999994e-28 < m

   1. Initial program 76.2%

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

      1. associate-/l* 76.2%

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

      2. remove-double-neg 76.2%

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

      3. distribute-frac-neg2 76.2%

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

      4. distribute-neg-frac2 76.2%

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

      5. remove-double-neg 76.2%

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

      6. sqr-neg 76.2%

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

      7. associate-+l+ 76.2%

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

      8. sqr-neg 76.2%

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

      9. distribute-rgt-out 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{-28}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{+18} \lor \neg \left(m \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2.1e+18) (not (<= m 2e-28)))
   (* a (pow k m))
   (/ a (+ (+ 1.0 (* k 10.0)) (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.1e+18) || !(m <= 2e-28)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-2.1d+18)) .or. (.not. (m <= 2d-28))) then
        tmp = a * (k ** m)
    else
        tmp = a / ((1.0d0 + (k * 10.0d0)) + (k * k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.1e+18) || !(m <= 2e-28)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -2.1e+18) or not (m <= 2e-28):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / ((1.0 + (k * 10.0)) + (k * k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2.1e+18) || !(m <= 2e-28))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -2.1e+18) || ~((m <= 2e-28)))
		tmp = a * (k ^ m);
	else
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2.1e+18], N[Not[LessEqual[m, 2e-28]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.1e18 or 1.99999999999999994e-28 < m

   1. Initial program 87.9%

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

      1. associate-/l* 87.9%

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

      2. remove-double-neg 87.9%

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

      3. distribute-frac-neg2 87.9%

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

      4. distribute-neg-frac2 87.9%

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

      5. remove-double-neg 87.9%

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

      6. sqr-neg 87.9%

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

      7. associate-+l+ 87.9%

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

      8. sqr-neg 87.9%

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

      9. distribute-rgt-out 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.1e18 < m < 1.99999999999999994e-28

   1. Initial program 93.6%

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

   3. Taylor expanded in m around 0 92.9%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.1 \cdot 10^{+18} \lor \neg \left(m \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 28.7% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -6 \cdot 10^{+34} \lor \neg \left(k \leq 0.064\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -6e+34) (not (<= k 0.064))) (* 0.1 (/ a k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -6e+34) || !(k <= 0.064)) {
		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 <= (-6d+34)) .or. (.not. (k <= 0.064d0))) 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 <= -6e+34) || !(k <= 0.064)) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -6e+34) or not (k <= 0.064):
		tmp = 0.1 * (a / k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -6e+34) || !(k <= 0.064))
		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 <= -6e+34) || ~((k <= 0.064)))
		tmp = 0.1 * (a / k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -6e+34], N[Not[LessEqual[k, 0.064]], $MachinePrecision]], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -6.00000000000000037e34 or 0.064000000000000001 < k

   1. Initial program 80.1%

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

      1. associate-/l* 80.1%

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

      2. remove-double-neg 80.1%

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

      3. distribute-frac-neg2 80.1%

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

      4. distribute-neg-frac2 80.1%

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

      5. remove-double-neg 80.1%

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

      6. sqr-neg 80.1%

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

      7. associate-+l+ 80.1%

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

      8. sqr-neg 80.1%

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

      9. distribute-rgt-out 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in m around 0 59.1%

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

   6. Taylor expanded in k around 0 24.9%

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

      1. *-commutative 24.9%

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

   8. Simplified 24.9%

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

   9. Taylor expanded in k around inf 24.9%

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

   if -6.00000000000000037e34 < k < 0.064000000000000001

   1. Initial program 99.9%

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

      1. associate-/l* 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-frac-neg2 99.9%

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

      4. distribute-neg-frac2 99.9%

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

      5. remove-double-neg 99.9%

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

      6. sqr-neg 99.9%

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

      7. associate-+l+ 99.9%

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

      8. sqr-neg 99.9%

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

      9. distribute-rgt-out 99.9%

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

   3. Simplified 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-*r/ 99.9%

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

      4. clear-num 99.9%

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

      5. associate-/r* 99.1%

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

      6. associate-+l+ 99.1%

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

      7. distribute-rgt-in 99.1%

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

      8. +-commutative 99.1%

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

      9. fma-define 99.1%

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

      10. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. associate-/r/ 99.1%

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

      2. associate-*l/ 99.1%

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

      3. *-lft-identity 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in k around 0 98.5%

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

   10. Taylor expanded in m around 0 33.9%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6 \cdot 10^{+34} \lor \neg \left(k \leq 0.064\right):\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 28.6% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.072:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.072) (+ a (* -10.0 (* a k))) (* 0.1 (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.072) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 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) :: tmp
    if (k <= 0.072d0) then
        tmp = a + ((-10.0d0) * (a * k))
    else
        tmp = 0.1d0 * (a / k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.072) {
		tmp = a + (-10.0 * (a * k));
	} else {
		tmp = 0.1 * (a / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.072:
		tmp = a + (-10.0 * (a * k))
	else:
		tmp = 0.1 * (a / k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.072)
		tmp = Float64(a + Float64(-10.0 * Float64(a * k)));
	else
		tmp = Float64(0.1 * Float64(a / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.072)
		tmp = a + (-10.0 * (a * k));
	else
		tmp = 0.1 * (a / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.072], N[(a + N[(-10.0 * N[(a * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0719999999999999946

   1. Initial program 96.9%

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

      1. associate-/l* 96.9%

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

      2. remove-double-neg 96.9%

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

      3. distribute-frac-neg2 96.9%

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

      4. distribute-neg-frac2 96.9%

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

      5. remove-double-neg 96.9%

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

      6. sqr-neg 96.9%

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

      7. associate-+l+ 96.9%

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

      8. sqr-neg 96.9%

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

      9. distribute-rgt-out 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in m around 0 39.9%

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

   6. Taylor expanded in k around 0 29.2%

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

   if 0.0719999999999999946 < k

   1. Initial program 77.5%

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

      1. associate-/l* 77.5%

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

      2. remove-double-neg 77.5%

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

      3. distribute-frac-neg2 77.5%

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

      4. distribute-neg-frac2 77.5%

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

      5. remove-double-neg 77.5%

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

      6. sqr-neg 77.5%

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

      7. associate-+l+ 77.5%

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

      8. sqr-neg 77.5%

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

      9. distribute-rgt-out 77.5%

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

   3. Simplified 77.5%

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

   5. Taylor expanded in m around 0 60.8%

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

   6. Taylor expanded in k around 0 27.1%

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

      1. *-commutative 27.1%

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

   8. Simplified 27.1%

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

   9. Taylor expanded in k around inf 27.1%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.072:\\ \;\;\;\;a + -10 \cdot \left(a \cdot k\right)\\ \mathbf{else}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 31.0% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e+21) (* 0.1 (/ a k)) (/ a (+ 1.0 (* k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e+21) {
		tmp = 0.1 * (a / k);
	} 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 (m <= (-1.7d+21)) then
        tmp = 0.1d0 * (a / k)
    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 (m <= -1.7e+21) {
		tmp = 0.1 * (a / k);
	} else {
		tmp = a / (1.0 + (k * 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.7e+21:
		tmp = 0.1 * (a / k)
	else:
		tmp = a / (1.0 + (k * 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e+21)
		tmp = Float64(0.1 * Float64(a / k));
	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 (m <= -1.7e+21)
		tmp = 0.1 * (a / k);
	else
		tmp = a / (1.0 + (k * 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e+21], N[(0.1 * N[(a / k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.7e21

   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%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 40.9%

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

   6. Taylor expanded in k around 0 16.3%

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

      1. *-commutative 16.3%

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

   8. Simplified 16.3%

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

   9. Taylor expanded in k around inf 22.9%

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

   if -1.7e21 < m

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. remove-double-neg 85.4%

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

      3. distribute-frac-neg2 85.4%

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

      4. distribute-neg-frac2 85.4%

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

      5. remove-double-neg 85.4%

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

      6. sqr-neg 85.4%

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

      7. associate-+l+ 85.4%

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

      8. sqr-neg 85.4%

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

      9. distribute-rgt-out 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in m around 0 50.3%

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

   6. Taylor expanded in k around 0 35.7%

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

      1. *-commutative 35.7%

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

   8. Simplified 35.7%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{+21}:\\ \;\;\;\;0.1 \cdot \frac{a}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.6% accurate, 10.4× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	return 1.0 / ((1.0 + (k * (k + 10.0))) / 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 = 1.0d0 / ((1.0d0 + (k * (k + 10.0d0))) / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

   1. distribute-rgt-in 89.9%

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

   2. associate-+l+ 89.9%

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

   3. associate-*r/ 89.9%

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

   4. clear-num 89.8%

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

   5. associate-/r* 88.6%

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

   6. associate-+l+ 88.6%

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

   7. distribute-rgt-in 88.6%

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

   8. +-commutative 88.6%

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

   9. fma-define 88.6%

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

   10. +-commutative 88.6%

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

6. Applied egg-rr 88.6%

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

   1. associate-/l/ 89.8%

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

8. Simplified 89.8%

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

9. Taylor expanded in m around 0 47.6%

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

10. Final simplification 47.6%

    \[\leadsto \frac{1}{\frac{1 + k \cdot \left(k + 10\right)}{a}} \]
11. Add Preprocessing

Alternative 8: 45.6% accurate, 12.7× speedup

Math

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

FPCore

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

C

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

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 / (1.0d0 + (k * (k + 10.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

5. Taylor expanded in m around 0 47.4%

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

6. Final simplification 47.4%

   \[\leadsto \frac{a}{1 + k \cdot \left(k + 10\right)} \]
7. Add Preprocessing

Alternative 9: 44.8% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1 + k \cdot k} \end{array} \]

FPCore

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

C

double code(double a, double k, double m) {
	return a / (1.0 + (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 / (1.0d0 + (k * k))
end function

Java

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

Python

def code(a, k, m):
	return a / (1.0 + (k * k))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

3. Taylor expanded in m around 0 47.4%

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

4. Taylor expanded in k around 0 45.8%

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

5. Final simplification 45.8%

   \[\leadsto \frac{a}{1 + k \cdot k} \]
6. Add Preprocessing

Alternative 10: 20.5% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.9%

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

   1. associate-/l* 89.9%

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

   2. remove-double-neg 89.9%

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

   3. distribute-frac-neg2 89.9%

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

   4. distribute-neg-frac2 89.9%

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

   5. remove-double-neg 89.9%

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

   6. sqr-neg 89.9%

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

   7. associate-+l+ 89.9%

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

   8. sqr-neg 89.9%

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

   9. distribute-rgt-out 89.9%

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

3. Simplified 89.9%

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

   1. distribute-rgt-in 89.9%

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

   2. associate-+l+ 89.9%

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

   3. associate-*r/ 89.9%

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

   4. clear-num 89.8%

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

   5. associate-/r* 88.6%

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

   6. associate-+l+ 88.6%

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

   7. distribute-rgt-in 88.6%

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

   8. +-commutative 88.6%

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

   9. fma-define 88.6%

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

   10. +-commutative 88.6%

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

6. Applied egg-rr 88.6%

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

   1. associate-/r/ 88.6%

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

   2. associate-*l/ 88.6%

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

   3. *-lft-identity 88.6%

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

8. Simplified 88.6%

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

9. Taylor expanded in k around 0 81.7%

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

10. Taylor expanded in m around 0 19.2%

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

11. Final simplification 19.2%

    \[\leadsto a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))