Falkner and Boettcher, Appendix A

Percentage Accurate: 90.2% → 97.4%

Time: 8.7s

Alternatives: 11

Speedup: 1.1×

• 23.4% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.2% 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: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 95.8%

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

   2. Taylor expanded in k around 0 98.7%

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

   if 1 < k

   1. Initial program 83.7%

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

   2. Taylor expanded in k around inf 83.1%

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

      1. unpow2 83.1%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r* 95.9%

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

      5. exp-prod 95.9%

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

      6. neg-mul-1 95.9%

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

      7. log-rec 95.9%

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

      8. remove-double-neg 95.9%

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

      9. exp-prod 95.9%

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

      10. exp-to-pow 95.9%

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

   4. Simplified 95.9%

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

      1. clear-num 95.9%

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

      2. un-div-inv 95.9%

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

   6. Applied egg-rr 95.9%

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

4. Final simplification 97.7%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 95.8%

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

   2. Taylor expanded in k around 0 98.7%

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

   if 1 < k

   1. Initial program 83.7%

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

   2. Taylor expanded in k around inf 83.1%

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

      1. unpow2 83.1%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r* 95.9%

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

      5. exp-prod 95.9%

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

      6. neg-mul-1 95.9%

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

      7. log-rec 95.9%

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

      8. remove-double-neg 95.9%

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

      9. exp-prod 95.9%

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

      10. exp-to-pow 95.9%

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

   4. Simplified 95.9%

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

4. Final simplification 97.7%

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

Alternative 3: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.74 \lor \neg \left(m \leq 7.5 \cdot 10^{-21}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.74) (not (<= m 7.5e-21)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.74) || !(m <= 7.5e-21)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-0.74d0)) .or. (.not. (m <= 7.5d-21))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.74) || !(m <= 7.5e-21)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.74) or not (m <= 7.5e-21):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.74) || !(m <= 7.5e-21))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.74) || ~((m <= 7.5e-21)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.74], N[Not[LessEqual[m, 7.5e-21]], $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 < -0.73999999999999999 or 7.50000000000000072e-21 < m

   1. Initial program 91.3%

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

   2. Taylor expanded in k around 0 99.4%

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

   if -0.73999999999999999 < m < 7.50000000000000072e-21

   1. Initial program 92.0%

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

   2. Taylor expanded in m around 0 91.1%

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

      1. +-commutative 91.1%

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

      2. unpow2 91.1%

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

      3. distribute-rgt-in 91.1%

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

      4. fma-udef 91.1%

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

      5. +-commutative 91.1%

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

   4. Simplified 91.1%

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

      1. fma-udef 91.1%

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

   6. Applied egg-rr 91.1%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.74 \lor \neg \left(m \leq 7.5 \cdot 10^{-21}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 4: 96.8% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* a (pow k m)) (/ a (pow k (- 2.0 m)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / pow(k, (2.0 - m));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.0d0) then
        tmp = a * (k ** m)
    else
        tmp = a / (k ** (2.0d0 - m))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / Math.pow(k, (2.0 - m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.0:
		tmp = a * math.pow(k, m)
	else:
		tmp = a / math.pow(k, (2.0 - m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.0)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / (k ^ Float64(2.0 - m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = a / (k ^ (2.0 - m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[Power[k, N[(2.0 - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 95.8%

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

   2. Taylor expanded in k around 0 98.7%

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

   if 1 < k

   1. Initial program 83.7%

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

   2. Taylor expanded in k around inf 83.1%

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

      1. unpow2 83.1%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r* 95.9%

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

      5. exp-prod 95.9%

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

      6. neg-mul-1 95.9%

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

      7. log-rec 95.9%

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

      8. remove-double-neg 95.9%

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

      9. exp-prod 95.9%

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

      10. exp-to-pow 95.9%

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

   4. Simplified 95.9%

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

      1. expm1-log1p-u 79.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\right)} \]

      2. expm1-udef 57.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)} - 1} \]

      3. frac-times 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}}\right)} - 1 \]

      4. associate-/l* 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{k \cdot k}{{k}^{m}}}}\right)} - 1 \]

      5. pow2 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      6. pow-div 58.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 91.9%

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

   8. Simplified 91.9%

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

4. Final simplification 96.3%

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

Alternative 5: 47.7% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.7e-306)
   (/ a (* k k))
   (if (<= k 0.1) (+ a (* -10.0 (* k a))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.7e-306) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 1.7d-306) then
        tmp = a / (k * k)
    else if (k <= 0.1d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.7e-306) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.7e-306:
		tmp = a / (k * k)
	elif k <= 0.1:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.7e-306)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.7e-306)
		tmp = a / (k * k);
	elseif (k <= 0.1)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.7e-306], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.6999999999999999e-306

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0 13.6%

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

      1. +-commutative 13.6%

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

      2. unpow2 13.6%

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

      3. distribute-rgt-in 13.6%

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

      4. fma-udef 13.6%

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

      5. +-commutative 13.6%

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

   4. Simplified 13.6%

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

   5. Taylor expanded in k around inf 23.6%

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

      1. unpow2 23.6%

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

   7. Simplified 23.6%

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

   if 1.6999999999999999e-306 < k < 0.10000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.8%

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

      1. +-commutative 39.8%

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

      2. unpow2 39.8%

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

      3. distribute-rgt-in 39.8%

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

      4. fma-udef 39.8%

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

      5. +-commutative 39.8%

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

   4. Simplified 39.8%

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

   5. Taylor expanded in k around 0 39.2%

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

   if 0.10000000000000001 < k

   1. Initial program 83.7%

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

   2. Taylor expanded in k around inf 83.1%

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

      1. unpow2 83.1%

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

      2. times-frac 95.9%

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

      3. *-commutative 95.9%

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

      4. associate-*r* 95.9%

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

      5. exp-prod 95.9%

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

      6. neg-mul-1 95.9%

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

      7. log-rec 95.9%

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

      8. remove-double-neg 95.9%

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

      9. exp-prod 95.9%

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

      10. exp-to-pow 95.9%

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

   4. Simplified 95.9%

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

      1. expm1-log1p-u 79.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\right)} \]

      2. expm1-udef 57.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)} - 1} \]

      3. frac-times 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}}\right)} - 1 \]

      4. associate-/l* 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{k \cdot k}{{k}^{m}}}}\right)} - 1 \]

      5. pow2 55.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      6. pow-div 58.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 58.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in m around 0 65.1%

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

       1. unpow2 65.1%

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

       2. associate-/r* 71.3%

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

   11. Simplified 71.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.7 \cdot 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 6: 47.6% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1060000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 3.2e-306)
   (/ a (* k k))
   (if (<= k 1060000000.0) (/ a (+ 1.0 (* k 10.0))) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 3.2e-306) {
		tmp = a / (k * k);
	} else if (k <= 1060000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 3.2d-306) then
        tmp = a / (k * k)
    else if (k <= 1060000000.0d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 3.2e-306) {
		tmp = a / (k * k);
	} else if (k <= 1060000000.0) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 3.2e-306:
		tmp = a / (k * k)
	elif k <= 1060000000.0:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 3.2e-306)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1060000000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 3.2e-306)
		tmp = a / (k * k);
	elseif (k <= 1060000000.0)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 3.2e-306], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1060000000.0], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.19999999999999971e-306

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0 13.6%

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

      1. +-commutative 13.6%

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

      2. unpow2 13.6%

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

      3. distribute-rgt-in 13.6%

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

      4. fma-udef 13.6%

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

      5. +-commutative 13.6%

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

   4. Simplified 13.6%

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

   5. Taylor expanded in k around inf 23.6%

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

      1. unpow2 23.6%

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

   7. Simplified 23.6%

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

   if 3.19999999999999971e-306 < k < 1.06e9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.4%

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

      1. +-commutative 39.4%

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

      2. unpow2 39.4%

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

      3. distribute-rgt-in 39.4%

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

      4. fma-udef 39.4%

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

      5. +-commutative 39.4%

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

   4. Simplified 39.4%

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

   5. Taylor expanded in k around 0 38.9%

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

   if 1.06e9 < k

   1. Initial program 83.5%

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

   2. Taylor expanded in k around inf 82.9%

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

      1. unpow2 82.9%

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

      2. times-frac 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*r* 95.8%

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

      5. exp-prod 95.8%

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

      6. neg-mul-1 95.8%

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

      7. log-rec 95.8%

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

      8. remove-double-neg 95.8%

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

      9. exp-prod 95.8%

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

      10. exp-to-pow 95.8%

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

   4. Simplified 95.8%

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

      1. expm1-log1p-u 80.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\right)} \]

      2. expm1-udef 58.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)} - 1} \]

      3. frac-times 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}}\right)} - 1 \]

      4. associate-/l* 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{k \cdot k}{{k}^{m}}}}\right)} - 1 \]

      5. pow2 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      6. pow-div 59.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in m around 0 65.9%

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

       1. unpow2 65.9%

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

       2. associate-/r* 72.1%

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

   11. Simplified 72.1%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1060000000:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 7: 53.0% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.02:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.02) (/ a (* k k)) (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-1.02d0)) then
        tmp = a / (k * k)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.02) {
		tmp = a / (k * k);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.02:
		tmp = a / (k * k)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.02)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.02)
		tmp = a / (k * k);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.02], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.02

   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 28.4%

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

      1. +-commutative 28.4%

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

      2. unpow2 28.4%

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

      3. distribute-rgt-in 28.4%

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

      4. fma-udef 28.4%

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

      5. +-commutative 28.4%

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

   4. Simplified 28.4%

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

   5. Taylor expanded in k around inf 54.2%

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

      1. unpow2 54.2%

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

   7. Simplified 54.2%

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

   if -1.02 < m

   1. Initial program 87.2%

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

   2. Taylor expanded in m around 0 47.9%

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

      1. +-commutative 47.9%

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

      2. unpow2 47.9%

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

      3. distribute-rgt-in 47.9%

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

      4. fma-udef 47.9%

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

      5. +-commutative 47.9%

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

   4. Simplified 47.9%

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

      1. fma-udef 47.9%

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

   6. Applied egg-rr 47.9%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.02:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 46.2% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-306} \lor \neg \left(k \leq 1060000000\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k 5.2e-306) (not (<= k 1060000000.0))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= 5.2e-306) || !(k <= 1060000000.0)) {
		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 <= 5.2d-306) .or. (.not. (k <= 1060000000.0d0))) 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 <= 5.2e-306) || !(k <= 1060000000.0)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= 5.2e-306) or not (k <= 1060000000.0):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= 5.2e-306) || !(k <= 1060000000.0))
		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 <= 5.2e-306) || ~((k <= 1060000000.0)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, 5.2e-306], N[Not[LessEqual[k, 1060000000.0]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < 5.2000000000000001e-306 or 1.06e9 < k

   1. Initial program 86.7%

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

   2. Taylor expanded in m around 0 42.5%

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

      1. +-commutative 42.5%

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

      2. unpow2 42.5%

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

      3. distribute-rgt-in 42.5%

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

      4. fma-udef 42.5%

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

      5. +-commutative 42.5%

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

   4. Simplified 42.5%

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

   5. Taylor expanded in k around inf 46.7%

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

      1. unpow2 46.7%

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

   7. Simplified 46.7%

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

   if 5.2000000000000001e-306 < k < 1.06e9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.4%

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

      1. +-commutative 39.4%

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

      2. unpow2 39.4%

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

      3. distribute-rgt-in 39.4%

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

      4. fma-udef 39.4%

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

      5. +-commutative 39.4%

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

   4. Simplified 39.4%

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

   5. Taylor expanded in k around 0 38.2%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.2 \cdot 10^{-306} \lor \neg \left(k \leq 1060000000\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 9: 47.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1060000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 3.4e-305) (/ a (* k k)) (if (<= k 1060000000.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 3.4e-305) {
		tmp = a / (k * k);
	} else if (k <= 1060000000.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 3.4d-305) then
        tmp = a / (k * k)
    else if (k <= 1060000000.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 3.4e-305) {
		tmp = a / (k * k);
	} else if (k <= 1060000000.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 3.4e-305:
		tmp = a / (k * k)
	elif k <= 1060000000.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 3.4e-305)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1060000000.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 3.4e-305)
		tmp = a / (k * k);
	elseif (k <= 1060000000.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 3.4e-305], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1060000000.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 3.4000000000000001e-305

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0 13.6%

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

      1. +-commutative 13.6%

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

      2. unpow2 13.6%

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

      3. distribute-rgt-in 13.6%

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

      4. fma-udef 13.6%

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

      5. +-commutative 13.6%

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

   4. Simplified 13.6%

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

   5. Taylor expanded in k around inf 23.6%

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

      1. unpow2 23.6%

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

   7. Simplified 23.6%

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

   if 3.4000000000000001e-305 < k < 1.06e9

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 39.4%

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

      1. +-commutative 39.4%

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

      2. unpow2 39.4%

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

      3. distribute-rgt-in 39.4%

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

      4. fma-udef 39.4%

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

      5. +-commutative 39.4%

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

   4. Simplified 39.4%

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

   5. Taylor expanded in k around 0 38.2%

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

   if 1.06e9 < k

   1. Initial program 83.5%

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

   2. Taylor expanded in k around inf 82.9%

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

      1. unpow2 82.9%

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

      2. times-frac 95.8%

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

      3. *-commutative 95.8%

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

      4. associate-*r* 95.8%

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

      5. exp-prod 95.8%

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

      6. neg-mul-1 95.8%

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

      7. log-rec 95.8%

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

      8. remove-double-neg 95.8%

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

      9. exp-prod 95.8%

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

      10. exp-to-pow 95.8%

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

   4. Simplified 95.8%

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

      1. expm1-log1p-u 80.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)\right)} \]

      2. expm1-udef 58.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{k} \cdot \frac{{k}^{m}}{k}\right)} - 1} \]

      3. frac-times 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{k \cdot k}}\right)} - 1 \]

      4. associate-/l* 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{k \cdot k}{{k}^{m}}}}\right)} - 1 \]

      5. pow2 55.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{\color{blue}{{k}^{2}}}{{k}^{m}}}\right)} - 1 \]

      6. pow-div 59.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\color{blue}{{k}^{\left(2 - m\right)}}}\right)} - 1 \]

   6. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 73.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{{k}^{\left(2 - m\right)}}\right)\right)} \]

      2. expm1-log1p 91.9%

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

   8. Simplified 91.9%

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

   9. Taylor expanded in m around 0 65.9%

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

       1. unpow2 65.9%

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

       2. associate-/r* 72.1%

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

   11. Simplified 72.1%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1060000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \]

Alternative 10: 26.2% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m -3.1e-23) (/ a (* k 10.0)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-23) {
		tmp = a / (k * 10.0);
	} 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 (m <= (-3.1d-23)) then
        tmp = a / (k * 10.0d0)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.1e-23) {
		tmp = a / (k * 10.0);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.1e-23:
		tmp = a / (k * 10.0)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.1e-23)
		tmp = Float64(a / Float64(k * 10.0));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.1e-23)
		tmp = a / (k * 10.0);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.1e-23], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if m < -3.0999999999999999e-23

   1. Initial program 99.0%

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

   2. Taylor expanded in m around 0 29.7%

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

      1. +-commutative 29.7%

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

      2. unpow2 29.7%

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

      3. distribute-rgt-in 29.7%

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

      4. fma-udef 29.7%

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

      5. +-commutative 29.7%

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

   4. Simplified 29.7%

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

   5. Taylor expanded in k around 0 10.5%

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

   6. Taylor expanded in k around inf 19.7%

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

      1. *-commutative 19.7%

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

   8. Simplified 19.7%

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

   if -3.0999999999999999e-23 < m

   1. Initial program 87.5%

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

   2. Taylor expanded in m around 0 47.7%

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

      1. +-commutative 47.7%

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

      2. unpow2 47.7%

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

      3. distribute-rgt-in 47.7%

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

      4. fma-udef 47.7%

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

      5. +-commutative 47.7%

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

   4. Simplified 47.7%

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

   5. Taylor expanded in k around 0 23.3%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 11: 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 91.5%

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

2. Taylor expanded in m around 0 41.4%

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

   1. +-commutative 41.4%

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

   2. unpow2 41.4%

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

   3. distribute-rgt-in 41.4%

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

   4. fma-udef 41.4%

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

   5. +-commutative 41.4%

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

4. Simplified 41.4%

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

5. Taylor expanded in k around 0 16.4%

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

6. Final simplification 16.4%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023283 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))