Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 97.2%

Time: 9.5s

Alternatives: 13

Speedup: 1.1×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{k}{{k}^{m}}\\ \mathbf{if}\;k \leq 3 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10, \frac{t_0}{a}, \frac{1}{a \cdot {k}^{m}}\right) + t_0 \cdot \frac{k}{a}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ k (pow k m))))
   (if (<= k 3e-67)
     (/ a (/ 1.0 (pow k m)))
     (pow
      (+ (fma 10.0 (/ t_0 a) (/ 1.0 (* a (pow k m)))) (* t_0 (/ k a)))
      -1.0))))

C

double code(double a, double k, double m) {
	double t_0 = k / pow(k, m);
	double tmp;
	if (k <= 3e-67) {
		tmp = a / (1.0 / pow(k, m));
	} else {
		tmp = pow((fma(10.0, (t_0 / a), (1.0 / (a * pow(k, m)))) + (t_0 * (k / a))), -1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(k / (k ^ m))
	tmp = 0.0
	if (k <= 3e-67)
		tmp = Float64(a / Float64(1.0 / (k ^ m)));
	else
		tmp = Float64(fma(10.0, Float64(t_0 / a), Float64(1.0 / Float64(a * (k ^ m)))) + Float64(t_0 * Float64(k / a))) ^ -1.0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k / N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3e-67], N[(a / N[(1.0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(10.0 * N[(t$95$0 / a), $MachinePrecision] + N[(1.0 / N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.00000000000000032e-67

   1. Initial program 92.7%

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

      1. associate-/l* 92.7%

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

      2. sqr-neg 92.7%

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

      3. +-commutative 92.7%

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

      4. sqr-neg 92.7%

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

      5. fma-def 92.7%

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

      6. +-commutative 92.7%

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

      7. *-commutative 92.7%

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

      8. fma-def 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in k around 0 100.0%

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

   if 3.00000000000000032e-67 < k

   1. Initial program 85.9%

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

      1. associate-/l* 85.9%

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

      2. sqr-neg 85.9%

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

      3. +-commutative 85.9%

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

      4. sqr-neg 85.9%

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

      5. fma-def 85.9%

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

      6. +-commutative 85.9%

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

      7. *-commutative 85.9%

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

      8. fma-def 85.9%

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

   3. Simplified 85.9%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

      3. fma-udef 85.9%

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

      4. fma-udef 85.9%

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

      5. *-commutative 85.9%

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

      6. associate-+r+ 85.9%

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

      7. +-commutative 85.9%

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

      8. distribute-rgt-out 85.9%

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

      9. fma-def 85.9%

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

   5. Applied egg-rr 85.9%

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

   6. Taylor expanded in k around 0 85.9%

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

      1. associate-+r+ 85.9%

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

      2. fma-def 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-/r* 85.9%

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

      5. unpow2 85.9%

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

      6. *-commutative 85.9%

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

      7. times-frac 98.0%

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

   8. Simplified 98.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10, \frac{\frac{k}{{k}^{m}}}{a}, \frac{1}{a \cdot {k}^{m}}\right) + \frac{k}{{k}^{m}} \cdot \frac{k}{a}\right)}^{-1}\\ \end{array} \]

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0019:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.0019)
   (* a (pow k m))
   (if (<= m 3.5e-36)
     (/ a (+ (+ 1.0 (* k 10.0)) (* k k)))
     (/ a (/ 1.0 (pow k m))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0019) {
		tmp = a * pow(k, m);
	} else if (m <= 3.5e-36) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a / (1.0 / pow(k, m));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.0019d0)) then
        tmp = a * (k ** m)
    else if (m <= 3.5d-36) then
        tmp = a / ((1.0d0 + (k * 10.0d0)) + (k * k))
    else
        tmp = a / (1.0d0 / (k ** m))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.0019) {
		tmp = a * Math.pow(k, m);
	} else if (m <= 3.5e-36) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a / (1.0 / Math.pow(k, m));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.0019:
		tmp = a * math.pow(k, m)
	elif m <= 3.5e-36:
		tmp = a / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = a / (1.0 / math.pow(k, m))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.0019)
		tmp = Float64(a * (k ^ m));
	elseif (m <= 3.5e-36)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(a / Float64(1.0 / (k ^ m)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.0019)
		tmp = a * (k ^ m);
	elseif (m <= 3.5e-36)
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = a / (1.0 / (k ^ m));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.0019], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.5e-36], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0019

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. sqr-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. sqr-neg 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -0.0019 < m < 3.5e-36

   1. Initial program 93.4%

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

   2. Taylor expanded in m around 0 92.9%

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

   if 3.5e-36 < m

   1. Initial program 75.6%

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

      1. associate-/l* 75.6%

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

      2. sqr-neg 75.6%

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

      3. +-commutative 75.6%

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

      4. sqr-neg 75.6%

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

      5. fma-def 75.6%

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

      6. +-commutative 75.6%

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

      7. *-commutative 75.6%

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

      8. fma-def 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0019:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 3.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \end{array} \]

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.7e-7) (/ a (/ 1.0 (pow k m))) (* (/ a k) (/ (pow k m) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.7e-7) {
		tmp = a / (1.0 / 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 <= 2.7d-7) then
        tmp = a / (1.0d0 / (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 <= 2.7e-7) {
		tmp = a / (1.0 / 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 <= 2.7e-7:
		tmp = a / (1.0 / 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 <= 2.7e-7)
		tmp = Float64(a / Float64(1.0 / (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 <= 2.7e-7)
		tmp = a / (1.0 / (k ^ m));
	else
		tmp = (a / k) * ((k ^ m) / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.7e-7], N[(a / N[(1.0 / N[Power[k, m], $MachinePrecision]), $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 < 2.70000000000000009e-7

   1. Initial program 93.3%

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

      1. associate-/l* 93.3%

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

      2. sqr-neg 93.3%

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

      3. +-commutative 93.3%

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

      4. sqr-neg 93.3%

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

      5. fma-def 93.3%

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

      6. +-commutative 93.3%

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

      7. *-commutative 93.3%

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

      8. fma-def 93.3%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in k around 0 99.7%

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

   if 2.70000000000000009e-7 < k

   1. Initial program 83.8%

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

      1. associate-/l* 83.8%

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

      2. sqr-neg 83.8%

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

      3. +-commutative 83.8%

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

      4. sqr-neg 83.8%

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

      5. fma-def 83.8%

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

      6. +-commutative 83.8%

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

      7. *-commutative 83.8%

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

      8. fma-def 83.8%

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

   3. Simplified 83.8%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. fma-udef 83.7%

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

      4. fma-udef 83.7%

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

      5. *-commutative 83.7%

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

      6. associate-+r+ 83.7%

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

      7. +-commutative 83.7%

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

      8. distribute-rgt-out 83.7%

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

      9. fma-def 83.7%

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

   5. Applied egg-rr 83.7%

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

   6. Taylor expanded in k around inf 81.6%

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

      1. unpow2 81.6%

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

      2. times-frac 96.5%

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

      3. associate-*r* 96.5%

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

      4. neg-mul-1 96.5%

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

      5. exp-prod 96.3%

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

      6. log-rec 96.3%

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

   8. Simplified 96.3%

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

   9. Taylor expanded in a around 0 81.6%

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

       1. associate-/l* 81.6%

          \[\leadsto \color{blue}{\frac{a}{\frac{{k}^{2}}{{k}^{m}}}} \]

       2. *-rgt-identity 81.6%

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

       3. unpow2 81.6%

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

       4. associate-*r/ 91.5%

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

       5. times-frac 96.4%

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

       6. associate-/l* 96.5%

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

       7. *-lft-identity 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{k}^{m}}{k}\\ \end{array} \]

Alternative 4: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-6} \lor \neg \left(m \leq 3.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -6e-6) (not (<= m 3.5e-36)))
   (* 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 <= -6e-6) || !(m <= 3.5e-36)) {
		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 <= (-6d-6)) .or. (.not. (m <= 3.5d-36))) 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 <= -6e-6) || !(m <= 3.5e-36)) {
		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 <= -6e-6) or not (m <= 3.5e-36):
		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 <= -6e-6) || !(m <= 3.5e-36))
		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 <= -6e-6) || ~((m <= 3.5e-36)))
		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, -6e-6], N[Not[LessEqual[m, 3.5e-36]], $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 < -6.0000000000000002e-6 or 3.5e-36 < m

   1. Initial program 88.1%

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

      1. associate-/l* 88.1%

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

      2. sqr-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. sqr-neg 88.1%

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

      5. fma-def 88.1%

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

      6. +-commutative 88.1%

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

      7. *-commutative 88.1%

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

      8. fma-def 88.1%

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

   3. Simplified 88.1%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -6.0000000000000002e-6 < m < 3.5e-36

   1. Initial program 93.4%

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

   2. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-6} \lor \neg \left(m \leq 3.5 \cdot 10^{-36}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \end{array} \]

Alternative 5: 60.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;m \leq -6 \cdot 10^{-81}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (+ 1.0 (* k 10.0)))))
   (if (<= m -6e-81)
     (/ a (* k k))
     (if (<= m 1.3e-124)
       t_0
       (if (<= m 1.06e-43) (/ (/ a k) k) (if (<= m 1.0) t_0 (* a (* k k))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -6e-81) {
		tmp = a / (k * k);
	} else if (m <= 1.3e-124) {
		tmp = t_0;
	} else if (m <= 1.06e-43) {
		tmp = (a / k) / k;
	} else if (m <= 1.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a / (1.0d0 + (k * 10.0d0))
    if (m <= (-6d-81)) then
        tmp = a / (k * k)
    else if (m <= 1.3d-124) then
        tmp = t_0
    else if (m <= 1.06d-43) then
        tmp = (a / k) / k
    else if (m <= 1.0d0) then
        tmp = t_0
    else
        tmp = a * (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -6e-81) {
		tmp = a / (k * k);
	} else if (m <= 1.3e-124) {
		tmp = t_0;
	} else if (m <= 1.06e-43) {
		tmp = (a / k) / k;
	} else if (m <= 1.0) {
		tmp = t_0;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (1.0 + (k * 10.0))
	tmp = 0
	if m <= -6e-81:
		tmp = a / (k * k)
	elif m <= 1.3e-124:
		tmp = t_0
	elif m <= 1.06e-43:
		tmp = (a / k) / k
	elif m <= 1.0:
		tmp = t_0
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(1.0 + Float64(k * 10.0)))
	tmp = 0.0
	if (m <= -6e-81)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.3e-124)
		tmp = t_0;
	elseif (m <= 1.06e-43)
		tmp = Float64(Float64(a / k) / k);
	elseif (m <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (1.0 + (k * 10.0));
	tmp = 0.0;
	if (m <= -6e-81)
		tmp = a / (k * k);
	elseif (m <= 1.3e-124)
		tmp = t_0;
	elseif (m <= 1.06e-43)
		tmp = (a / k) / k;
	elseif (m <= 1.0)
		tmp = t_0;
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -6e-81], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3e-124], t$95$0, If[LessEqual[m, 1.06e-43], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 1.0], t$95$0, N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -5.9999999999999998e-81

   1. Initial program 98.1%

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

   2. Taylor expanded in m around 0 45.0%

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

   3. Taylor expanded in k around inf 65.6%

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

      1. unpow2 65.6%

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

   5. Simplified 65.6%

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

   if -5.9999999999999998e-81 < m < 1.3e-124 or 1.05999999999999994e-43 < m < 1

   1. Initial program 97.1%

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

   2. Taylor expanded in k around 0 69.1%

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

   3. Taylor expanded in m around 0 68.3%

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

   if 1.3e-124 < m < 1.05999999999999994e-43

   1. Initial program 84.2%

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

   2. Taylor expanded in m around 0 84.2%

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

   3. Taylor expanded in k around inf 60.4%

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

      1. unpow2 60.4%

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

      2. distribute-rgt-in 60.4%

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

      3. +-commutative 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in k around inf 60.1%

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

      1. unpow2 60.1%

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

      2. associate-/r* 75.6%

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

   8. Simplified 75.6%

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

   if 1 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

       \[\leadsto \color{blue}{a \cdot \left(k \cdot k\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6 \cdot 10^{-81}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 1.06 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;m \leq 1:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 6: 72.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.4e-9)
   (/ a (* k k))
   (if (<= m 0.85) (/ a (+ (+ 1.0 (* k 10.0)) (* k k))) (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.4e-9) {
		tmp = a / (k * k);
	} else if (m <= 0.85) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} 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 (m <= (-6.4d-9)) then
        tmp = a / (k * k)
    else if (m <= 0.85d0) then
        tmp = a / ((1.0d0 + (k * 10.0d0)) + (k * k))
    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 (m <= -6.4e-9) {
		tmp = a / (k * k);
	} else if (m <= 0.85) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.4e-9:
		tmp = a / (k * k)
	elif m <= 0.85:
		tmp = a / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.4e-9)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.85)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.4e-9)
		tmp = a / (k * k);
	elseif (m <= 0.85)
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.4e-9], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.85], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.40000000000000023e-9

   1. Initial program 98.9%

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

   2. Taylor expanded in m around 0 39.2%

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

   3. Taylor expanded in k around inf 65.7%

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

      1. unpow2 65.7%

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

   5. Simplified 65.7%

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

   if -6.40000000000000023e-9 < m < 0.849999999999999978

   1. Initial program 94.7%

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

   2. Taylor expanded in m around 0 94.1%

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

   if 0.849999999999999978 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.85:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 7: 76.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot \left(k + 10\right)\right) + -1}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.4e-9)
   (/ a (+ (+ 1.0 (* k (+ k 10.0))) -1.0))
   (if (<= m 0.95) (/ a (+ (+ 1.0 (* k 10.0)) (* k k))) (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.4e-9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) + -1.0);
	} else if (m <= 0.95) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} 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 (m <= (-6.4d-9)) then
        tmp = a / ((1.0d0 + (k * (k + 10.0d0))) + (-1.0d0))
    else if (m <= 0.95d0) then
        tmp = a / ((1.0d0 + (k * 10.0d0)) + (k * k))
    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 (m <= -6.4e-9) {
		tmp = a / ((1.0 + (k * (k + 10.0))) + -1.0);
	} else if (m <= 0.95) {
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.4e-9:
		tmp = a / ((1.0 + (k * (k + 10.0))) + -1.0)
	elif m <= 0.95:
		tmp = a / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.4e-9)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * Float64(k + 10.0))) + -1.0));
	elseif (m <= 0.95)
		tmp = Float64(a / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.4e-9)
		tmp = a / ((1.0 + (k * (k + 10.0))) + -1.0);
	elseif (m <= 0.95)
		tmp = a / ((1.0 + (k * 10.0)) + (k * k));
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.4e-9], N[(a / N[(N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.40000000000000023e-9

   1. Initial program 98.9%

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

   2. Taylor expanded in m around 0 39.2%

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

   3. Taylor expanded in k around inf 47.6%

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

      1. unpow2 47.6%

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

      2. distribute-rgt-in 47.6%

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

      3. +-commutative 47.6%

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

   5. Simplified 47.6%

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

      1. expm1-log1p-u 47.6%

         \[\leadsto \frac{a}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)\right)}} \]

      2. expm1-udef 78.6%

         \[\leadsto \frac{a}{\color{blue}{e^{\mathsf{log1p}\left(k \cdot \left(k + 10\right)\right)} - 1}} \]

      3. log1p-udef 78.6%

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

      4. add-exp-log 78.6%

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

   7. Applied egg-rr 78.6%

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

   if -6.40000000000000023e-9 < m < 0.94999999999999996

   1. Initial program 94.7%

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

   2. Taylor expanded in m around 0 94.1%

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

   if 0.94999999999999996 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot \left(k + 10\right)\right) + -1}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 8: 72.1% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.4e-9)
   (/ a (* k k))
   (if (<= m 1.05) (/ a (+ 1.0 (* k (+ k 10.0)))) (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.4e-9) {
		tmp = a / (k * k);
	} else if (m <= 1.05) {
		tmp = a / (1.0 + (k * (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 (m <= (-6.4d-9)) then
        tmp = a / (k * k)
    else if (m <= 1.05d0) then
        tmp = a / (1.0d0 + (k * (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 (m <= -6.4e-9) {
		tmp = a / (k * k);
	} else if (m <= 1.05) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.4e-9:
		tmp = a / (k * k)
	elif m <= 1.05:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.4e-9)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.05)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.4e-9)
		tmp = a / (k * k);
	elseif (m <= 1.05)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.4e-9], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.40000000000000023e-9

   1. Initial program 98.9%

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

   2. Taylor expanded in m around 0 39.2%

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

   3. Taylor expanded in k around inf 65.7%

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

      1. unpow2 65.7%

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

   5. Simplified 65.7%

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

   if -6.40000000000000023e-9 < m < 1.05000000000000004

   1. Initial program 94.7%

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

      1. associate-/l* 94.7%

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

      2. sqr-neg 94.7%

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

      3. +-commutative 94.7%

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

      4. sqr-neg 94.7%

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

      5. fma-def 94.7%

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

      6. +-commutative 94.7%

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

      7. *-commutative 94.7%

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

      8. fma-def 94.7%

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

   3. Simplified 94.7%

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

      1. clear-num 94.5%

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

      2. inv-pow 94.5%

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

      3. fma-udef 94.5%

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

      4. fma-udef 94.5%

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

      5. *-commutative 94.5%

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

      6. associate-+r+ 94.5%

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

      7. +-commutative 94.5%

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

      8. distribute-rgt-out 94.5%

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

      9. fma-def 94.5%

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

   5. Applied egg-rr 94.5%

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

   6. Taylor expanded in m around 0 94.1%

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

   if 1.05000000000000004 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 9: 71.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.49)
   (/ a (* k k))
   (if (<= m 1.05) (/ a (+ 1.0 (* k k))) (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.49) {
		tmp = a / (k * k);
	} else if (m <= 1.05) {
		tmp = a / (1.0 + (k * k));
	} 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 (m <= (-0.49d0)) then
        tmp = a / (k * k)
    else if (m <= 1.05d0) then
        tmp = a / (1.0d0 + (k * k))
    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 (m <= -0.49) {
		tmp = a / (k * k);
	} else if (m <= 1.05) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.49:
		tmp = a / (k * k)
	elif m <= 1.05:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.49)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.05)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.49)
		tmp = a / (k * k);
	elseif (m <= 1.05)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.49], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.48999999999999999

   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 39.5%

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

   3. Taylor expanded in k around inf 66.6%

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

      1. unpow2 66.6%

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

   5. Simplified 66.6%

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

   if -0.48999999999999999 < m < 1.05000000000000004

   1. Initial program 93.8%

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

   2. Taylor expanded in m around 0 92.6%

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

   3. Taylor expanded in k around 0 90.1%

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

   if 1.05000000000000004 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.49:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.05:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 10: 44.4% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.45:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2e-83) (/ 0.1 (/ k a)) (if (<= m 0.45) a (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2e-83) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.45) {
		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 (m <= (-2d-83)) then
        tmp = 0.1d0 / (k / a)
    else if (m <= 0.45d0) 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 (m <= -2e-83) {
		tmp = 0.1 / (k / a);
	} else if (m <= 0.45) {
		tmp = a;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2e-83:
		tmp = 0.1 / (k / a)
	elif m <= 0.45:
		tmp = a
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2e-83)
		tmp = Float64(0.1 / Float64(k / a));
	elseif (m <= 0.45)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2e-83)
		tmp = 0.1 / (k / a);
	elseif (m <= 0.45)
		tmp = a;
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2e-83], N[(0.1 / N[(k / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.45], a, N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.0000000000000001e-83

   1. Initial program 98.1%

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

   2. Taylor expanded in k around 0 93.5%

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

   3. Taylor expanded in k around inf 77.8%

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

      1. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in m around 0 28.7%

      \[\leadsto \color{blue}{0.1 \cdot \frac{a}{k}} \]
   7. Step-by-step derivation

      1. associate-*r/ 28.7%

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

      2. associate-/l* 28.7%

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

   8. Simplified 28.7%

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

   if -2.0000000000000001e-83 < m < 0.450000000000000011

   1. Initial program 95.0%

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

   2. Taylor expanded in m around 0 94.3%

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

   3. Taylor expanded in k around 0 52.6%

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

   if 0.450000000000000011 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{-83}:\\ \;\;\;\;\frac{0.1}{\frac{k}{a}}\\ \mathbf{elif}\;m \leq 0.45:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 11: 57.5% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-84}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.74:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8e-84) (/ a (* k k)) (if (<= m 0.74) a (* a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8e-84) {
		tmp = a / (k * k);
	} else if (m <= 0.74) {
		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 (m <= (-8d-84)) then
        tmp = a / (k * k)
    else if (m <= 0.74d0) 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 (m <= -8e-84) {
		tmp = a / (k * k);
	} else if (m <= 0.74) {
		tmp = a;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -8e-84:
		tmp = a / (k * k)
	elif m <= 0.74:
		tmp = a
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8e-84)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.74)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -8e-84)
		tmp = a / (k * k);
	elseif (m <= 0.74)
		tmp = a;
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8e-84], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.74], a, N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.0000000000000003e-84

   1. Initial program 98.1%

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

   2. Taylor expanded in m around 0 46.1%

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

   3. Taylor expanded in k around inf 66.3%

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

      1. unpow2 66.3%

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

   5. Simplified 66.3%

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

   if -8.0000000000000003e-84 < m < 0.73999999999999999

   1. Initial program 95.0%

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

   2. Taylor expanded in m around 0 94.3%

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

   3. Taylor expanded in k around 0 52.6%

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

   if 0.73999999999999999 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -8 \cdot 10^{-84}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.74:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 12: 39.3% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.25:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.25) a (* a (* k k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.25) {
		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 (m <= 0.25d0) 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 (m <= 0.25) {
		tmp = a;
	} else {
		tmp = a * (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.25:
		tmp = a
	else:
		tmp = a * (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.25)
		tmp = a;
	else
		tmp = Float64(a * Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.25)
		tmp = a;
	else
		tmp = a * (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.25], a, N[(a * N[(k * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.25

   1. Initial program 96.8%

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

   2. Taylor expanded in m around 0 67.1%

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

   3. Taylor expanded in k around 0 26.8%

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

   if 0.25 < m

   1. Initial program 74.0%

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

   2. Taylor expanded in m around 0 3.0%

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

   3. Taylor expanded in k around inf 2.1%

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

      1. unpow2 2.1%

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

   5. Simplified 2.1%

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

      1. expm1-log1p-u 1.5%

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

      2. expm1-udef 6.4%

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

   7. Applied egg-rr 65.6%

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

      1. expm1-def 48.0%

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

      2. expm1-log1p 57.5%

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

      3. *-commutative 57.5%

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

   9. Simplified 57.5%

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

   10. Taylor expanded in k around 0 68.8%

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

       1. unpow2 68.8%

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

   12. Simplified 68.8%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.25:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot k\right)\\ \end{array} \]

Alternative 13: 20.0% 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. Taylor expanded in m around 0 47.8%

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

3. Taylor expanded in k around 0 20.0%

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

4. Final simplification 20.0%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023293 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))