Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 99.9%

Time: 14.9s

Alternatives: 16

Speedup: 1.0×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, k \cdot \mathsf{hypot}\left(1, \sqrt{\frac{10}{k}}\right)\right)\\ \mathbf{if}\;k \leq 10^{-99}:\\ \;\;\;\;\frac{a}{\frac{1}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a \cdot {k}^{m}}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot 1.0 (* k (hypot 1.0 (sqrt (/ 10.0 k)))))))
   (if (<= k 1e-99) (/ a (/ 1.0 (pow k m))) (/ (/ (* a (pow k m)) t_0) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(1.0, (k * hypot(1.0, sqrt((10.0 / k)))));
	double tmp;
	if (k <= 1e-99) {
		tmp = a / (1.0 / pow(k, m));
	} else {
		tmp = ((a * pow(k, m)) / t_0) / t_0;
	}
	return tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.hypot(1.0, (k * Math.hypot(1.0, Math.sqrt((10.0 / k)))));
	double tmp;
	if (k <= 1e-99) {
		tmp = a / (1.0 / Math.pow(k, m));
	} else {
		tmp = ((a * Math.pow(k, m)) / t_0) / t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.hypot(1.0, (k * math.hypot(1.0, math.sqrt((10.0 / k)))))
	tmp = 0
	if k <= 1e-99:
		tmp = a / (1.0 / math.pow(k, m))
	else:
		tmp = ((a * math.pow(k, m)) / t_0) / t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = hypot(1.0, Float64(k * hypot(1.0, sqrt(Float64(10.0 / k)))))
	tmp = 0.0
	if (k <= 1e-99)
		tmp = Float64(a / Float64(1.0 / (k ^ m)));
	else
		tmp = Float64(Float64(Float64(a * (k ^ m)) / t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = hypot(1.0, (k * hypot(1.0, sqrt((10.0 / k)))));
	tmp = 0.0;
	if (k <= 1e-99)
		tmp = a / (1.0 / (k ^ m));
	else
		tmp = ((a * (k ^ m)) / t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[1.0 ^ 2 + N[(k * N[Sqrt[1.0 ^ 2 + N[Sqrt[N[(10.0 / k), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, 1e-99], N[(a / N[(1.0 / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1e-99

   1. Initial program 96.1%

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

      1. associate-/l* 96.1%

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

      2. remove-double-neg 96.1%

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

      3. distribute-frac-neg2 96.1%

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

      4. distribute-neg-frac2 96.1%

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

      5. remove-double-neg 96.1%

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

      6. sqr-neg 96.1%

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

      7. associate-+l+ 96.1%

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

      8. sqr-neg 96.1%

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

      9. distribute-rgt-out 96.1%

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

   3. Simplified 96.1%

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

      1. clear-num 96.1%

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

      2. un-div-inv 96.1%

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

      3. +-commutative 96.1%

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

      4. fma-define 96.1%

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

      5. +-commutative 96.1%

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

   6. Applied egg-rr 96.1%

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

   7. Taylor expanded in k around 0 100.0%

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

   if 1e-99 < k

   1. Initial program 88.6%

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

      1. associate-/l* 88.5%

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

      2. remove-double-neg 88.5%

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

      3. distribute-frac-neg2 88.5%

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

      4. distribute-neg-frac2 88.5%

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

      5. remove-double-neg 88.5%

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

      6. sqr-neg 88.5%

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

      7. associate-+l+ 88.5%

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

      8. sqr-neg 88.5%

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

      9. distribute-rgt-out 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in k around inf 88.5%

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

      1. associate-*r/ 88.5%

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

      2. metadata-eval 88.5%

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

   7. Simplified 88.5%

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

      1. associate-*r/ 88.6%

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

      2. add-sqr-sqrt 88.5%

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

      3. associate-/r* 88.5%

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

   9. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{a \cdot {k}^{m}}{\mathsf{hypot}\left(1, k \cdot \mathsf{hypot}\left(1, \sqrt{\frac{10}{k}}\right)\right)}}{\mathsf{hypot}\left(1, k \cdot \mathsf{hypot}\left(1, \sqrt{\frac{10}{k}}\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)\\ \mathbf{if}\;k \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \mathsf{fma}\left(k, -10, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot 1.0 (* k (sqrt (+ (/ 10.0 k) 1.0))))))
   (if (<= k 1.1e-9)
     (* a (* (pow k m) (fma k -10.0 1.0)))
     (/ (* a (/ (pow k m) t_0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(1.0, (k * sqrt(((10.0 / k) + 1.0))));
	double tmp;
	if (k <= 1.1e-9) {
		tmp = a * (pow(k, m) * fma(k, -10.0, 1.0));
	} else {
		tmp = (a * (pow(k, m) / t_0)) / t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(1.0, Float64(k * sqrt(Float64(Float64(10.0 / k) + 1.0))))
	tmp = 0.0
	if (k <= 1.1e-9)
		tmp = Float64(a * Float64((k ^ m) * fma(k, -10.0, 1.0)));
	else
		tmp = Float64(Float64(a * Float64((k ^ m) / t_0)) / t_0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[1.0 ^ 2 + N[(k * N[Sqrt[N[(N[(10.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, 1.1e-9], N[(a * N[(N[Power[k, m], $MachinePrecision] * N[(k * -10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.0999999999999999e-9

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in k around 0 86.7%

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

      1. associate-*r* 86.7%

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

      2. *-lft-identity 86.7%

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

      3. distribute-rgt-out 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-define 100.0%

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

   7. Simplified 100.0%

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

   if 1.0999999999999999e-9 < k

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. remove-double-neg 85.2%

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

      3. distribute-frac-neg2 85.2%

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

      4. distribute-neg-frac2 85.2%

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

      5. remove-double-neg 85.2%

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

      6. sqr-neg 85.2%

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

      7. associate-+l+ 85.2%

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

      8. sqr-neg 85.2%

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

      9. distribute-rgt-out 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in k around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. metadata-eval 85.2%

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

   7. Simplified 85.2%

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

      1. *-un-lft-identity 85.2%

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

      2. add-sqr-sqrt 85.2%

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

      3. times-frac 85.1%

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

      4. add-sqr-sqrt 85.1%

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

      5. hypot-1-def 85.1%

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

      6. associate-*r* 85.1%

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

      7. sqrt-prod 85.1%

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

      8. sqrt-unprod 85.0%

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

      9. add-sqr-sqrt 85.1%

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

      10. add-sqr-sqrt 85.1%

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

      11. hypot-1-def 85.1%

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

   9. Applied egg-rr 95.2%

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

       1. associate-*l/ 95.2%

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

       2. *-lft-identity 95.2%

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

       3. hypot-undefine 95.2%

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

       4. metadata-eval 95.2%

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

       5. rem-square-sqrt 95.2%

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

       6. hypot-undefine 95.2%

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

       7. metadata-eval 95.2%

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

       8. rem-square-sqrt 95.2%

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

   11. Simplified 95.2%

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

       1. associate-*r/ 99.8%

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

   13. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-9}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \mathsf{fma}\left(k, -10, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \frac{{k}^{m}}{\mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)}}{\mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)\\ \mathbf{if}\;k \leq 10^{-9}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \mathsf{fma}\left(k, -10, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{{k}^{m}}{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot 1.0 (* k (sqrt (+ (/ 10.0 k) 1.0))))))
   (if (<= k 1e-9)
     (* a (* (pow k m) (fma k -10.0 1.0)))
     (* a (/ (/ (pow k m) t_0) t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(1.0, (k * sqrt(((10.0 / k) + 1.0))));
	double tmp;
	if (k <= 1e-9) {
		tmp = a * (pow(k, m) * fma(k, -10.0, 1.0));
	} else {
		tmp = a * ((pow(k, m) / t_0) / t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(1.0, Float64(k * sqrt(Float64(Float64(10.0 / k) + 1.0))))
	tmp = 0.0
	if (k <= 1e-9)
		tmp = Float64(a * Float64((k ^ m) * fma(k, -10.0, 1.0)));
	else
		tmp = Float64(a * Float64(Float64((k ^ m) / t_0) / t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[1.0 ^ 2 + N[(k * N[Sqrt[N[(N[(10.0 / k), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[k, 1e-9], N[(a * N[(N[Power[k, m], $MachinePrecision] * N[(k * -10.0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000006e-9

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in k around 0 86.7%

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

      1. associate-*r* 86.7%

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

      2. *-lft-identity 86.7%

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

      3. distribute-rgt-out 100.0%

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

      4. *-commutative 100.0%

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

      5. fma-define 100.0%

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

   7. Simplified 100.0%

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

   if 1.00000000000000006e-9 < k

   1. Initial program 85.2%

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

      1. associate-/l* 85.2%

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

      2. remove-double-neg 85.2%

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

      3. distribute-frac-neg2 85.2%

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

      4. distribute-neg-frac2 85.2%

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

      5. remove-double-neg 85.2%

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

      6. sqr-neg 85.2%

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

      7. associate-+l+ 85.2%

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

      8. sqr-neg 85.2%

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

      9. distribute-rgt-out 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in k around inf 85.2%

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

      1. associate-*r/ 85.2%

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

      2. metadata-eval 85.2%

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

   7. Simplified 85.2%

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

      1. *-un-lft-identity 85.2%

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

      2. add-sqr-sqrt 85.2%

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

      3. times-frac 85.1%

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

      4. add-sqr-sqrt 85.1%

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

      5. hypot-1-def 85.1%

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

      6. associate-*r* 85.1%

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

      7. sqrt-prod 85.1%

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

      8. sqrt-unprod 85.0%

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

      9. add-sqr-sqrt 85.1%

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

      10. add-sqr-sqrt 85.1%

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

      11. hypot-1-def 85.1%

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

   9. Applied egg-rr 95.2%

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

       1. associate-*l/ 95.2%

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

       2. *-lft-identity 95.2%

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

       3. hypot-undefine 95.2%

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

       4. metadata-eval 95.2%

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

       5. rem-square-sqrt 95.2%

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

       6. hypot-undefine 95.2%

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

       7. metadata-eval 95.2%

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

       8. rem-square-sqrt 95.2%

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

   11. Simplified 95.2%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-9}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \mathsf{fma}\left(k, -10, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{\frac{{k}^{m}}{\mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)}}{\mathsf{hypot}\left(1, k \cdot \sqrt{\frac{10}{k} + 1}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{\frac{\mathsf{fma}\left(k, k + 10, 1\right)}{{k}^{m}}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.75) (/ a (/ (fma k (+ k 10.0) 1.0) (pow k m))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / (fma(k, (k + 10.0), 1.0) / pow(k, m));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(a / Float64(fma(k, Float64(k + 10.0), 1.0) / (k ^ m)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.75], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision] / N[Power[k, m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

      1. clear-num 96.8%

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

      2. un-div-inv 96.8%

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

      3. +-commutative 96.8%

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

      4. fma-define 96.9%

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

      5. +-commutative 96.9%

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

   6. Applied egg-rr 96.9%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in k around 0 100.0%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{t\_0}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 0.75) (/ t_0 (+ (* k k) (+ (* k 10.0) 1.0))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 0.75) {
		tmp = t_0 / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (m <= 0.75d0) then
        tmp = t_0 / ((k * k) + ((k * 10.0d0) + 1.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (m <= 0.75) {
		tmp = t_0 / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 0.75:
		tmp = t_0 / ((k * k) + ((k * 10.0) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(t_0 / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (m <= 0.75)
		tmp = t_0 / ((k * k) + ((k * 10.0) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, 0.75], N[(t$95$0 / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.75) (* a (/ (pow k m) (+ (* k (+ k 10.0)) 1.0))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a * (pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.75d0) then
        tmp = a * ((k ** m) / ((k * (k + 10.0d0)) + 1.0d0))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a * (Math.pow(k, m) / ((k * (k + 10.0)) + 1.0));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.75:
		tmp = a * (math.pow(k, m) / ((k * (k + 10.0)) + 1.0))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(a * Float64((k ^ m) / Float64(Float64(k * Float64(k + 10.0)) + 1.0)));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.75)
		tmp = a * ((k ^ m) / ((k * (k + 10.0)) + 1.0));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.75], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in k around 0 100.0%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00055 \lor \neg \left(m \leq 0.0045\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.00055) (not (<= m 0.0045)))
   (* a (pow k m))
   (/ a (+ (* k k) (+ (* k 10.0) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00055) || !(m <= 0.0045)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	}
	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 <= (-0.00055d0)) .or. (.not. (m <= 0.0045d0))) then
        tmp = a * (k ** m)
    else
        tmp = a / ((k * k) + ((k * 10.0d0) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.00055) || !(m <= 0.0045)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.00055) or not (m <= 0.0045):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / ((k * k) + ((k * 10.0) + 1.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.00055) || !(m <= 0.0045))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.00055) || ~((m <= 0.0045)))
		tmp = a * (k ^ m);
	else
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.00055], N[Not[LessEqual[m, 0.0045]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.50000000000000033e-4 or 0.00449999999999999966 < m

   1. Initial program 91.1%

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

      1. associate-/l* 91.1%

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

      2. remove-double-neg 91.1%

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

      3. distribute-frac-neg2 91.1%

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

      4. distribute-neg-frac2 91.1%

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

      5. remove-double-neg 91.1%

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

      6. sqr-neg 91.1%

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

      7. associate-+l+ 91.1%

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

      8. sqr-neg 91.1%

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

      9. distribute-rgt-out 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in k around 0 100.0%

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

   if -5.50000000000000033e-4 < m < 0.00449999999999999966

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in m around 0 93.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.00055 \lor \neg \left(m \leq 0.0045\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.65:\\ \;\;\;\;\frac{\frac{0.001 \cdot \frac{a}{k} - a \cdot 0.01}{k} - a \cdot -0.1}{k}\\ \mathbf{elif}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.65)
   (/ (- (/ (- (* 0.001 (/ a k)) (* a 0.01)) k) (* a -0.1)) k)
   (if (<= m 0.75)
     (/ a (+ (* k k) (+ (* k 10.0) 1.0)))
     (* a (+ (* k (* k 99.0)) 1.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.65) {
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	} else if (m <= 0.75) {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	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 <= (-1.65d0)) then
        tmp = ((((0.001d0 * (a / k)) - (a * 0.01d0)) / k) - (a * (-0.1d0))) / k
    else if (m <= 0.75d0) then
        tmp = a / ((k * k) + ((k * 10.0d0) + 1.0d0))
    else
        tmp = a * ((k * (k * 99.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.65) {
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	} else if (m <= 0.75) {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.65:
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k
	elif m <= 0.75:
		tmp = a / ((k * k) + ((k * 10.0) + 1.0))
	else:
		tmp = a * ((k * (k * 99.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.65)
		tmp = Float64(Float64(Float64(Float64(Float64(0.001 * Float64(a / k)) - Float64(a * 0.01)) / k) - Float64(a * -0.1)) / k);
	elseif (m <= 0.75)
		tmp = Float64(a / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * 99.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.65)
		tmp = ((((0.001 * (a / k)) - (a * 0.01)) / k) - (a * -0.1)) / k;
	elseif (m <= 0.75)
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	else
		tmp = a * ((k * (k * 99.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.65], N[(N[(N[(N[(N[(0.001 * N[(a / k), $MachinePrecision]), $MachinePrecision] - N[(a * 0.01), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] - N[(a * -0.1), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 0.75], N[(a / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.6499999999999999

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 80.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\right)\right)} \]

      2. expm1-undefine 80.5%

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

      3. +-commutative 80.5%

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

      4. fma-define 80.5%

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

   9. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 80.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)\right)} \]

       2. associate-*r/ 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}}\right)\right) \]

       3. *-rgt-identity 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) \cdot 1}}\right)\right) \]

       4. times-frac 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \frac{{k}^{m}}{1}}\right)\right) \]

       5. /-rgt-identity 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \color{blue}{{k}^{m}}\right)\right) \]

   11. Simplified 80.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot {k}^{m}\right)\right)} \]

   12. Taylor expanded in m around 0 37.7%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{a}{1 + 10 \cdot k}\right)}\right) \]

   13. Taylor expanded in k around -inf 41.4%

       \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.001 \cdot \frac{a}{k} - 0.01 \cdot a}{k} + -0.1 \cdot a}{k}} \]

   if -1.6499999999999999 < m < 0.75

   1. Initial program 94.4%

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

      1. *-commutative 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in m around 0 93.0%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 27.6%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.65:\\ \;\;\;\;\frac{\frac{0.001 \cdot \frac{a}{k} - a \cdot 0.01}{k} - a \cdot -0.1}{k}\\ \mathbf{elif}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.225:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 78000000000000:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.225)
   (* (/ a k) 0.1)
   (if (<= m 78000000000000.0) (/ a (+ (* k 10.0) 1.0)) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.225) {
		tmp = (a / k) * 0.1;
	} else if (m <= 78000000000000.0) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = -10.0 * (k * 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 <= (-0.225d0)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 78000000000000.0d0) then
        tmp = a / ((k * 10.0d0) + 1.0d0)
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.225) {
		tmp = (a / k) * 0.1;
	} else if (m <= 78000000000000.0) {
		tmp = a / ((k * 10.0) + 1.0);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.225:
		tmp = (a / k) * 0.1
	elif m <= 78000000000000.0:
		tmp = a / ((k * 10.0) + 1.0)
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.225)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 78000000000000.0)
		tmp = Float64(a / Float64(Float64(k * 10.0) + 1.0));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.225)
		tmp = (a / k) * 0.1;
	elseif (m <= 78000000000000.0)
		tmp = a / ((k * 10.0) + 1.0);
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.225], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 78000000000000.0], N[(a / N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.225000000000000006

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

      1. expm1-log1p-u 80.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\right)\right)} \]

      2. expm1-undefine 80.5%

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

      3. +-commutative 80.5%

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

      4. fma-define 80.5%

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

   9. Applied egg-rr 80.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 80.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)\right)} \]

       2. associate-*r/ 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}}\right)\right) \]

       3. *-rgt-identity 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) \cdot 1}}\right)\right) \]

       4. times-frac 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \frac{{k}^{m}}{1}}\right)\right) \]

       5. /-rgt-identity 80.5%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \color{blue}{{k}^{m}}\right)\right) \]

   11. Simplified 80.5%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot {k}^{m}\right)\right)} \]

   12. Taylor expanded in m around 0 37.7%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{a}{1 + 10 \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 29.1%

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

       1. *-commutative 29.1%

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

   15. Simplified 29.1%

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

   if -0.225000000000000006 < m < 7.8e13

   1. Initial program 94.6%

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

      1. associate-/l* 94.5%

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

      2. remove-double-neg 94.5%

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

      3. distribute-frac-neg2 94.5%

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

      4. distribute-neg-frac2 94.5%

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

      5. remove-double-neg 94.5%

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

      6. sqr-neg 94.5%

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

      7. associate-+l+ 94.5%

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

      8. sqr-neg 94.5%

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

      9. distribute-rgt-out 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in m around 0 89.5%

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

   6. Taylor expanded in k around 0 59.7%

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

      1. *-commutative 64.3%

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

   8. Simplified 59.7%

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

   if 7.8e13 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 5.4%

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

      1. *-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in k around inf 15.7%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.225:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 78000000000000:\\ \;\;\;\;\frac{a}{k \cdot 10 + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.75)
   (/ a (+ (* k k) (+ (* k 10.0) 1.0)))
   (* a (+ (* k (* k 99.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	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 <= 0.75d0) then
        tmp = a / ((k * k) + ((k * 10.0d0) + 1.0d0))
    else
        tmp = a * ((k * (k * 99.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.75:
		tmp = a / ((k * k) + ((k * 10.0) + 1.0))
	else:
		tmp = a * ((k * (k * 99.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(a / Float64(Float64(k * k) + Float64(Float64(k * 10.0) + 1.0)));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * 99.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.75)
		tmp = a / ((k * k) + ((k * 10.0) + 1.0));
	else
		tmp = a * ((k * (k * 99.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.75], N[(a / N[(N[(k * k), $MachinePrecision] + N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

      1. *-commutative 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in m around 0 66.0%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 27.6%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + \left(k \cdot 10 + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 80000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.85e-12)
   (* (/ a k) 0.1)
   (if (<= m 80000000000000.0) a (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.85e-12) {
		tmp = (a / k) * 0.1;
	} else if (m <= 80000000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * 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 <= (-2.85d-12)) then
        tmp = (a / k) * 0.1d0
    else if (m <= 80000000000000.0d0) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.85e-12) {
		tmp = (a / k) * 0.1;
	} else if (m <= 80000000000000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.85e-12:
		tmp = (a / k) * 0.1
	elif m <= 80000000000000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.85e-12)
		tmp = Float64(Float64(a / k) * 0.1);
	elseif (m <= 80000000000000.0)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -2.85e-12)
		tmp = (a / k) * 0.1;
	elseif (m <= 80000000000000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.85e-12], N[(N[(a / k), $MachinePrecision] * 0.1), $MachinePrecision], If[LessEqual[m, 80000000000000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.8500000000000002e-12

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-frac-neg2 100.0%

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

      4. distribute-neg-frac2 100.0%

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

      5. remove-double-neg 100.0%

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

      6. sqr-neg 100.0%

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

      7. associate-+l+ 100.0%

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

      8. sqr-neg 100.0%

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

      9. distribute-rgt-out 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in k around 0 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. expm1-log1p-u 80.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{1 + k \cdot 10}\right)\right)} \]

      2. expm1-undefine 80.1%

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

      3. +-commutative 80.1%

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

      4. fma-define 80.1%

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

   9. Applied egg-rr 80.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-define 80.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}\right)\right)} \]

       2. associate-*r/ 80.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(k, 10, 1\right)}}\right)\right) \]

       3. *-rgt-identity 80.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot {k}^{m}}{\color{blue}{\mathsf{fma}\left(k, 10, 1\right) \cdot 1}}\right)\right) \]

       4. times-frac 80.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \frac{{k}^{m}}{1}}\right)\right) \]

       5. /-rgt-identity 80.0%

          \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot \color{blue}{{k}^{m}}\right)\right) \]

   11. Simplified 80.0%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\mathsf{fma}\left(k, 10, 1\right)} \cdot {k}^{m}\right)\right)} \]

   12. Taylor expanded in m around 0 38.8%

       \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \frac{a}{1 + 10 \cdot k}\right)}\right) \]

   13. Taylor expanded in k around inf 29.4%

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

       1. *-commutative 29.4%

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

   15. Simplified 29.4%

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

   if -2.8500000000000002e-12 < m < 8e13

   1. Initial program 94.4%

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

      1. associate-/l* 94.4%

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

      2. remove-double-neg 94.4%

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

      3. distribute-frac-neg2 94.4%

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

      4. distribute-neg-frac2 94.4%

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

      5. remove-double-neg 94.4%

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

      6. sqr-neg 94.4%

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

      7. associate-+l+ 94.4%

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

      8. sqr-neg 94.4%

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

      9. distribute-rgt-out 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in m around 0 89.7%

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

   6. Taylor expanded in k around 0 51.3%

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

   if 8e13 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 5.4%

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

      1. *-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in k around inf 15.7%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.85 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{k} \cdot 0.1\\ \mathbf{elif}\;m \leq 80000000000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.75)
   (/ a (+ (* k (+ k 10.0)) 1.0))
   (* a (+ (* k (* k 99.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	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 <= 0.75d0) then
        tmp = a / ((k * (k + 10.0d0)) + 1.0d0)
    else
        tmp = a * ((k * (k * 99.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * (k + 10.0)) + 1.0);
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.75:
		tmp = a / ((k * (k + 10.0)) + 1.0)
	else:
		tmp = a * ((k * (k * 99.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(a / Float64(Float64(k * Float64(k + 10.0)) + 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * 99.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.75)
		tmp = a / ((k * (k + 10.0)) + 1.0);
	else
		tmp = a * ((k * (k * 99.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.75], N[(a / N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in m around 0 66.0%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 27.6%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.3% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.75) (/ a (+ (* k k) 1.0)) (* a (+ (* k (* k 99.0)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * k) + 1.0);
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	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 <= 0.75d0) then
        tmp = a / ((k * k) + 1.0d0)
    else
        tmp = a * ((k * (k * 99.0d0)) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.75) {
		tmp = a / ((k * k) + 1.0);
	} else {
		tmp = a * ((k * (k * 99.0)) + 1.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.75:
		tmp = a / ((k * k) + 1.0)
	else:
		tmp = a * ((k * (k * 99.0)) + 1.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.75)
		tmp = Float64(a / Float64(Float64(k * k) + 1.0));
	else
		tmp = Float64(a * Float64(Float64(k * Float64(k * 99.0)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.75)
		tmp = a / ((k * k) + 1.0);
	else
		tmp = a * ((k * (k * 99.0)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.75], N[(a / N[(N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(k * N[(k * 99.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.75

   1. Initial program 96.8%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in m around 0 66.0%

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

   6. Taylor expanded in k around inf 63.8%

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

   if 0.75 < m

   1. Initial program 82.7%

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

      1. associate-/l* 82.7%

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

      2. remove-double-neg 82.7%

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

      3. distribute-frac-neg2 82.7%

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

      4. distribute-neg-frac2 82.7%

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

      5. remove-double-neg 82.7%

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

      6. sqr-neg 82.7%

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

      7. associate-+l+ 82.7%

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

      8. sqr-neg 82.7%

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

      9. distribute-rgt-out 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 27.6%

      \[\leadsto a \cdot \color{blue}{\left(1 + k \cdot \left(99 \cdot k - 10\right)\right)} \]

   7. Taylor expanded in k around inf 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(99 \cdot k\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 27.6%

         \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]

   9. Simplified 27.6%

      \[\leadsto a \cdot \left(1 + k \cdot \color{blue}{\left(k \cdot 99\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(k \cdot \left(k \cdot 99\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.3% accurate, 9.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.15) (/ a (+ (* k k) 1.0)) (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.15) {
		tmp = a / ((k * k) + 1.0);
	} else {
		tmp = -10.0 * (k * 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 <= 1.15d0) then
        tmp = a / ((k * k) + 1.0d0)
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.15) {
		tmp = a / ((k * k) + 1.0);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.15:
		tmp = a / ((k * k) + 1.0)
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.15)
		tmp = Float64(a / Float64(Float64(k * k) + 1.0));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.15)
		tmp = a / ((k * k) + 1.0);
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.15], N[(a / N[(N[(k * k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.1499999999999999

   1. Initial program 96.9%

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

      1. associate-/l* 96.8%

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

      2. remove-double-neg 96.8%

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

      3. distribute-frac-neg2 96.8%

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

      4. distribute-neg-frac2 96.8%

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

      5. remove-double-neg 96.8%

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

      6. sqr-neg 96.8%

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

      7. associate-+l+ 96.8%

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

      8. sqr-neg 96.8%

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

      9. distribute-rgt-out 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in m around 0 65.6%

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

   6. Taylor expanded in k around inf 63.5%

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

   if 1.1499999999999999 < m

   1. Initial program 82.5%

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

      1. associate-/l* 82.5%

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

      2. remove-double-neg 82.5%

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

      3. distribute-frac-neg2 82.5%

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

      4. distribute-neg-frac2 82.5%

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

      5. remove-double-neg 82.5%

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

      6. sqr-neg 82.5%

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

      7. associate-+l+ 82.5%

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

      8. sqr-neg 82.5%

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

      9. distribute-rgt-out 82.5%

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

   3. Simplified 82.5%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 5.3%

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

      1. *-commutative 5.3%

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

   8. Simplified 5.3%

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

   9. Taylor expanded in k around inf 15.3%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.15:\\ \;\;\;\;\frac{a}{k \cdot k + 1}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.06 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 1.06e+14) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.06e+14) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * 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 <= 1.06d+14) then
        tmp = a
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.06e+14) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.06e+14:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.06e+14)
		tmp = a;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.06e+14)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.06e+14], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.06e14

   1. Initial program 96.9%

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

      1. associate-/l* 96.9%

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

      2. remove-double-neg 96.9%

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

      3. distribute-frac-neg2 96.9%

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

      4. distribute-neg-frac2 96.9%

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

      5. remove-double-neg 96.9%

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

      6. sqr-neg 96.9%

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

      7. associate-+l+ 96.9%

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

      8. sqr-neg 96.9%

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

      9. distribute-rgt-out 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in m around 0 64.6%

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

   6. Taylor expanded in k around 0 30.4%

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

   if 1.06e14 < m

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. remove-double-neg 81.8%

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

      3. distribute-frac-neg2 81.8%

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

      4. distribute-neg-frac2 81.8%

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

      5. remove-double-neg 81.8%

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

      6. sqr-neg 81.8%

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

      7. associate-+l+ 81.8%

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

      8. sqr-neg 81.8%

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

      9. distribute-rgt-out 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in m around 0 2.9%

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

   6. Taylor expanded in k around 0 5.4%

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

      1. *-commutative 5.4%

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

   8. Simplified 5.4%

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

   9. Taylor expanded in k around inf 15.7%

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

4. Final simplification 26.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.06 \cdot 10^{+14}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 20.4% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 92.4%

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

   1. associate-/l* 92.4%

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

   2. remove-double-neg 92.4%

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

   3. distribute-frac-neg2 92.4%

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

   4. distribute-neg-frac2 92.4%

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

   5. remove-double-neg 92.4%

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

   6. sqr-neg 92.4%

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

   7. associate-+l+ 92.4%

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

   8. sqr-neg 92.4%

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

   9. distribute-rgt-out 92.4%

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

3. Simplified 92.4%

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

5. Taylor expanded in m around 0 46.0%

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

6. Taylor expanded in k around 0 22.3%

   \[\leadsto \color{blue}{a} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))