Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.7%

Time: 11.3s

Alternatives: 12

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 1.934500610124868e-295
  2. k = -9.575414810816206e-61
  3. m = 1.0886641513040529e+251

• 22.4% of points produce a very large (infinite) output. You may want to add a precondition.

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: 90.4% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \frac{1}{t\_0}\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \left(1 - k \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_1 + k \cdot \left(10 \cdot t\_1 + \frac{k}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (/ 1.0 t_0)))
   (if (<= k 2.4e-11)
     (* a (* (pow k m) (- 1.0 (* k 10.0))))
     (/ 1.0 (+ t_1 (* k (+ (* 10.0 t_1) (/ k t_0))))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (k <= 2.4e-11) {
		tmp = a * (pow(k, m) * (1.0 - (k * 10.0)));
	} else {
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a * (k ** m)
    t_1 = 1.0d0 / t_0
    if (k <= 2.4d-11) then
        tmp = a * ((k ** m) * (1.0d0 - (k * 10.0d0)))
    else
        tmp = 1.0d0 / (t_1 + (k * ((10.0d0 * t_1) + (k / t_0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (k <= 2.4e-11) {
		tmp = a * (Math.pow(k, m) * (1.0 - (k * 10.0)));
	} else {
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	t_1 = 1.0 / t_0
	tmp = 0
	if k <= 2.4e-11:
		tmp = a * (math.pow(k, m) * (1.0 - (k * 10.0)))
	else:
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (k <= 2.4e-11)
		tmp = Float64(a * Float64((k ^ m) * Float64(1.0 - Float64(k * 10.0))));
	else
		tmp = Float64(1.0 / Float64(t_1 + Float64(k * Float64(Float64(10.0 * t_1) + Float64(k / t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (k <= 2.4e-11)
		tmp = a * ((k ^ m) * (1.0 - (k * 10.0)));
	else
		tmp = 1.0 / (t_1 + (k * ((10.0 * t_1) + (k / t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[k, 2.4e-11], N[(a * N[(N[Power[k, m], $MachinePrecision] * N[(1.0 - N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$1 + N[(k * N[(N[(10.0 * t$95$1), $MachinePrecision] + N[(k / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-11

   1. Initial program 95.1%

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

      1. associate-/l* 95.1%

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

      2. remove-double-neg 95.1%

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

      3. distribute-frac-neg2 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. remove-double-neg 95.1%

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

      6. sqr-neg 95.1%

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

      7. associate-+l+ 95.1%

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

      8. sqr-neg 95.1%

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

      9. distribute-rgt-out 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in k around 0 88.3%

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

      1. associate-*r* 88.3%

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

      2. *-lft-identity 88.3%

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

      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%

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

      1. fma-undefine 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. cancel-sign-sub-inv 100.0%

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

      6. *-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2.4000000000000001e-11 < k

   1. Initial program 74.7%

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

      1. associate-/l* 74.7%

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

      2. remove-double-neg 74.7%

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

      3. distribute-frac-neg2 74.7%

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

      4. distribute-neg-frac2 74.7%

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

      5. remove-double-neg 74.7%

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

      6. sqr-neg 74.7%

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

      7. associate-+l+ 74.7%

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

      8. sqr-neg 74.7%

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

      9. distribute-rgt-out 74.7%

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

   3. Simplified 74.7%

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

      1. distribute-lft-in 74.7%

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

      2. associate-+l+ 74.7%

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

      3. associate-*r/ 74.7%

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

      4. clear-num 74.7%

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

      5. associate-+l+ 74.7%

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

      6. distribute-lft-in 74.7%

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

      7. +-commutative 74.7%

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

      8. fma-define 74.7%

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

      9. +-commutative 74.7%

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

      10. *-commutative 74.7%

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

   6. Applied egg-rr 74.7%

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

   7. Taylor expanded in k around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;a \cdot \left({k}^{m} \cdot \left(1 - k \cdot 10\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{a \cdot {k}^{m}} + k \cdot \left(10 \cdot \frac{1}{a \cdot {k}^{m}} + \frac{k}{a \cdot {k}^{m}}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (pow (/ (sqrt (* a (pow k m))) (hypot 1.0 k)) 2.0))

C

double code(double a, double k, double m) {
	return pow((sqrt((a * pow(k, m))) / hypot(1.0, k)), 2.0);
}

Java

public static double code(double a, double k, double m) {
	return Math.pow((Math.sqrt((a * Math.pow(k, m))) / Math.hypot(1.0, k)), 2.0);
}

Python

def code(a, k, m):
	return math.pow((math.sqrt((a * math.pow(k, m))) / math.hypot(1.0, k)), 2.0)

Julia

function code(a, k, m)
	return Float64(sqrt(Float64(a * (k ^ m))) / hypot(1.0, k)) ^ 2.0
end

MATLAB

function tmp = code(a, k, m)
	tmp = (sqrt((a * (k ^ m))) / hypot(1.0, k)) ^ 2.0;
end

Wolfram

code[a_, k_, m_] := N[Power[N[(N[Sqrt[N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + k ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 87.6%

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

   1. associate-/l* 87.6%

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

   2. remove-double-neg 87.6%

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

   3. distribute-frac-neg2 87.6%

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

   4. distribute-neg-frac2 87.6%

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

   5. remove-double-neg 87.6%

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

   6. sqr-neg 87.6%

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

   7. associate-+l+ 87.6%

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

   8. sqr-neg 87.6%

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

   9. distribute-rgt-out 87.6%

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

3. Simplified 87.6%

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

5. Taylor expanded in k around inf 87.4%

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

   1. add-sqr-sqrt 64.0%

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

   2. pow2 64.0%

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

   3. associate-*r/ 64.0%

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

   4. *-commutative 64.0%

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

   5. sqrt-div 59.7%

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

   6. *-commutative 59.7%

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

   7. hypot-1-def 64.7%

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

7. Applied egg-rr 64.7%

   \[\leadsto \color{blue}{{\left(\frac{\sqrt{a \cdot {k}^{m}}}{\mathsf{hypot}\left(1, k\right)}\right)}^{2}} \]
8. Add Preprocessing

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq 3.7:\\ \;\;\;\;\frac{t\_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m 3.7) (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k))) t_0)))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= 3.7) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} 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 <= 3.7d0) then
        tmp = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    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 <= 3.7) {
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if m <= 3.7:
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= 3.7)
		tmp = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	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 <= 3.7)
		tmp = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	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, 3.7], N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.7000000000000002

   1. Initial program 97.2%

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

   if 3.7000000000000002 < m

   1. Initial program 68.6%

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

      1. associate-/l* 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg2 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. remove-double-neg 68.6%

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

      6. sqr-neg 68.6%

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

      7. associate-+l+ 68.6%

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

      8. sqr-neg 68.6%

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

      9. distribute-rgt-out 68.6%

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

   3. Simplified 68.6%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.7:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.9) (* a (/ (pow k m) (+ 1.0 (* k (+ k 10.0))))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a * (pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2.9d0) then
        tmp = a * ((k ** m) / (1.0d0 + (k * (k + 10.0d0))))
    else
        tmp = a * (k ** m)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.9) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.9:
		tmp = a * (math.pow(k, m) / (1.0 + (k * (k + 10.0))))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.9)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.9)
		tmp = a * ((k ^ m) / (1.0 + (k * (k + 10.0))));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.9], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.89999999999999991

   1. Initial program 97.2%

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

      1. associate-/l* 97.2%

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

      2. remove-double-neg 97.2%

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

      3. distribute-frac-neg2 97.2%

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

      4. distribute-neg-frac2 97.2%

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

      5. remove-double-neg 97.2%

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

      6. sqr-neg 97.2%

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

      7. associate-+l+ 97.2%

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

      8. sqr-neg 97.2%

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

      9. distribute-rgt-out 97.2%

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

   3. Simplified 97.2%

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

   if 2.89999999999999991 < m

   1. Initial program 68.6%

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

      1. associate-/l* 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg2 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. remove-double-neg 68.6%

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

      6. sqr-neg 68.6%

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

      7. associate-+l+ 68.6%

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

      8. sqr-neg 68.6%

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

      9. distribute-rgt-out 68.6%

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

   3. Simplified 68.6%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 2.9:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.5:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.5) (* a (/ (pow k m) (+ 1.0 (* k k)))) (* a (pow k m))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.5) {
		tmp = a * (pow(k, m) / (1.0 + (k * k)));
	} 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 <= 2.5d0) then
        tmp = a * ((k ** m) / (1.0d0 + (k * k)))
    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 <= 2.5) {
		tmp = a * (Math.pow(k, m) / (1.0 + (k * k)));
	} else {
		tmp = a * Math.pow(k, m);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2.5:
		tmp = a * (math.pow(k, m) / (1.0 + (k * k)))
	else:
		tmp = a * math.pow(k, m)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.5)
		tmp = Float64(a * Float64((k ^ m) / Float64(1.0 + Float64(k * k))));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2.5)
		tmp = a * ((k ^ m) / (1.0 + (k * k)));
	else
		tmp = a * (k ^ m);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.5], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.5

   1. Initial program 97.2%

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

      1. associate-/l* 97.2%

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

      2. remove-double-neg 97.2%

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

      3. distribute-frac-neg2 97.2%

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

      4. distribute-neg-frac2 97.2%

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

      5. remove-double-neg 97.2%

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

      6. sqr-neg 97.2%

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

      7. associate-+l+ 97.2%

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

      8. sqr-neg 97.2%

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

      9. distribute-rgt-out 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in k around inf 96.9%

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

   if 2.5 < m

   1. Initial program 68.6%

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

      1. associate-/l* 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg2 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. remove-double-neg 68.6%

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

      6. sqr-neg 68.6%

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

      7. associate-+l+ 68.6%

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

      8. sqr-neg 68.6%

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

      9. distribute-rgt-out 68.6%

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

   3. Simplified 68.6%

      \[\leadsto \color{blue}{a \cdot \frac{{k}^{m}}{1 + k \cdot \left(10 + k\right)}} \]
   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 6: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.0039 \lor \neg \left(m \leq 1.35 \cdot 10^{-33}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -0.0039) (not (<= m 1.35e-33)))
   (* a (pow k m))
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.0039) || !(m <= 1.35e-33)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-0.0039d0)) .or. (.not. (m <= 1.35d-33))) then
        tmp = a * (k ** m)
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -0.0039) || !(m <= 1.35e-33)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -0.0039) or not (m <= 1.35e-33):
		tmp = a * math.pow(k, m)
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -0.0039) || !(m <= 1.35e-33))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -0.0039) || ~((m <= 1.35e-33)))
		tmp = a * (k ^ m);
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -0.0039], N[Not[LessEqual[m, 1.35e-33]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0038999999999999998 or 1.35e-33 < m

   1. Initial program 83.8%

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

      1. associate-/l* 83.8%

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

      2. remove-double-neg 83.8%

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

      3. distribute-frac-neg2 83.8%

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

      4. distribute-neg-frac2 83.8%

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

      5. remove-double-neg 83.8%

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

      6. sqr-neg 83.8%

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

      7. associate-+l+ 83.8%

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

      8. sqr-neg 83.8%

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

      9. distribute-rgt-out 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in k around 0 99.4%

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

   if -0.0038999999999999998 < m < 1.35e-33

   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* 94.6%

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

      2. remove-double-neg 94.6%

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

      3. distribute-frac-neg2 94.6%

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

      4. distribute-neg-frac2 94.6%

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

      5. remove-double-neg 94.6%

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

      6. sqr-neg 94.6%

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

      7. associate-+l+ 94.6%

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

      8. sqr-neg 94.6%

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

      9. distribute-rgt-out 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in m around 0 94.5%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0039 \lor \neg \left(m \leq 1.35 \cdot 10^{-33}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.055:\\ \;\;\;\;\frac{a}{t\_0}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.055)
     (/ a t_0)
     (if (<= m 2.2) (/ a (+ 1.0 t_0)) (+ a (* a (* k (- (* k 99.0) 10.0))))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.055) {
		tmp = a / t_0;
	} else if (m <= 2.2) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-0.055d0)) then
        tmp = a / t_0
    else if (m <= 2.2d0) then
        tmp = a / (1.0d0 + t_0)
    else
        tmp = a + (a * (k * ((k * 99.0d0) - 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.055) {
		tmp = a / t_0;
	} else if (m <= 2.2) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.055:
		tmp = a / t_0
	elif m <= 2.2:
		tmp = a / (1.0 + t_0)
	else:
		tmp = a + (a * (k * ((k * 99.0) - 10.0)))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.055)
		tmp = Float64(a / t_0);
	elseif (m <= 2.2)
		tmp = Float64(a / Float64(1.0 + t_0));
	else
		tmp = Float64(a + Float64(a * Float64(k * Float64(Float64(k * 99.0) - 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.055)
		tmp = a / t_0;
	elseif (m <= 2.2)
		tmp = a / (1.0 + t_0);
	else
		tmp = a + (a * (k * ((k * 99.0) - 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.055], N[(a / t$95$0), $MachinePrecision], If[LessEqual[m, 2.2], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(a + N[(a * N[(k * N[(N[(k * 99.0), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0550000000000000003

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 31.2%

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

   6. Taylor expanded in k around inf 31.2%

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

   7. Taylor expanded in k around inf 41.9%

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

      1. unpow2 41.9%

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

      2. associate-*l* 36.9%

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

      3. +-commutative 36.9%

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

      4. distribute-rgt-in 36.9%

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

      5. associate-*l* 36.9%

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

      6. lft-mult-inverse 36.9%

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

      7. metadata-eval 36.9%

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

      8. *-lft-identity 36.9%

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

      9. +-commutative 36.9%

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

   9. Simplified 36.9%

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

   if -0.0550000000000000003 < m < 2.2000000000000002

   1. Initial program 94.9%

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

      1. associate-/l* 94.9%

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

      2. remove-double-neg 94.9%

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

      3. distribute-frac-neg2 94.9%

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

      4. distribute-neg-frac2 94.9%

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

      5. remove-double-neg 94.9%

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

      6. sqr-neg 94.9%

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

      7. associate-+l+ 94.9%

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

      8. sqr-neg 94.9%

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

      9. distribute-rgt-out 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in m around 0 92.4%

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

   if 2.2000000000000002 < m

   1. Initial program 68.6%

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

      1. associate-/l* 68.6%

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

      2. remove-double-neg 68.6%

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

      3. distribute-frac-neg2 68.6%

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

      4. distribute-neg-frac2 68.6%

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

      5. remove-double-neg 68.6%

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

      6. sqr-neg 68.6%

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

      7. associate-+l+ 68.6%

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

      8. sqr-neg 68.6%

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

      9. distribute-rgt-out 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in m around 0 3.0%

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

   6. Taylor expanded in k around 0 31.2%

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]

   7. Taylor expanded in a around 0 37.8%

      \[\leadsto a + \color{blue}{a \cdot \left(k \cdot \left(99 \cdot k - 10\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.055:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.2:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a + a \cdot \left(k \cdot \left(k \cdot 99 - 10\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -0.245:\\ \;\;\;\;\frac{a}{t\_0}\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{a}{1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* k (+ k 10.0))))
   (if (<= m -0.245)
     (/ a t_0)
     (if (<= m 6.5e+37) (/ a (+ 1.0 t_0)) (* a (+ 1.0 (* k -10.0)))))))

C

double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.245) {
		tmp = a / t_0;
	} else if (m <= 6.5e+37) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = k * (k + 10.0d0)
    if (m <= (-0.245d0)) then
        tmp = a / t_0
    else if (m <= 6.5d+37) then
        tmp = a / (1.0d0 + t_0)
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = k * (k + 10.0);
	double tmp;
	if (m <= -0.245) {
		tmp = a / t_0;
	} else if (m <= 6.5e+37) {
		tmp = a / (1.0 + t_0);
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = k * (k + 10.0)
	tmp = 0
	if m <= -0.245:
		tmp = a / t_0
	elif m <= 6.5e+37:
		tmp = a / (1.0 + t_0)
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(k * Float64(k + 10.0))
	tmp = 0.0
	if (m <= -0.245)
		tmp = Float64(a / t_0);
	elseif (m <= 6.5e+37)
		tmp = Float64(a / Float64(1.0 + t_0));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = k * (k + 10.0);
	tmp = 0.0;
	if (m <= -0.245)
		tmp = a / t_0;
	elseif (m <= 6.5e+37)
		tmp = a / (1.0 + t_0);
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.245], N[(a / t$95$0), $MachinePrecision], If[LessEqual[m, 6.5e+37], N[(a / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.245

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 31.2%

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

   6. Taylor expanded in k around inf 31.2%

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

   7. Taylor expanded in k around inf 41.9%

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

      1. unpow2 41.9%

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

      2. associate-*l* 36.9%

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

      3. +-commutative 36.9%

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

      4. distribute-rgt-in 36.9%

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

      5. associate-*l* 36.9%

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

      6. lft-mult-inverse 36.9%

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

      7. metadata-eval 36.9%

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

      8. *-lft-identity 36.9%

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

      9. +-commutative 36.9%

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

   9. Simplified 36.9%

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

   if -0.245 < m < 6.4999999999999998e37

   1. Initial program 93.4%

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

      1. associate-/l* 93.4%

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

      2. remove-double-neg 93.4%

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

      3. distribute-frac-neg2 93.4%

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

      4. distribute-neg-frac2 93.4%

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

      5. remove-double-neg 93.4%

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

      6. sqr-neg 93.4%

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

      7. associate-+l+ 93.4%

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

      8. sqr-neg 93.4%

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

      9. distribute-rgt-out 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in m around 0 85.4%

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

   if 6.4999999999999998e37 < m

   1. Initial program 67.9%

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

      1. associate-/l* 67.9%

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

      2. remove-double-neg 67.9%

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

      3. distribute-frac-neg2 67.9%

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

      4. distribute-neg-frac2 67.9%

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

      5. remove-double-neg 67.9%

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

      6. sqr-neg 67.9%

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

      7. associate-+l+ 67.9%

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

      8. sqr-neg 67.9%

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

      9. distribute-rgt-out 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-lft-identity 66.7%

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

      3. distribute-rgt-out 66.7%

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

      4. *-commutative 66.7%

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

      5. fma-define 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in m around 0 11.5%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.245:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.9% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{+111} \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -9.2e+111) (not (<= k 0.075)))
   (/ a (* k (+ k 10.0)))
   (* a (+ 1.0 (* k -10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9.2e+111) || !(k <= 0.075)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-9.2d+111)) .or. (.not. (k <= 0.075d0))) then
        tmp = a / (k * (k + 10.0d0))
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -9.2e+111) || !(k <= 0.075)) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -9.2e+111) or not (k <= 0.075):
		tmp = a / (k * (k + 10.0))
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -9.2e+111) || !(k <= 0.075))
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -9.2e+111) || ~((k <= 0.075)))
		tmp = a / (k * (k + 10.0));
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -9.2e+111], N[Not[LessEqual[k, 0.075]], $MachinePrecision]], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -9.20000000000000008e111 or 0.0749999999999999972 < k

   1. Initial program 73.7%

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

      1. *-commutative 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in m around 0 53.2%

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

   6. Taylor expanded in k around inf 53.2%

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

   7. Taylor expanded in k around inf 53.3%

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

      1. unpow2 53.3%

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

      2. associate-*l* 53.3%

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

      3. +-commutative 53.3%

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

      4. distribute-rgt-in 53.3%

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

      5. associate-*l* 53.3%

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

      6. lft-mult-inverse 53.3%

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

      7. metadata-eval 53.3%

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

      8. *-lft-identity 53.3%

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

      9. +-commutative 53.3%

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

   9. Simplified 53.3%

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

   if -9.20000000000000008e111 < k < 0.0749999999999999972

   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 85.9%

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

      1. associate-*r* 85.9%

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

      2. *-lft-identity 85.9%

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

      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%

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

   8. Taylor expanded in m around 0 37.9%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{+111} \lor \neg \left(k \leq 0.075\right):\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.31)
   (/ a (* k (+ k 10.0)))
   (if (<= m 2.9e+35) (/ a (+ 1.0 (* k k))) (* a (+ 1.0 (* k -10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.31) {
		tmp = a / (k * (k + 10.0));
	} else if (m <= 2.9e+35) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= (-0.31d0)) then
        tmp = a / (k * (k + 10.0d0))
    else if (m <= 2.9d+35) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a * (1.0d0 + (k * (-10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.31) {
		tmp = a / (k * (k + 10.0));
	} else if (m <= 2.9e+35) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a * (1.0 + (k * -10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.31:
		tmp = a / (k * (k + 10.0))
	elif m <= 2.9e+35:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a * (1.0 + (k * -10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.31)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	elseif (m <= 2.9e+35)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a * Float64(1.0 + Float64(k * -10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -0.31)
		tmp = a / (k * (k + 10.0));
	elseif (m <= 2.9e+35)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a * (1.0 + (k * -10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.31], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.9e+35], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 + N[(k * -10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.309999999999999998

   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. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in m around 0 31.2%

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

   6. Taylor expanded in k around inf 31.2%

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

   7. Taylor expanded in k around inf 41.9%

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

      1. unpow2 41.9%

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

      2. associate-*l* 36.9%

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

      3. +-commutative 36.9%

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

      4. distribute-rgt-in 36.9%

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

      5. associate-*l* 36.9%

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

      6. lft-mult-inverse 36.9%

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

      7. metadata-eval 36.9%

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

      8. *-lft-identity 36.9%

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

      9. +-commutative 36.9%

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

   9. Simplified 36.9%

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

   if -0.309999999999999998 < m < 2.89999999999999995e35

   1. Initial program 93.4%

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

      1. associate-/l* 93.4%

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

      2. remove-double-neg 93.4%

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

      3. distribute-frac-neg2 93.4%

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

      4. distribute-neg-frac2 93.4%

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

      5. remove-double-neg 93.4%

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

      6. sqr-neg 93.4%

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

      7. associate-+l+ 93.4%

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

      8. sqr-neg 93.4%

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

      9. distribute-rgt-out 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in m around 0 85.4%

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

   6. Taylor expanded in k around inf 84.9%

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

   if 2.89999999999999995e35 < m

   1. Initial program 67.9%

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

      1. associate-/l* 67.9%

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

      2. remove-double-neg 67.9%

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

      3. distribute-frac-neg2 67.9%

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

      4. distribute-neg-frac2 67.9%

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

      5. remove-double-neg 67.9%

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

      6. sqr-neg 67.9%

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

      7. associate-+l+ 67.9%

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

      8. sqr-neg 67.9%

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

      9. distribute-rgt-out 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in k around 0 66.7%

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

      1. associate-*r* 66.7%

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

      2. *-lft-identity 66.7%

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

      3. distribute-rgt-out 66.7%

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

      4. *-commutative 66.7%

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

      5. fma-define 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in m around 0 11.5%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{elif}\;m \leq 2.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(1 + k \cdot -10\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 21.1% accurate, 16.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

   1. associate-/l* 87.6%

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

   2. remove-double-neg 87.6%

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

   3. distribute-frac-neg2 87.6%

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

   4. distribute-neg-frac2 87.6%

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

   5. remove-double-neg 87.6%

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

   6. sqr-neg 87.6%

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

   7. associate-+l+ 87.6%

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

   8. sqr-neg 87.6%

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

   9. distribute-rgt-out 87.6%

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

3. Simplified 87.6%

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

5. Taylor expanded in k around 0 63.3%

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

   1. associate-*r* 62.9%

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

   2. *-lft-identity 62.9%

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

   3. distribute-rgt-out 70.3%

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

   4. *-commutative 70.3%

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

   5. fma-define 70.3%

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

7. Simplified 70.3%

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

8. Taylor expanded in m around 0 22.8%

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

9. Final simplification 22.8%

   \[\leadsto a \cdot \left(1 + k \cdot -10\right) \]
10. Add Preprocessing

Alternative 12: 20.2% accurate, 114.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 87.6%

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

   1. associate-/l* 87.6%

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

   2. remove-double-neg 87.6%

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

   3. distribute-frac-neg2 87.6%

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

   4. distribute-neg-frac2 87.6%

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

   5. remove-double-neg 87.6%

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

   6. sqr-neg 87.6%

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

   7. associate-+l+ 87.6%

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

   8. sqr-neg 87.6%

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

   9. distribute-rgt-out 87.6%

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

3. Simplified 87.6%

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

5. Taylor expanded in m around 0 44.4%

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

6. Taylor expanded in k around 0 21.1%

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

Reproduce

herbie shell --seed 2024180 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))