Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.1%

Time: 12.2s

Alternatives: 16

Speedup: 1.1×

• 23.3% 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: 90.4% 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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ \mathbf{if}\;m \leq -1.8 \cdot 10^{-254}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 40:\\ \;\;\;\;\frac{{k}^{m}}{t_0} \cdot \frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= m -1.8e-254)
     (* a (/ (pow k m) (fma k (+ k 10.0) 1.0)))
     (if (<= m 40.0) (* (/ (pow k m) t_0) (/ a t_0)) (* a (pow k m))))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (m <= -1.8e-254) {
		tmp = a * (pow(k, m) / fma(k, (k + 10.0), 1.0));
	} else if (m <= 40.0) {
		tmp = (pow(k, m) / t_0) * (a / t_0);
	} else {
		tmp = a * pow(k, m);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (m <= -1.8e-254)
		tmp = Float64(a * Float64((k ^ m) / fma(k, Float64(k + 10.0), 1.0)));
	elseif (m <= 40.0)
		tmp = Float64(Float64((k ^ m) / t_0) * Float64(a / t_0));
	else
		tmp = Float64(a * (k ^ m));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, If[LessEqual[m, -1.8e-254], N[(a * N[(N[Power[k, m], $MachinePrecision] / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 40.0], N[(N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.79999999999999992e-254

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. fma-def 99.9%

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

      6. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if -1.79999999999999992e-254 < m < 40

   1. Initial program 85.9%

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

      1. add-cbrt-cube 46.1%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(a \cdot {k}^{m}\right) \cdot \left(a \cdot {k}^{m}\right)\right) \cdot \left(a \cdot {k}^{m}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      2. pow3 46.0%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(a \cdot {k}^{m}\right)}^{3}}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. *-commutative 46.0%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left({k}^{m} \cdot a\right)}}^{3}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. Applied egg-rr 46.0%

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

      1. rem-cbrt-cube 85.9%

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

      2. add-sqr-sqrt 85.8%

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

      3. times-frac 85.8%

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

      4. +-commutative 85.8%

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

      5. add-sqr-sqrt 85.8%

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

      6. hypot-def 85.8%

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

      7. +-commutative 85.8%

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

      8. *-commutative 85.8%

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

      9. fma-def 85.8%

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

      10. +-commutative 85.8%

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

      11. add-sqr-sqrt 85.8%

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

      12. hypot-def 99.7%

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

      13. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 40 < m

   1. Initial program 76.4%

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

      1. associate-*r/ 76.4%

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

      2. associate-+l+ 76.4%

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

      3. +-commutative 76.4%

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

      4. distribute-rgt-out 76.4%

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

      5. fma-def 76.4%

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

      6. +-commutative 76.4%

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

   3. Simplified 76.4%

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

   4. Taylor expanded in k around 0 44.9%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.8 \cdot 10^{-254}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{elif}\;m \leq 40:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ t_1 := a \cdot {k}^{m}\\ t_2 := \sqrt[3]{t_1}\\ \mathbf{if}\;k \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_2}^{2}}{t_0} \cdot \frac{t_2}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0))))
        (t_1 (* a (pow k m)))
        (t_2 (cbrt t_1)))
   (if (<= k 3.2e-17) t_1 (* (/ (pow t_2 2.0) t_0) (/ t_2 t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double t_1 = a * pow(k, m);
	double t_2 = cbrt(t_1);
	double tmp;
	if (k <= 3.2e-17) {
		tmp = t_1;
	} else {
		tmp = (pow(t_2, 2.0) / t_0) * (t_2 / t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	t_1 = Float64(a * (k ^ m))
	t_2 = cbrt(t_1)
	tmp = 0.0
	if (k <= 3.2e-17)
		tmp = t_1;
	else
		tmp = Float64(Float64((t_2 ^ 2.0) / t_0) * Float64(t_2 / t_0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 1/3], $MachinePrecision]}, If[LessEqual[k, 3.2e-17], t$95$1, N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.2000000000000002e-17

   1. Initial program 91.6%

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

      1. associate-*r/ 91.6%

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

      2. associate-+l+ 91.6%

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

      3. +-commutative 91.6%

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

      4. distribute-rgt-out 91.6%

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

      5. fma-def 91.6%

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

      6. +-commutative 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in k around 0 50.0%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if 3.2000000000000002e-17 < k

   1. Initial program 83.2%

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

      1. add-cbrt-cube 70.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(a \cdot {k}^{m}\right) \cdot \left(a \cdot {k}^{m}\right)\right) \cdot \left(a \cdot {k}^{m}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      2. pow3 70.5%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(a \cdot {k}^{m}\right)}^{3}}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. *-commutative 70.5%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left({k}^{m} \cdot a\right)}}^{3}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. Applied egg-rr 70.5%

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

      1. rem-cbrt-cube 83.2%

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

      2. add-cube-cbrt 82.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. add-sqr-sqrt 82.9%

         \[\leadsto \frac{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}{\color{blue}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      4. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      5. pow2 82.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\sqrt{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. add-sqr-sqrt 82.9%

         \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\sqrt{k \cdot k + \color{blue}{\sqrt{1 + 10 \cdot k} \cdot \sqrt{1 + 10 \cdot k}}}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. hypot-def 82.9%

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

      9. +-commutative 82.9%

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

      10. *-commutative 82.9%

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

      11. fma-def 82.9%

          \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}}\right)} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   5. Applied egg-rr 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \end{array} \]

Alternative 3: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ t_1 := \sqrt[3]{t_0}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{{t_1}^{2}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{t_1}{\mathsf{hypot}\left(k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))) (t_1 (cbrt t_0)))
   (if (<= k 4.2e-17)
     t_0
     (*
      (/ (pow t_1 2.0) (hypot k (sqrt (fma k 10.0 1.0))))
      (/ t_1 (hypot k 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double t_1 = cbrt(t_0);
	double tmp;
	if (k <= 4.2e-17) {
		tmp = t_0;
	} else {
		tmp = (pow(t_1, 2.0) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * (t_1 / hypot(k, 1.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	t_1 = cbrt(t_0)
	tmp = 0.0
	if (k <= 4.2e-17)
		tmp = t_0;
	else
		tmp = Float64(Float64((t_1 ^ 2.0) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * Float64(t_1 / hypot(k, 1.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[k, 4.2e-17], t$95$0, N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[k ^ 2 + 1.0 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.19999999999999984e-17

   1. Initial program 91.6%

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

      1. associate-*r/ 91.6%

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

      2. associate-+l+ 91.6%

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

      3. +-commutative 91.6%

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

      4. distribute-rgt-out 91.6%

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

      5. fma-def 91.6%

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

      6. +-commutative 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in k around 0 50.0%

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

      1. exp-to-pow 100.0%

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

   6. Simplified 100.0%

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

   if 4.19999999999999984e-17 < k

   1. Initial program 83.2%

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

      1. add-cbrt-cube 70.5%

         \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(a \cdot {k}^{m}\right) \cdot \left(a \cdot {k}^{m}\right)\right) \cdot \left(a \cdot {k}^{m}\right)}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      2. pow3 70.5%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(a \cdot {k}^{m}\right)}^{3}}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. *-commutative 70.5%

         \[\leadsto \frac{\sqrt[3]{{\color{blue}{\left({k}^{m} \cdot a\right)}}^{3}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   3. Applied egg-rr 70.5%

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

      1. rem-cbrt-cube 83.2%

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

      2. add-cube-cbrt 82.9%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

      3. add-sqr-sqrt 82.9%

         \[\leadsto \frac{\left(\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}\right) \cdot \sqrt[3]{{k}^{m} \cdot a}}{\color{blue}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k} \cdot \sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      4. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{{k}^{m} \cdot a} \cdot \sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}}} \]

      5. pow2 82.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      6. +-commutative 82.9%

         \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\sqrt{\color{blue}{k \cdot k + \left(1 + 10 \cdot k\right)}}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      7. add-sqr-sqrt 82.9%

         \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\sqrt{k \cdot k + \color{blue}{\sqrt{1 + 10 \cdot k} \cdot \sqrt{1 + 10 \cdot k}}}} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

      8. hypot-def 82.9%

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

      9. +-commutative 82.9%

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

      10. *-commutative 82.9%

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

      11. fma-def 82.9%

          \[\leadsto \frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\color{blue}{\mathsf{fma}\left(k, 10, 1\right)}}\right)} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\sqrt{\left(1 + 10 \cdot k\right) + k \cdot k}} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{{k}^{m} \cdot a}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{\sqrt[3]{{k}^{m} \cdot a}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}} \]

   6. Taylor expanded in k around 0 98.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{a \cdot {k}^{m}}\right)}^{2}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{\sqrt[3]{a \cdot {k}^{m}}}{\mathsf{hypot}\left(k, 1\right)}\\ \end{array} \]

Alternative 4: 93.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+157}:\\ \;\;\;\;\frac{t_0}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= k 6.5e-12)
     t_0
     (if (<= k 2.7e+157) (/ t_0 (* k k)) (* (/ a k) (/ 1.0 k))))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 6.5e-12) {
		tmp = t_0;
	} else if (k <= 2.7e+157) {
		tmp = t_0 / (k * k);
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a * (k ** m)
    if (k <= 6.5d-12) then
        tmp = t_0
    else if (k <= 2.7d+157) then
        tmp = t_0 / (k * k)
    else
        tmp = (a / k) * (1.0d0 / k)
    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 (k <= 6.5e-12) {
		tmp = t_0;
	} else if (k <= 2.7e+157) {
		tmp = t_0 / (k * k);
	} else {
		tmp = (a / k) * (1.0 / k);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 6.5e-12:
		tmp = t_0
	elif k <= 2.7e+157:
		tmp = t_0 / (k * k)
	else:
		tmp = (a / k) * (1.0 / k)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 6.5e-12)
		tmp = t_0;
	elseif (k <= 2.7e+157)
		tmp = Float64(t_0 / Float64(k * k));
	else
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 6.5e-12)
		tmp = t_0;
	elseif (k <= 2.7e+157)
		tmp = t_0 / (k * k);
	else
		tmp = (a / k) * (1.0 / k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 6.5e-12], t$95$0, If[LessEqual[k, 2.7e+157], N[(t$95$0 / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < 6.5000000000000002e-12

   1. Initial program 91.7%

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

      1. associate-*r/ 91.7%

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

      2. associate-+l+ 91.7%

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

      3. +-commutative 91.7%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in k around 0 50.5%

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

      1. exp-to-pow 99.9%

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

   6. Simplified 99.9%

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

   if 6.5000000000000002e-12 < k < 2.7e157

   1. Initial program 99.6%

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

   2. Taylor expanded in k around inf 98.7%

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

      1. unpow2 98.7%

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

   4. Simplified 98.7%

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

   if 2.7e157 < k

   1. Initial program 59.6%

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

      1. associate-*r/ 59.6%

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

      2. associate-+l+ 59.6%

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

      3. +-commutative 59.6%

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

      4. distribute-rgt-out 59.6%

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

      5. fma-def 59.6%

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

      6. +-commutative 59.6%

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

   3. Simplified 59.6%

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

   4. Taylor expanded in m around 0 59.9%

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

   5. Taylor expanded in k around inf 59.9%

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

      1. unpow2 59.9%

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

   7. Simplified 59.9%

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

      1. clear-num 59.9%

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

      2. inv-pow 59.9%

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

   9. Applied egg-rr 59.9%

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

       1. unpow-1 59.9%

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

       2. associate-/l* 77.4%

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

   11. Simplified 77.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. inv-pow 77.4%

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

       2. associate-/r/ 77.5%

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

       3. unpow-prod-down 77.7%

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

       4. inv-pow 77.7%

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

       5. clear-num 77.7%

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

       6. inv-pow 77.7%

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

   13. Applied egg-rr 77.7%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+157}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \end{array} \]

Alternative 5: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{{k}^{m}} \cdot \frac{k}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 6.5e-12) (* a (pow k m)) (/ 1.0 (* (/ k (pow k m)) (/ k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.5e-12) {
		tmp = a * pow(k, m);
	} else {
		tmp = 1.0 / ((k / pow(k, m)) * (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 (k <= 6.5d-12) then
        tmp = a * (k ** m)
    else
        tmp = 1.0d0 / ((k / (k ** m)) * (k / a))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 6.5e-12) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = 1.0 / ((k / Math.pow(k, m)) * (k / a));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 6.5e-12:
		tmp = a * math.pow(k, m)
	else:
		tmp = 1.0 / ((k / math.pow(k, m)) * (k / a))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 6.5e-12)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(1.0 / Float64(Float64(k / (k ^ m)) * Float64(k / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 6.5e-12)
		tmp = a * (k ^ m);
	else
		tmp = 1.0 / ((k / (k ^ m)) * (k / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 6.5e-12], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(k / N[Power[k, m], $MachinePrecision]), $MachinePrecision] * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.5000000000000002e-12

   1. Initial program 91.7%

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

      1. associate-*r/ 91.7%

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

      2. associate-+l+ 91.7%

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

      3. +-commutative 91.7%

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

      4. distribute-rgt-out 91.7%

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

      5. fma-def 91.7%

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

      6. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in k around 0 50.5%

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

      1. exp-to-pow 99.9%

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

   6. Simplified 99.9%

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

   if 6.5000000000000002e-12 < k

   1. Initial program 82.8%

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

      1. associate-*r/ 82.8%

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

      2. associate-+l+ 82.8%

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

      3. +-commutative 82.8%

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

      4. distribute-rgt-out 82.8%

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

      5. fma-def 82.8%

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

      6. +-commutative 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 82.8%

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

      2. add-sqr-sqrt 82.8%

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

      3. associate-/r* 82.8%

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

      4. *-commutative 82.8%

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

   5. Applied egg-rr 82.8%

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

      1. clear-num 82.8%

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

      2. inv-pow 82.8%

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

      3. associate-/l* 80.5%

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

   7. Applied egg-rr 80.5%

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

      1. unpow-1 80.5%

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

      2. associate-/r/ 80.6%

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

   9. Simplified 80.6%

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

   10. Taylor expanded in k around inf 82.3%

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

       1. mul-1-neg 82.3%

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

       2. log-rec 82.3%

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

       3. distribute-lft-neg-in 82.3%

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

       4. remove-double-neg 82.3%

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

       5. exp-to-pow 82.3%

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

       6. unpow2 82.3%

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

       7. times-frac 93.1%

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

   12. Simplified 93.1%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-12}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{{k}^{m}} \cdot \frac{k}{a}}\\ \end{array} \]

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-6} \lor \neg \left(m \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -2.8e-6) (not (<= m 2.9e-5)))
   (* a (pow k m))
   (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -2.8e-6) || !(m <= 2.9e-5)) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * (1.0 / (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 <= (-2.8d-6)) .or. (.not. (m <= 2.9d-5))) then
        tmp = a * (k ** m)
    else
        tmp = a * (1.0d0 / (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 <= -2.8e-6) || !(m <= 2.9e-5)) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -2.8e-6) or not (m <= 2.9e-5):
		tmp = a * math.pow(k, m)
	else:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -2.8e-6) || !(m <= 2.9e-5))
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * Float64(1.0 / 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 <= -2.8e-6) || ~((m <= 2.9e-5)))
		tmp = a * (k ^ m);
	else
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -2.8e-6], N[Not[LessEqual[m, 2.9e-5]], $MachinePrecision]], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.79999999999999987e-6 or 2.9e-5 < m

   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-*r/ 87.6%

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

      2. associate-+l+ 87.6%

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

      3. +-commutative 87.6%

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

      4. distribute-rgt-out 87.6%

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

      5. fma-def 87.6%

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

      6. +-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in k around 0 50.6%

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

      1. exp-to-pow 99.4%

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

   6. Simplified 99.4%

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

   if -2.79999999999999987e-6 < m < 2.9e-5

   1. Initial program 90.5%

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

      1. associate-*r/ 90.5%

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

      2. associate-+l+ 90.5%

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

      3. +-commutative 90.5%

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

      4. distribute-rgt-out 90.5%

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

      5. fma-def 90.5%

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

      6. +-commutative 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in m around 0 89.2%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.8 \cdot 10^{-6} \lor \neg \left(m \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 7: 43.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{if}\;m \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -1.6 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-108}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 19000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (/ a k) (/ 1.0 k))))
   (if (<= m -9.5e-29)
     (* a (/ 1.0 (* k k)))
     (if (<= m -1.6e-177)
       a
       (if (<= m 1.25e-226)
         t_0
         (if (<= m 8e-108) a (if (<= m 19000.0) t_0 (* -10.0 (* k a)))))))))

C

double code(double a, double k, double m) {
	double t_0 = (a / k) * (1.0 / k);
	double tmp;
	if (m <= -9.5e-29) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -1.6e-177) {
		tmp = a;
	} else if (m <= 1.25e-226) {
		tmp = t_0;
	} else if (m <= 8e-108) {
		tmp = a;
	} else if (m <= 19000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (a / k) * (1.0d0 / k)
    if (m <= (-9.5d-29)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= (-1.6d-177)) then
        tmp = a
    else if (m <= 1.25d-226) then
        tmp = t_0
    else if (m <= 8d-108) then
        tmp = a
    else if (m <= 19000.0d0) then
        tmp = t_0
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = (a / k) * (1.0 / k);
	double tmp;
	if (m <= -9.5e-29) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -1.6e-177) {
		tmp = a;
	} else if (m <= 1.25e-226) {
		tmp = t_0;
	} else if (m <= 8e-108) {
		tmp = a;
	} else if (m <= 19000.0) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = (a / k) * (1.0 / k)
	tmp = 0
	if m <= -9.5e-29:
		tmp = a * (1.0 / (k * k))
	elif m <= -1.6e-177:
		tmp = a
	elif m <= 1.25e-226:
		tmp = t_0
	elif m <= 8e-108:
		tmp = a
	elif m <= 19000.0:
		tmp = t_0
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a / k) * Float64(1.0 / k))
	tmp = 0.0
	if (m <= -9.5e-29)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= -1.6e-177)
		tmp = a;
	elseif (m <= 1.25e-226)
		tmp = t_0;
	elseif (m <= 8e-108)
		tmp = a;
	elseif (m <= 19000.0)
		tmp = t_0;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (a / k) * (1.0 / k);
	tmp = 0.0;
	if (m <= -9.5e-29)
		tmp = a * (1.0 / (k * k));
	elseif (m <= -1.6e-177)
		tmp = a;
	elseif (m <= 1.25e-226)
		tmp = t_0;
	elseif (m <= 8e-108)
		tmp = a;
	elseif (m <= 19000.0)
		tmp = t_0;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -9.5e-29], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -1.6e-177], a, If[LessEqual[m, 1.25e-226], t$95$0, If[LessEqual[m, 8e-108], a, If[LessEqual[m, 19000.0], t$95$0, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -9.50000000000000023e-29

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 36.0%

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

   5. Taylor expanded in k around inf 65.9%

      \[\leadsto a \cdot \color{blue}{\frac{1}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 65.9%

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

   7. Simplified 65.9%

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

   if -9.50000000000000023e-29 < m < -1.5999999999999999e-177 or 1.2499999999999999e-226 < m < 8.00000000000000032e-108

   1. Initial program 94.8%

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

      1. associate-*r/ 94.9%

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

      2. associate-+l+ 94.9%

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

      3. +-commutative 94.9%

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

      4. distribute-rgt-out 94.9%

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

      5. fma-def 94.9%

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

      6. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in k around 0 74.7%

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

      1. exp-to-pow 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in m around 0 74.7%

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

   if -1.5999999999999999e-177 < m < 1.2499999999999999e-226 or 8.00000000000000032e-108 < m < 19000

   1. Initial program 84.0%

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

      1. associate-*r/ 84.0%

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

      2. associate-+l+ 84.0%

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

      3. +-commutative 84.0%

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

      4. distribute-rgt-out 84.0%

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

      5. fma-def 84.0%

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

      6. +-commutative 84.0%

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

   3. Simplified 84.0%

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

   4. Taylor expanded in m around 0 83.6%

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

   5. Taylor expanded in k around inf 57.4%

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

      1. unpow2 57.4%

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

   7. Simplified 57.4%

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

      1. clear-num 57.5%

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

      2. inv-pow 57.5%

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

   9. Applied egg-rr 57.5%

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

       1. unpow-1 57.5%

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

       2. associate-/l* 69.8%

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

   11. Simplified 69.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. inv-pow 69.8%

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

       2. associate-/r/ 70.0%

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

       3. unpow-prod-down 70.1%

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

       4. inv-pow 70.1%

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

       5. clear-num 70.0%

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

       6. inv-pow 70.0%

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

   13. Applied egg-rr 70.0%

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

   if 19000 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

      \[\leadsto \color{blue}{-10 \cdot \left(k \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -1.6 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{-226}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{elif}\;m \leq 8 \cdot 10^{-108}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 19000:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 8: 43.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -7 \cdot 10^{-178}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-106}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 210000:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.05e-25)
   (* a (/ 1.0 (* k k)))
   (if (<= m -7e-178)
     a
     (if (<= m 3.3e-227)
       (/ 1.0 (* k (/ k a)))
       (if (<= m 1.4e-106)
         a
         (if (<= m 210000.0) (* (/ a k) (/ 1.0 k)) (* -10.0 (* k a))))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.05e-25) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -7e-178) {
		tmp = a;
	} else if (m <= 3.3e-227) {
		tmp = 1.0 / (k * (k / a));
	} else if (m <= 1.4e-106) {
		tmp = a;
	} else if (m <= 210000.0) {
		tmp = (a / k) * (1.0 / k);
	} 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 <= (-2.05d-25)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= (-7d-178)) then
        tmp = a
    else if (m <= 3.3d-227) then
        tmp = 1.0d0 / (k * (k / a))
    else if (m <= 1.4d-106) then
        tmp = a
    else if (m <= 210000.0d0) then
        tmp = (a / k) * (1.0d0 / k)
    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 <= -2.05e-25) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -7e-178) {
		tmp = a;
	} else if (m <= 3.3e-227) {
		tmp = 1.0 / (k * (k / a));
	} else if (m <= 1.4e-106) {
		tmp = a;
	} else if (m <= 210000.0) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -2.05e-25:
		tmp = a * (1.0 / (k * k))
	elif m <= -7e-178:
		tmp = a
	elif m <= 3.3e-227:
		tmp = 1.0 / (k * (k / a))
	elif m <= 1.4e-106:
		tmp = a
	elif m <= 210000.0:
		tmp = (a / k) * (1.0 / k)
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.05e-25)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= -7e-178)
		tmp = a;
	elseif (m <= 3.3e-227)
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	elseif (m <= 1.4e-106)
		tmp = a;
	elseif (m <= 210000.0)
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	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 <= -2.05e-25)
		tmp = a * (1.0 / (k * k));
	elseif (m <= -7e-178)
		tmp = a;
	elseif (m <= 3.3e-227)
		tmp = 1.0 / (k * (k / a));
	elseif (m <= 1.4e-106)
		tmp = a;
	elseif (m <= 210000.0)
		tmp = (a / k) * (1.0 / k);
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.05e-25], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -7e-178], a, If[LessEqual[m, 3.3e-227], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.4e-106], a, If[LessEqual[m, 210000.0], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if m < -2.04999999999999994e-25

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 36.0%

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

   5. Taylor expanded in k around inf 65.9%

      \[\leadsto a \cdot \color{blue}{\frac{1}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 65.9%

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

   7. Simplified 65.9%

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

   if -2.04999999999999994e-25 < m < -6.99999999999999966e-178 or 3.2999999999999999e-227 < m < 1.39999999999999994e-106

   1. Initial program 94.8%

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

      1. associate-*r/ 94.9%

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

      2. associate-+l+ 94.9%

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

      3. +-commutative 94.9%

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

      4. distribute-rgt-out 94.9%

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

      5. fma-def 94.9%

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

      6. +-commutative 94.9%

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

   3. Simplified 94.9%

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

   4. Taylor expanded in k around 0 74.7%

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

      1. exp-to-pow 74.7%

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

   6. Simplified 74.7%

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

   7. Taylor expanded in m around 0 74.7%

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

   if -6.99999999999999966e-178 < m < 3.2999999999999999e-227

   1. Initial program 85.6%

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

      1. associate-*r/ 85.5%

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

      2. associate-+l+ 85.5%

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

      3. +-commutative 85.5%

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

      4. distribute-rgt-out 85.5%

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

      5. fma-def 85.5%

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

      6. +-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in m around 0 85.6%

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

   5. Taylor expanded in k around inf 52.8%

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

      1. unpow2 52.8%

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

   7. Simplified 52.8%

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

      1. clear-num 52.8%

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

      2. inv-pow 52.8%

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

   9. Applied egg-rr 52.8%

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

       1. unpow-1 52.8%

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

       2. associate-/l* 66.8%

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

   11. Simplified 66.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 67.1%

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

   13. Applied egg-rr 67.1%

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

   if 1.39999999999999994e-106 < m < 2.1e5

   1. Initial program 79.8%

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

      1. associate-*r/ 79.8%

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

      2. associate-+l+ 79.8%

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

      3. +-commutative 79.8%

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

      4. distribute-rgt-out 79.8%

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

      5. fma-def 79.8%

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

      6. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in m around 0 78.0%

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

   5. Taylor expanded in k around inf 70.2%

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

      1. unpow2 70.2%

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

   7. Simplified 70.2%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. unpow-1 70.3%

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

       2. associate-/l* 78.0%

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

   11. Simplified 78.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. inv-pow 78.0%

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

       2. associate-/r/ 78.0%

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

       3. unpow-prod-down 78.9%

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

       4. inv-pow 78.9%

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

       5. clear-num 78.9%

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

       6. inv-pow 78.9%

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

   13. Applied egg-rr 78.9%

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

   if 2.1e5 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

      \[\leadsto \color{blue}{-10 \cdot \left(k \cdot a\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.05 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -7 \cdot 10^{-178}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 3.3 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-106}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 210000:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 9: 45.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{1 + k \cdot 10}\\ \mathbf{if}\;m \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -2.3 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 210000:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (+ 1.0 (* k 10.0)))))
   (if (<= m -5.2e-26)
     (* a (/ 1.0 (* k k)))
     (if (<= m -2.3e-175)
       t_0
       (if (<= m -2e-308)
         (/ 1.0 (* k (/ k a)))
         (if (<= m 2.3e-102)
           t_0
           (if (<= m 210000.0) (* (/ a k) (/ 1.0 k)) (* -10.0 (* k a)))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -5.2e-26) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -2.3e-175) {
		tmp = t_0;
	} else if (m <= -2e-308) {
		tmp = 1.0 / (k * (k / a));
	} else if (m <= 2.3e-102) {
		tmp = t_0;
	} else if (m <= 210000.0) {
		tmp = (a / k) * (1.0 / k);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = a / (1.0d0 + (k * 10.0d0))
    if (m <= (-5.2d-26)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= (-2.3d-175)) then
        tmp = t_0
    else if (m <= (-2d-308)) then
        tmp = 1.0d0 / (k * (k / a))
    else if (m <= 2.3d-102) then
        tmp = t_0
    else if (m <= 210000.0d0) then
        tmp = (a / k) * (1.0d0 / k)
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (1.0 + (k * 10.0));
	double tmp;
	if (m <= -5.2e-26) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -2.3e-175) {
		tmp = t_0;
	} else if (m <= -2e-308) {
		tmp = 1.0 / (k * (k / a));
	} else if (m <= 2.3e-102) {
		tmp = t_0;
	} else if (m <= 210000.0) {
		tmp = (a / k) * (1.0 / k);
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (1.0 + (k * 10.0))
	tmp = 0
	if m <= -5.2e-26:
		tmp = a * (1.0 / (k * k))
	elif m <= -2.3e-175:
		tmp = t_0
	elif m <= -2e-308:
		tmp = 1.0 / (k * (k / a))
	elif m <= 2.3e-102:
		tmp = t_0
	elif m <= 210000.0:
		tmp = (a / k) * (1.0 / k)
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(1.0 + Float64(k * 10.0)))
	tmp = 0.0
	if (m <= -5.2e-26)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= -2.3e-175)
		tmp = t_0;
	elseif (m <= -2e-308)
		tmp = Float64(1.0 / Float64(k * Float64(k / a)));
	elseif (m <= 2.3e-102)
		tmp = t_0;
	elseif (m <= 210000.0)
		tmp = Float64(Float64(a / k) * Float64(1.0 / k));
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (1.0 + (k * 10.0));
	tmp = 0.0;
	if (m <= -5.2e-26)
		tmp = a * (1.0 / (k * k));
	elseif (m <= -2.3e-175)
		tmp = t_0;
	elseif (m <= -2e-308)
		tmp = 1.0 / (k * (k / a));
	elseif (m <= 2.3e-102)
		tmp = t_0;
	elseif (m <= 210000.0)
		tmp = (a / k) * (1.0 / k);
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -5.2e-26], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -2.3e-175], t$95$0, If[LessEqual[m, -2e-308], N[(1.0 / N[(k * N[(k / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.3e-102], t$95$0, If[LessEqual[m, 210000.0], N[(N[(a / k), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if m < -5.2000000000000002e-26

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 36.0%

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

   5. Taylor expanded in k around inf 65.9%

      \[\leadsto a \cdot \color{blue}{\frac{1}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 65.9%

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

   7. Simplified 65.9%

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

   if -5.2000000000000002e-26 < m < -2.3e-175 or -1.9999999999999998e-308 < m < 2.29999999999999987e-102

   1. Initial program 92.8%

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

      1. associate-*r/ 92.8%

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

      2. associate-+l+ 92.8%

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

      3. +-commutative 92.8%

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

      4. distribute-rgt-out 92.8%

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

      5. fma-def 92.8%

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

      6. +-commutative 92.8%

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

   3. Simplified 92.8%

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

   4. Taylor expanded in m around 0 92.8%

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

   5. Taylor expanded in k around 0 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

   if -2.3e-175 < m < -1.9999999999999998e-308

   1. Initial program 83.4%

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

      1. associate-*r/ 83.4%

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

      2. associate-+l+ 83.4%

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

      3. +-commutative 83.4%

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

      4. distribute-rgt-out 83.4%

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

      5. fma-def 83.4%

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

      6. +-commutative 83.4%

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

   3. Simplified 83.4%

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

   4. Taylor expanded in m around 0 83.4%

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

   5. Taylor expanded in k around inf 61.2%

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

      1. unpow2 61.2%

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

   7. Simplified 61.2%

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

      1. clear-num 61.1%

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

      2. inv-pow 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. unpow-1 61.1%

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

       2. associate-/l* 77.4%

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

   11. Simplified 77.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. associate-/r/ 77.6%

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

   13. Applied egg-rr 77.6%

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

   if 2.29999999999999987e-102 < m < 2.1e5

   1. Initial program 79.8%

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

      1. associate-*r/ 79.8%

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

      2. associate-+l+ 79.8%

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

      3. +-commutative 79.8%

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

      4. distribute-rgt-out 79.8%

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

      5. fma-def 79.8%

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

      6. +-commutative 79.8%

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

   3. Simplified 79.8%

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

   4. Taylor expanded in m around 0 78.0%

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

   5. Taylor expanded in k around inf 70.2%

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

      1. unpow2 70.2%

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

   7. Simplified 70.2%

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

      1. clear-num 70.3%

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

      2. inv-pow 70.3%

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

   9. Applied egg-rr 70.3%

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

       1. unpow-1 70.3%

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

       2. associate-/l* 78.0%

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

   11. Simplified 78.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{k}{\frac{a}{k}}}} \]
   12. Step-by-step derivation

       1. inv-pow 78.0%

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

       2. associate-/r/ 78.0%

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

       3. unpow-prod-down 78.9%

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

       4. inv-pow 78.9%

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

       5. clear-num 78.9%

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

       6. inv-pow 78.9%

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

   13. Applied egg-rr 78.9%

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

   if 2.1e5 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

      \[\leadsto \color{blue}{-10 \cdot \left(k \cdot a\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.2 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -2.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{k \cdot \frac{k}{a}}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{elif}\;m \leq 210000:\\ \;\;\;\;\frac{a}{k} \cdot \frac{1}{k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 10: 42.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq -9.8 \cdot 10^{-178}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-306}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 19000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.6e-25)
     t_0
     (if (<= m -9.8e-178)
       a
       (if (<= m 1.05e-306)
         t_0
         (if (<= m 4.6e-102) a (if (<= m 19000.0) t_0 (* -10.0 (* k a)))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.6e-25) {
		tmp = t_0;
	} else if (m <= -9.8e-178) {
		tmp = a;
	} else if (m <= 1.05e-306) {
		tmp = t_0;
	} else if (m <= 4.6e-102) {
		tmp = a;
	} else if (m <= 19000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-1.6d-25)) then
        tmp = t_0
    else if (m <= (-9.8d-178)) then
        tmp = a
    else if (m <= 1.05d-306) then
        tmp = t_0
    else if (m <= 4.6d-102) then
        tmp = a
    else if (m <= 19000.0d0) then
        tmp = t_0
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.6e-25) {
		tmp = t_0;
	} else if (m <= -9.8e-178) {
		tmp = a;
	} else if (m <= 1.05e-306) {
		tmp = t_0;
	} else if (m <= 4.6e-102) {
		tmp = a;
	} else if (m <= 19000.0) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -1.6e-25:
		tmp = t_0
	elif m <= -9.8e-178:
		tmp = a
	elif m <= 1.05e-306:
		tmp = t_0
	elif m <= 4.6e-102:
		tmp = a
	elif m <= 19000.0:
		tmp = t_0
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.6e-25)
		tmp = t_0;
	elseif (m <= -9.8e-178)
		tmp = a;
	elseif (m <= 1.05e-306)
		tmp = t_0;
	elseif (m <= 4.6e-102)
		tmp = a;
	elseif (m <= 19000.0)
		tmp = t_0;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -1.6e-25)
		tmp = t_0;
	elseif (m <= -9.8e-178)
		tmp = a;
	elseif (m <= 1.05e-306)
		tmp = t_0;
	elseif (m <= 4.6e-102)
		tmp = a;
	elseif (m <= 19000.0)
		tmp = t_0;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.6e-25], t$95$0, If[LessEqual[m, -9.8e-178], a, If[LessEqual[m, 1.05e-306], t$95$0, If[LessEqual[m, 4.6e-102], a, If[LessEqual[m, 19000.0], t$95$0, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.6000000000000001e-25 or -9.80000000000000041e-178 < m < 1.0500000000000001e-306 or 4.59999999999999973e-102 < m < 19000

   1. Initial program 95.4%

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

      1. associate-*r/ 95.4%

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

      2. associate-+l+ 95.4%

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

      3. +-commutative 95.4%

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

      4. distribute-rgt-out 95.4%

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

      5. fma-def 95.4%

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

      6. +-commutative 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in m around 0 48.0%

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

   5. Taylor expanded in k around inf 66.0%

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

      1. unpow2 66.0%

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

   7. Simplified 66.0%

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

   if -1.6000000000000001e-25 < m < -9.80000000000000041e-178 or 1.0500000000000001e-306 < m < 4.59999999999999973e-102

   1. Initial program 92.7%

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

      1. associate-*r/ 92.7%

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

      2. associate-+l+ 92.7%

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

      3. +-commutative 92.7%

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

      4. distribute-rgt-out 92.7%

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

      5. fma-def 92.7%

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

      6. +-commutative 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in k around 0 67.4%

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

      1. exp-to-pow 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in m around 0 67.4%

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

   if 19000 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq -9.8 \cdot 10^{-178}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-306}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-102}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 19000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 11: 42.8% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.75 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -5.1 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 9.8 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-103}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 20000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.75e-25)
     (* a (/ 1.0 (* k k)))
     (if (<= m -5.1e-177)
       a
       (if (<= m 9.8e-308)
         t_0
         (if (<= m 4.3e-103) a (if (<= m 20000.0) t_0 (* -10.0 (* k a)))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.75e-25) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -5.1e-177) {
		tmp = a;
	} else if (m <= 9.8e-308) {
		tmp = t_0;
	} else if (m <= 4.3e-103) {
		tmp = a;
	} else if (m <= 20000.0) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-1.75d-25)) then
        tmp = a * (1.0d0 / (k * k))
    else if (m <= (-5.1d-177)) then
        tmp = a
    else if (m <= 9.8d-308) then
        tmp = t_0
    else if (m <= 4.3d-103) then
        tmp = a
    else if (m <= 20000.0d0) then
        tmp = t_0
    else
        tmp = (-10.0d0) * (k * a)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.75e-25) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= -5.1e-177) {
		tmp = a;
	} else if (m <= 9.8e-308) {
		tmp = t_0;
	} else if (m <= 4.3e-103) {
		tmp = a;
	} else if (m <= 20000.0) {
		tmp = t_0;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -1.75e-25:
		tmp = a * (1.0 / (k * k))
	elif m <= -5.1e-177:
		tmp = a
	elif m <= 9.8e-308:
		tmp = t_0
	elif m <= 4.3e-103:
		tmp = a
	elif m <= 20000.0:
		tmp = t_0
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.75e-25)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= -5.1e-177)
		tmp = a;
	elseif (m <= 9.8e-308)
		tmp = t_0;
	elseif (m <= 4.3e-103)
		tmp = a;
	elseif (m <= 20000.0)
		tmp = t_0;
	else
		tmp = Float64(-10.0 * Float64(k * a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -1.75e-25)
		tmp = a * (1.0 / (k * k));
	elseif (m <= -5.1e-177)
		tmp = a;
	elseif (m <= 9.8e-308)
		tmp = t_0;
	elseif (m <= 4.3e-103)
		tmp = a;
	elseif (m <= 20000.0)
		tmp = t_0;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.75e-25], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, -5.1e-177], a, If[LessEqual[m, 9.8e-308], t$95$0, If[LessEqual[m, 4.3e-103], a, If[LessEqual[m, 20000.0], t$95$0, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if m < -1.7500000000000001e-25

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 36.0%

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

   5. Taylor expanded in k around inf 65.9%

      \[\leadsto a \cdot \color{blue}{\frac{1}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 65.9%

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

   7. Simplified 65.9%

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

   if -1.7500000000000001e-25 < m < -5.10000000000000008e-177 or 9.7999999999999997e-308 < m < 4.30000000000000023e-103

   1. Initial program 92.7%

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

      1. associate-*r/ 92.7%

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

      2. associate-+l+ 92.7%

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

      3. +-commutative 92.7%

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

      4. distribute-rgt-out 92.7%

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

      5. fma-def 92.7%

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

      6. +-commutative 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in k around 0 67.4%

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

      1. exp-to-pow 67.4%

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

   6. Simplified 67.4%

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

   7. Taylor expanded in m around 0 67.4%

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

   if -5.10000000000000008e-177 < m < 9.7999999999999997e-308 or 4.30000000000000023e-103 < m < 2e4

   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-*r/ 82.5%

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

      2. associate-+l+ 82.5%

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

      3. +-commutative 82.5%

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

      4. distribute-rgt-out 82.5%

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

      5. fma-def 82.5%

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

      6. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 81.8%

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

   5. Taylor expanded in k around inf 66.1%

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

      1. unpow2 66.1%

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

   7. Simplified 66.1%

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

   if 2e4 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

      \[\leadsto \color{blue}{-10 \cdot \left(k \cdot a\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.75 \cdot 10^{-25}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq -5.1 \cdot 10^{-177}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 9.8 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-103}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 20000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 12: 58.2% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.122:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 21000:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.122)
   (/ a (* k k))
   (if (<= m 21000.0)
     (* a (/ 1.0 (+ 1.0 (* k (+ k 10.0)))))
     (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.122) {
		tmp = a / (k * k);
	} else if (m <= 21000.0) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} 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 <= (-0.122d0)) then
        tmp = a / (k * k)
    else if (m <= 21000.0d0) then
        tmp = a * (1.0d0 / (1.0d0 + (k * (k + 10.0d0))))
    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 <= -0.122) {
		tmp = a / (k * k);
	} else if (m <= 21000.0) {
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.122:
		tmp = a / (k * k)
	elif m <= 21000.0:
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))))
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.122)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 21000.0)
		tmp = Float64(a * Float64(1.0 / Float64(1.0 + Float64(k * Float64(k + 10.0)))));
	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 <= -0.122)
		tmp = a / (k * k);
	elseif (m <= 21000.0)
		tmp = a * (1.0 / (1.0 + (k * (k + 10.0))));
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.122], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 21000.0], N[(a * N[(1.0 / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.122

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 34.3%

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

   5. Taylor expanded in k around inf 67.4%

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

      1. unpow2 67.4%

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

   7. Simplified 67.4%

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

   if -0.122 < m < 21000

   1. Initial program 89.7%

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

      1. associate-*r/ 89.7%

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

      2. associate-+l+ 89.7%

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

      3. +-commutative 89.7%

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

      4. distribute-rgt-out 89.7%

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

      5. fma-def 89.7%

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

      6. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in m around 0 86.7%

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

   if 21000 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.122:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 21000:\\ \;\;\;\;a \cdot \frac{1}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 13: 58.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.13:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 20000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.13)
   (/ a (* k k))
   (if (<= m 20000.0) (/ a (+ 1.0 (* k (+ k 10.0)))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.13) {
		tmp = a / (k * k);
	} else if (m <= 20000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} 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 <= (-0.13d0)) then
        tmp = a / (k * k)
    else if (m <= 20000.0d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    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 <= -0.13) {
		tmp = a / (k * k);
	} else if (m <= 20000.0) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.13:
		tmp = a / (k * k)
	elif m <= 20000.0:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.13)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 20000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	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 <= -0.13)
		tmp = a / (k * k);
	elseif (m <= 20000.0)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.13], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 20000.0], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.13

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 34.3%

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

   5. Taylor expanded in k around inf 67.4%

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

      1. unpow2 67.4%

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

   7. Simplified 67.4%

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

   if -0.13 < m < 2e4

   1. Initial program 89.7%

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

      1. associate-*r/ 89.7%

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

      2. associate-+l+ 89.7%

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

      3. +-commutative 89.7%

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

      4. distribute-rgt-out 89.7%

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

      5. fma-def 89.7%

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

      6. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in m around 0 86.7%

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

   if 2e4 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

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

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.13:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 20000:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 14: 57.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.47:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 680000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.47)
   (/ a (* k k))
   (if (<= m 680000.0) (/ a (+ 1.0 (* k k))) (* -10.0 (* k a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.47) {
		tmp = a / (k * k);
	} else if (m <= 680000.0) {
		tmp = a / (1.0 + (k * k));
	} 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 <= (-0.47d0)) then
        tmp = a / (k * k)
    else if (m <= 680000.0d0) then
        tmp = a / (1.0d0 + (k * k))
    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 <= -0.47) {
		tmp = a / (k * k);
	} else if (m <= 680000.0) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -0.47:
		tmp = a / (k * k)
	elif m <= 680000.0:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.47)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 680000.0)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	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 <= -0.47)
		tmp = a / (k * k);
	elseif (m <= 680000.0)
		tmp = a / (1.0 + (k * k));
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.47], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 680000.0], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.46999999999999997

   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-*r/ 100.0%

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

      2. associate-+l+ 100.0%

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

      3. +-commutative 100.0%

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

      4. distribute-rgt-out 100.0%

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

      5. fma-def 100.0%

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

      6. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 34.3%

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

   5. Taylor expanded in k around inf 67.4%

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

      1. unpow2 67.4%

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

   7. Simplified 67.4%

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

   if -0.46999999999999997 < m < 6.8e5

   1. Initial program 89.7%

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

      1. associate-*r/ 89.7%

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

      2. associate-+l+ 89.7%

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

      3. +-commutative 89.7%

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

      4. distribute-rgt-out 89.7%

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

      5. fma-def 89.7%

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

      6. +-commutative 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in m around 0 86.7%

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

   5. Taylor expanded in k around inf 86.0%

      \[\leadsto \frac{a}{1 + \color{blue}{{k}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 86.0%

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

   7. Simplified 86.0%

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

   if 6.8e5 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.47:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 680000:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 15: 25.6% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 19000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 19000.0) a (* -10.0 (* k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 19000.0) {
		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 <= 19000.0d0) 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 <= 19000.0) {
		tmp = a;
	} else {
		tmp = -10.0 * (k * a);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 19000.0:
		tmp = a
	else:
		tmp = -10.0 * (k * a)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 19000.0)
		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 <= 19000.0)
		tmp = a;
	else
		tmp = -10.0 * (k * a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 19000.0], a, N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 19000

   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-*r/ 94.6%

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

      2. associate-+l+ 94.6%

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

      3. +-commutative 94.6%

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

      4. distribute-rgt-out 94.6%

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

      5. fma-def 94.6%

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

      6. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in k around 0 53.5%

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

      1. exp-to-pow 73.7%

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

   6. Simplified 73.7%

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

   7. Taylor expanded in m around 0 27.2%

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

   if 19000 < m

   1. Initial program 77.3%

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

      1. associate-*r/ 77.3%

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

      2. associate-+l+ 77.3%

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

      3. +-commutative 77.3%

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

      4. distribute-rgt-out 77.3%

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

      5. fma-def 77.3%

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

      6. +-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in m around 0 2.8%

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

   5. Taylor expanded in k around 0 9.4%

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

      1. *-commutative 9.4%

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

   7. Simplified 9.4%

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

   8. Taylor expanded in k around inf 21.7%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 19000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;-10 \cdot \left(k \cdot a\right)\\ \end{array} \]

Alternative 16: 20.3% 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 88.6%

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

   1. associate-*r/ 88.6%

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

   2. associate-+l+ 88.6%

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

   3. +-commutative 88.6%

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

   4. distribute-rgt-out 88.6%

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

   5. fma-def 88.6%

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

   6. +-commutative 88.6%

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

3. Simplified 88.6%

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

4. Taylor expanded in k around 0 50.3%

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

   1. exp-to-pow 82.8%

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

6. Simplified 82.8%

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

7. Taylor expanded in m around 0 19.1%

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

8. Final simplification 19.1%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023230 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))