Falkner and Boettcher, Appendix A

Percentage Accurate: 90.8% → 99.4%

Time: 10.9s

Alternatives: 18

Speedup: 1.1×

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

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.8% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)\\ t_1 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -2.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{t_1}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;m \leq 0.009:\\ \;\;\;\;\frac{\frac{a}{t_0}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))) (t_1 (* (pow k m) a)))
   (if (<= m -2.55e-14)
     (/ t_1 (+ (+ 1.0 (* k 10.0)) (* k k)))
     (if (<= m 0.009) (/ (/ a t_0) t_0) t_1))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double t_1 = pow(k, m) * a;
	double tmp;
	if (m <= -2.55e-14) {
		tmp = t_1 / ((1.0 + (k * 10.0)) + (k * k));
	} else if (m <= 0.009) {
		tmp = (a / t_0) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	t_1 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -2.55e-14)
		tmp = Float64(t_1 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)));
	elseif (m <= 0.009)
		tmp = Float64(Float64(a / t_0) / t_0);
	else
		tmp = t_1;
	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[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -2.55e-14], N[(t$95$1 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.009], N[(N[(a / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.5499999999999999e-14

   1. Initial program 100.0%

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

   if -2.5499999999999999e-14 < m < 0.00899999999999999932

   1. Initial program 90.8%

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

      1. associate-*r/ 90.8%

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

      2. *-commutative 90.8%

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

      3. sqr-neg 90.8%

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

      4. associate-+l+ 90.8%

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

      5. +-commutative 90.8%

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

      6. sqr-neg 90.8%

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

      7. distribute-rgt-out 90.8%

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

      8. fma-def 90.8%

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

      9. +-commutative 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in m around 0 90.8%

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

      1. *-un-lft-identity 90.8%

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

      2. distribute-rgt-in 90.8%

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

      3. associate-+l+ 90.8%

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

      4. add-sqr-sqrt 90.8%

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

      5. times-frac 90.8%

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

      6. +-commutative 90.8%

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

      7. add-sqr-sqrt 90.8%

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

      8. hypot-def 90.8%

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

      9. +-commutative 90.8%

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

      10. *-commutative 90.8%

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

      11. fma-def 90.8%

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

      12. +-commutative 90.8%

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

      13. add-sqr-sqrt 90.8%

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

      14. hypot-def 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

   8. Simplified 99.6%

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

   if 0.00899999999999999932 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.55 \cdot 10^{-14}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{elif}\;m \leq 0.009:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 2: 98.0% accurate, 0.2× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (hypot k (sqrt (fma k 10.0 1.0)))))
   (if (<= k 1e-10)
     (* (* (fma k -10.0 1.0) (pow k m)) a)
     (* (/ (pow k m) t_0) (/ a t_0)))))

C

double code(double a, double k, double m) {
	double t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)));
	double tmp;
	if (k <= 1e-10) {
		tmp = (fma(k, -10.0, 1.0) * pow(k, m)) * a;
	} else {
		tmp = (pow(k, m) / t_0) * (a / t_0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = hypot(k, sqrt(fma(k, 10.0, 1.0)))
	tmp = 0.0
	if (k <= 1e-10)
		tmp = Float64(Float64(fma(k, -10.0, 1.0) * (k ^ m)) * a);
	else
		tmp = Float64(Float64((k ^ m) / t_0) * Float64(a / 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]}, If[LessEqual[k, 1e-10], N[(N[(N[(k * -10.0 + 1.0), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / t$95$0), $MachinePrecision] * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.00000000000000004e-10

   1. Initial program 95.6%

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

      1. associate-*r/ 95.6%

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

      2. *-commutative 95.6%

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

      3. sqr-neg 95.6%

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

      4. associate-+l+ 95.6%

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

      5. +-commutative 95.6%

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

      6. sqr-neg 95.6%

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

      7. distribute-rgt-out 95.6%

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

      8. fma-def 95.6%

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

      9. +-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in k around 0 86.2%

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

      1. associate-*r* 86.2%

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

      2. distribute-lft1-in 100.0%

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

      3. *-commutative 100.0%

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

      4. fma-def 100.0%

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

   6. Simplified 100.0%

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

   if 1.00000000000000004e-10 < k

   1. Initial program 80.9%

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

      1. add-cube-cbrt 80.8%

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

      2. pow3 80.8%

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

      3. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

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

      1. unpow3 80.8%

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

      2. add-cube-cbrt 80.9%

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

      3. add-sqr-sqrt 80.9%

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

      4. times-frac 80.9%

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

      5. +-commutative 80.9%

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

      6. add-sqr-sqrt 80.9%

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

      7. hypot-def 80.9%

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

      8. +-commutative 80.9%

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

      9. *-commutative 80.9%

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

      10. fma-def 80.9%

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

      11. +-commutative 80.9%

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

      12. add-sqr-sqrt 80.9%

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

   5. Applied egg-rr 95.8%

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

4. Final simplification 98.4%

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

Alternative 3: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.096:\\ \;\;\;\;\left(\mathsf{fma}\left(k, -10, 1\right) \cdot {k}^{m}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.096)
   (* (* (fma k -10.0 1.0) (pow k m)) a)
   (* (/ (pow k m) (hypot k (sqrt (fma k 10.0 1.0)))) (/ a k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.096) {
		tmp = (fma(k, -10.0, 1.0) * pow(k, m)) * a;
	} else {
		tmp = (pow(k, m) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * (a / k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.096)
		tmp = Float64(Float64(fma(k, -10.0, 1.0) * (k ^ m)) * a);
	else
		tmp = Float64(Float64((k ^ m) / hypot(k, sqrt(fma(k, 10.0, 1.0)))) * Float64(a / k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.096], N[(N[(N[(k * -10.0 + 1.0), $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[Power[k, m], $MachinePrecision] / N[Sqrt[k ^ 2 + N[Sqrt[N[(k * 10.0 + 1.0), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.096000000000000002

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-+l+ 95.7%

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

      5. +-commutative 95.7%

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

      6. sqr-neg 95.7%

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

      7. distribute-rgt-out 95.7%

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

      8. fma-def 95.7%

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

      9. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in k around 0 85.8%

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

      1. associate-*r* 85.8%

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

      2. distribute-lft1-in 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-def 99.2%

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

   6. Simplified 99.2%

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

   if 0.096000000000000002 < k

   1. Initial program 80.1%

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

      1. add-cube-cbrt 80.0%

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

      2. pow3 80.0%

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

      3. *-commutative 80.0%

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

   3. Applied egg-rr 80.0%

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

      1. unpow3 80.0%

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

      2. add-cube-cbrt 80.1%

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

      3. add-sqr-sqrt 80.1%

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

      4. times-frac 80.1%

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

      5. +-commutative 80.1%

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

      6. add-sqr-sqrt 80.1%

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

      7. hypot-def 80.1%

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

      8. +-commutative 80.1%

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

      9. *-commutative 80.1%

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

      10. fma-def 80.1%

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

      11. +-commutative 80.1%

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

      12. add-sqr-sqrt 80.1%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in k around inf 95.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.096:\\ \;\;\;\;\left(\mathsf{fma}\left(k, -10, 1\right) \cdot {k}^{m}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{hypot}\left(k, \sqrt{\mathsf{fma}\left(k, 10, 1\right)}\right)} \cdot \frac{a}{k}\\ \end{array} \]

Alternative 4: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(k, -10, 1\right) \cdot {k}^{m}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{\left(e^{-m}\right)}^{\left(-\log k\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.1)
   (* (* (fma k -10.0 1.0) (pow k m)) a)
   (* (/ a k) (/ (pow (exp (- m)) (- (log k))) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.1) {
		tmp = (fma(k, -10.0, 1.0) * pow(k, m)) * a;
	} else {
		tmp = (a / k) * (pow(exp(-m), -log(k)) / k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 0.1)
		tmp = Float64(Float64(fma(k, -10.0, 1.0) * (k ^ m)) * a);
	else
		tmp = Float64(Float64(a / k) * Float64((exp(Float64(-m)) ^ Float64(-log(k))) / k));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-+l+ 95.7%

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

      5. +-commutative 95.7%

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

      6. sqr-neg 95.7%

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

      7. distribute-rgt-out 95.7%

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

      8. fma-def 95.7%

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

      9. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in k around 0 85.8%

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

      1. associate-*r* 85.8%

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

      2. distribute-lft1-in 99.2%

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

      3. *-commutative 99.2%

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

      4. fma-def 99.2%

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

   6. Simplified 99.2%

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

   if 0.10000000000000001 < k

   1. Initial program 80.1%

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

      1. associate-*r/ 80.1%

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

      2. *-commutative 80.1%

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

      3. sqr-neg 80.1%

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

      4. associate-+l+ 80.1%

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

      5. +-commutative 80.1%

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

      6. sqr-neg 80.1%

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

      7. distribute-rgt-out 80.1%

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

      8. fma-def 80.1%

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

      9. +-commutative 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in k around inf 80.1%

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

      1. *-commutative 80.1%

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

      2. unpow2 80.1%

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

      3. times-frac 95.6%

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

      4. associate-*r* 95.6%

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

      5. exp-prod 95.4%

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

      6. mul-1-neg 95.4%

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

      7. log-rec 95.4%

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

   6. Simplified 95.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.1:\\ \;\;\;\;\left(\mathsf{fma}\left(k, -10, 1\right) \cdot {k}^{m}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k} \cdot \frac{{\left(e^{-m}\right)}^{\left(-\log k\right)}}{k}\\ \end{array} \]

Alternative 5: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ t_1 := \frac{t_0}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{if}\;t_1 \leq 4 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)) (t_1 (/ t_0 (+ (+ 1.0 (* k 10.0)) (* k k)))))
   (if (<= t_1 4e+63) t_1 t_0)))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+63) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (k ** m) * a
    t_1 = t_0 / ((1.0d0 + (k * 10.0d0)) + (k * k))
    if (t_1 <= 4d+63) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	double tmp;
	if (t_1 <= 4e+63) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k))
	tmp = 0
	if t_1 <= 4e+63:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(k * 10.0)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 4e+63)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	t_1 = t_0 / ((1.0 + (k * 10.0)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 4e+63)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+63], t$95$1, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k))) < 4.00000000000000023e63

   1. Initial program 96.0%

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

   if 4.00000000000000023e63 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 1 (*.f64 10 k)) (*.f64 k k)))

   1. Initial program 63.0%

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

      1. associate-*r/ 63.0%

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

      2. *-commutative 63.0%

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

      3. sqr-neg 63.0%

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

      4. associate-+l+ 63.0%

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

      5. +-commutative 63.0%

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

      6. sqr-neg 63.0%

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

      7. distribute-rgt-out 63.0%

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

      8. fma-def 63.0%

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

      9. +-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in k around 0 100.0%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{{k}^{m} \cdot a}{\left(1 + k \cdot 10\right) + k \cdot k} \leq 4 \cdot 10^{+63}:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\left(1 + k \cdot 10\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]

Alternative 6: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-5} \lor \neg \left(m \leq 0.0142\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.4e-5) (not (<= m 0.0142)))
   (* (pow k m) a)
   (/ a (fma k (+ k 10.0) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.4e-5) || !(m <= 0.0142)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma(k, (k + 10.0), 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5.4e-5) || !(m <= 0.0142))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5.4e-5], N[Not[LessEqual[m, 0.0142]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.3999999999999998e-5 or 0.014200000000000001 < m

   1. Initial program 89.6%

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

      1. associate-*r/ 89.6%

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

      2. *-commutative 89.6%

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

      3. sqr-neg 89.6%

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

      4. associate-+l+ 89.6%

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

      5. +-commutative 89.6%

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

      6. sqr-neg 89.6%

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

      7. distribute-rgt-out 89.6%

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

      8. fma-def 89.6%

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

      9. +-commutative 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -5.3999999999999998e-5 < m < 0.014200000000000001

   1. Initial program 91.0%

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

      1. associate-*r/ 91.0%

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

      2. *-commutative 91.0%

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

      3. sqr-neg 91.0%

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

      4. associate-+l+ 91.0%

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

      5. +-commutative 91.0%

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

      6. sqr-neg 91.0%

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

      7. distribute-rgt-out 91.0%

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

      8. fma-def 91.0%

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

      9. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. +-commutative 90.7%

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

      3. fma-udef 90.7%

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

   6. Simplified 90.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-5} \lor \neg \left(m \leq 0.0142\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \end{array} \]

Alternative 7: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-7} \lor \neg \left(m \leq 0.088\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -3.8e-7) (not (<= m 0.088)))
   (* (pow k m) a)
   (/ a (+ 1.0 (* k (+ k 10.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.8e-7) || !(m <= 0.088)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((m <= (-3.8d-7)) .or. (.not. (m <= 0.088d0))) then
        tmp = (k ** m) * a
    else
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((m <= -3.8e-7) || !(m <= 0.088)) {
		tmp = Math.pow(k, m) * a;
	} else {
		tmp = a / (1.0 + (k * (k + 10.0)));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (m <= -3.8e-7) or not (m <= 0.088):
		tmp = math.pow(k, m) * a
	else:
		tmp = a / (1.0 + (k * (k + 10.0)))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -3.8e-7) || !(m <= 0.088))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((m <= -3.8e-7) || ~((m <= 0.088)))
		tmp = (k ^ m) * a;
	else
		tmp = a / (1.0 + (k * (k + 10.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -3.8e-7], N[Not[LessEqual[m, 0.088]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -3.80000000000000015e-7 or 0.087999999999999995 < m

   1. Initial program 89.6%

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

      1. associate-*r/ 89.6%

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

      2. *-commutative 89.6%

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

      3. sqr-neg 89.6%

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

      4. associate-+l+ 89.6%

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

      5. +-commutative 89.6%

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

      6. sqr-neg 89.6%

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

      7. distribute-rgt-out 89.6%

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

      8. fma-def 89.6%

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

      9. +-commutative 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in k around 0 100.0%

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

   if -3.80000000000000015e-7 < m < 0.087999999999999995

   1. Initial program 91.0%

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

      1. associate-*r/ 91.0%

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

      2. *-commutative 91.0%

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

      3. sqr-neg 91.0%

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

      4. associate-+l+ 91.0%

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

      5. +-commutative 91.0%

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

      6. sqr-neg 91.0%

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

      7. distribute-rgt-out 91.0%

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

      8. fma-def 91.0%

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

      9. +-commutative 91.0%

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

   3. Simplified 91.0%

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

   4. Taylor expanded in m around 0 90.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-7} \lor \neg \left(m \leq 0.088\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 8: 67.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{t_0 + -1}\\ \mathbf{elif}\;m \leq 1.68:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(-99 + \frac{-1000}{k}\right) - \frac{10000}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m -4.1e+15)
     (/ a (+ t_0 -1.0))
     (if (<= m 1.68)
       (/ a t_0)
       (/ a (- (+ -99.0 (/ -1000.0 k)) (/ 10000.0 (* k k))))))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 + (k * (k + 10.0));
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (t_0 + -1.0);
	} else if (m <= 1.68) {
		tmp = a / t_0;
	} else {
		tmp = a / ((-99.0 + (-1000.0 / k)) - (10000.0 / (k * k)));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (k * (k + 10.0d0))
    if (m <= (-4.1d+15)) then
        tmp = a / (t_0 + (-1.0d0))
    else if (m <= 1.68d0) then
        tmp = a / t_0
    else
        tmp = a / (((-99.0d0) + ((-1000.0d0) / k)) - (10000.0d0 / (k * k)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = 1.0 + (k * (k + 10.0));
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (t_0 + -1.0);
	} else if (m <= 1.68) {
		tmp = a / t_0;
	} else {
		tmp = a / ((-99.0 + (-1000.0 / k)) - (10000.0 / (k * k)));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = 1.0 + (k * (k + 10.0))
	tmp = 0
	if m <= -4.1e+15:
		tmp = a / (t_0 + -1.0)
	elif m <= 1.68:
		tmp = a / t_0
	else:
		tmp = a / ((-99.0 + (-1000.0 / k)) - (10000.0 / (k * k)))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(1.0 + Float64(k * Float64(k + 10.0)))
	tmp = 0.0
	if (m <= -4.1e+15)
		tmp = Float64(a / Float64(t_0 + -1.0));
	elseif (m <= 1.68)
		tmp = Float64(a / t_0);
	else
		tmp = Float64(a / Float64(Float64(-99.0 + Float64(-1000.0 / k)) - Float64(10000.0 / Float64(k * k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = 1.0 + (k * (k + 10.0));
	tmp = 0.0;
	if (m <= -4.1e+15)
		tmp = a / (t_0 + -1.0);
	elseif (m <= 1.68)
		tmp = a / t_0;
	else
		tmp = a / ((-99.0 + (-1000.0 / k)) - (10000.0 / (k * k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.1e+15], N[(a / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.68], N[(a / t$95$0), $MachinePrecision], N[(a / N[(N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision] - N[(10000.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.1e15

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 30.4%

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

   3. Taylor expanded in k around inf 35.0%

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

      1. unpow2 35.0%

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

      2. +-commutative 35.0%

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

      3. distribute-rgt-in 35.0%

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

   5. Simplified 35.0%

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

      1. expm1-log1p-u 35.0%

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

      2. expm1-udef 75.5%

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

      3. log1p-udef 75.5%

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

      4. add-exp-log 75.5%

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

   7. Applied egg-rr 75.5%

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

   if -4.1e15 < m < 1.67999999999999994

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

      3. sqr-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. +-commutative 91.4%

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

      6. sqr-neg 91.4%

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

      7. distribute-rgt-out 91.4%

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

      8. fma-def 91.4%

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

      9. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in m around 0 90.3%

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

   if 1.67999999999999994 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. +-commutative 2.9%

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

      4. flip-+ 2.9%

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

      5. associate-*l/ 2.9%

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

      6. metadata-eval 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in k around 0 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 33.1%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + \left(1000 \cdot \frac{1}{k} + 10000 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]
   11. Step-by-step derivation

       1. neg-sub0 33.1%

          \[\leadsto \frac{a}{\color{blue}{0 - \left(99 + \left(1000 \cdot \frac{1}{k} + 10000 \cdot \frac{1}{{k}^{2}}\right)\right)}} \]

       2. metadata-eval 33.1%

          \[\leadsto \frac{a}{\color{blue}{\log 1} - \left(99 + \left(1000 \cdot \frac{1}{k} + 10000 \cdot \frac{1}{{k}^{2}}\right)\right)} \]

       3. associate-+r+ 33.1%

          \[\leadsto \frac{a}{\log 1 - \color{blue}{\left(\left(99 + 1000 \cdot \frac{1}{k}\right) + 10000 \cdot \frac{1}{{k}^{2}}\right)}} \]

       4. associate--r+ 33.1%

          \[\leadsto \frac{a}{\color{blue}{\left(\log 1 - \left(99 + 1000 \cdot \frac{1}{k}\right)\right) - 10000 \cdot \frac{1}{{k}^{2}}}} \]

       5. metadata-eval 33.1%

          \[\leadsto \frac{a}{\left(\color{blue}{0} - \left(99 + 1000 \cdot \frac{1}{k}\right)\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       6. neg-sub0 33.1%

          \[\leadsto \frac{a}{\color{blue}{\left(-\left(99 + 1000 \cdot \frac{1}{k}\right)\right)} - 10000 \cdot \frac{1}{{k}^{2}}} \]

       7. distribute-neg-in 33.1%

          \[\leadsto \frac{a}{\color{blue}{\left(\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)\right)} - 10000 \cdot \frac{1}{{k}^{2}}} \]

       8. metadata-eval 33.1%

          \[\leadsto \frac{a}{\left(\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       9. associate-*r/ 33.1%

          \[\leadsto \frac{a}{\left(-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       10. metadata-eval 33.1%

           \[\leadsto \frac{a}{\left(-99 + \left(-\frac{\color{blue}{1000}}{k}\right)\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       11. distribute-neg-frac 33.1%

           \[\leadsto \frac{a}{\left(-99 + \color{blue}{\frac{-1000}{k}}\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       12. metadata-eval 33.1%

           \[\leadsto \frac{a}{\left(-99 + \frac{\color{blue}{-1000}}{k}\right) - 10000 \cdot \frac{1}{{k}^{2}}} \]

       13. associate-*r/ 33.1%

           \[\leadsto \frac{a}{\left(-99 + \frac{-1000}{k}\right) - \color{blue}{\frac{10000 \cdot 1}{{k}^{2}}}} \]

       14. metadata-eval 33.1%

           \[\leadsto \frac{a}{\left(-99 + \frac{-1000}{k}\right) - \frac{\color{blue}{10000}}{{k}^{2}}} \]

       15. unpow2 33.1%

           \[\leadsto \frac{a}{\left(-99 + \frac{-1000}{k}\right) - \frac{10000}{\color{blue}{k \cdot k}}} \]

   12. Simplified 33.1%

       \[\leadsto \frac{a}{\color{blue}{\left(-99 + \frac{-1000}{k}\right) - \frac{10000}{k \cdot k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot \left(k + 10\right)\right) + -1}\\ \mathbf{elif}\;m \leq 1.68:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(-99 + \frac{-1000}{k}\right) - \frac{10000}{k \cdot k}}\\ \end{array} \]

Alternative 9: 46.3% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -6.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k -6.2e-130)
   (/ a (* k k))
   (if (<= k 5.5e-304)
     (/ a (+ -99.0 (/ -1000.0 k)))
     (if (<= k 0.075) (+ a (* -10.0 (* k a))) (/ a (* k (+ k 10.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.2e-130) {
		tmp = a / (k * k);
	} else if (k <= 5.5e-304) {
		tmp = a / (-99.0 + (-1000.0 / k));
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = a / (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 (k <= (-6.2d-130)) then
        tmp = a / (k * k)
    else if (k <= 5.5d-304) then
        tmp = a / ((-99.0d0) + ((-1000.0d0) / k))
    else if (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= -6.2e-130) {
		tmp = a / (k * k);
	} else if (k <= 5.5e-304) {
		tmp = a / (-99.0 + (-1000.0 / k));
	} else if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = a / (k * (k + 10.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= -6.2e-130:
		tmp = a / (k * k)
	elif k <= 5.5e-304:
		tmp = a / (-99.0 + (-1000.0 / k))
	elif k <= 0.075:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= -6.2e-130)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 5.5e-304)
		tmp = Float64(a / Float64(-99.0 + Float64(-1000.0 / k)));
	elseif (k <= 0.075)
		tmp = Float64(a + Float64(-10.0 * Float64(k * a)));
	else
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= -6.2e-130)
		tmp = a / (k * k);
	elseif (k <= 5.5e-304)
		tmp = a / (-99.0 + (-1000.0 / k));
	elseif (k <= 0.075)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, -6.2e-130], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.5e-304], N[(a / N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if k < -6.20000000000000021e-130

   1. Initial program 87.3%

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

      1. associate-*r/ 87.3%

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

      2. *-commutative 87.3%

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

      3. sqr-neg 87.3%

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

      4. associate-+l+ 87.3%

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

      5. +-commutative 87.3%

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

      6. sqr-neg 87.3%

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

      7. distribute-rgt-out 87.3%

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

      8. fma-def 87.3%

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

      9. +-commutative 87.3%

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

   3. Simplified 87.3%

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

   4. Taylor expanded in m around 0 17.2%

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

   5. Taylor expanded in k around inf 20.4%

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

      1. unpow2 20.4%

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

   7. Simplified 20.4%

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

   if -6.20000000000000021e-130 < k < 5.50000000000000035e-304

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 3.0%

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

      1. +-commutative 3.0%

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

      2. *-commutative 3.0%

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

      3. +-commutative 3.0%

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

      4. flip-+ 3.0%

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

      5. associate-*l/ 3.0%

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

      6. metadata-eval 3.0%

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

   6. Applied egg-rr 3.0%

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

   7. Taylor expanded in k around 0 3.0%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 3.0%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 3.0%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 35.6%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + 1000 \cdot \frac{1}{k}\right)}} \]
   11. Step-by-step derivation

       1. distribute-neg-in 35.6%

          \[\leadsto \frac{a}{\color{blue}{\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)}} \]

       2. metadata-eval 35.6%

          \[\leadsto \frac{a}{\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)} \]

       3. associate-*r/ 35.6%

          \[\leadsto \frac{a}{-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)} \]

       4. metadata-eval 35.6%

          \[\leadsto \frac{a}{-99 + \left(-\frac{\color{blue}{1000}}{k}\right)} \]

       5. distribute-neg-frac 35.6%

          \[\leadsto \frac{a}{-99 + \color{blue}{\frac{-1000}{k}}} \]

       6. metadata-eval 35.6%

          \[\leadsto \frac{a}{-99 + \frac{\color{blue}{-1000}}{k}} \]

   12. Simplified 35.6%

       \[\leadsto \frac{a}{\color{blue}{-99 + \frac{-1000}{k}}} \]

   if 5.50000000000000035e-304 < k < 0.0749999999999999972

   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. *-commutative 99.9%

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

      3. sqr-neg 99.9%

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

      4. associate-+l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. sqr-neg 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 53.6%

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

   5. Taylor expanded in k around 0 52.2%

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

   if 0.0749999999999999972 < k

   1. Initial program 80.1%

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

   2. Taylor expanded in m around 0 54.9%

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

   3. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

      2. +-commutative 54.9%

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

      3. distribute-rgt-in 54.9%

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

   5. Simplified 54.9%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6.2 \cdot 10^{-130}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \mathbf{elif}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 10: 56.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.82:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.1e+15)
   (/ a (* k k))
   (if (<= m 0.82)
     (/ a (+ 1.0 (* k (+ k 10.0))))
     (/ a (+ -99.0 (/ -1000.0 k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (k * k);
	} else if (m <= 0.82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (-99.0 + (-1000.0 / 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) :: tmp
    if (m <= (-4.1d+15)) then
        tmp = a / (k * k)
    else if (m <= 0.82d0) then
        tmp = a / (1.0d0 + (k * (k + 10.0d0)))
    else
        tmp = a / ((-99.0d0) + ((-1000.0d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (k * k);
	} else if (m <= 0.82) {
		tmp = a / (1.0 + (k * (k + 10.0)));
	} else {
		tmp = a / (-99.0 + (-1000.0 / k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.1e+15:
		tmp = a / (k * k)
	elif m <= 0.82:
		tmp = a / (1.0 + (k * (k + 10.0)))
	else:
		tmp = a / (-99.0 + (-1000.0 / k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.1e+15)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.82)
		tmp = Float64(a / Float64(1.0 + Float64(k * Float64(k + 10.0))));
	else
		tmp = Float64(a / Float64(-99.0 + Float64(-1000.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.1e+15)
		tmp = a / (k * k);
	elseif (m <= 0.82)
		tmp = a / (1.0 + (k * (k + 10.0)));
	else
		tmp = a / (-99.0 + (-1000.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.1e+15], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.82], N[(a / N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.1e15

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 30.4%

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

   5. Taylor expanded in k around inf 58.5%

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

      1. unpow2 58.5%

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

   7. Simplified 58.5%

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

   if -4.1e15 < m < 0.819999999999999951

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

      3. sqr-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. +-commutative 91.4%

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

      6. sqr-neg 91.4%

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

      7. distribute-rgt-out 91.4%

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

      8. fma-def 91.4%

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

      9. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in m around 0 90.3%

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

   if 0.819999999999999951 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. +-commutative 2.9%

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

      4. flip-+ 2.9%

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

      5. associate-*l/ 2.9%

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

      6. metadata-eval 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in k around 0 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 13.5%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + 1000 \cdot \frac{1}{k}\right)}} \]
   11. Step-by-step derivation

       1. distribute-neg-in 13.5%

          \[\leadsto \frac{a}{\color{blue}{\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)}} \]

       2. metadata-eval 13.5%

          \[\leadsto \frac{a}{\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)} \]

       3. associate-*r/ 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)} \]

       4. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\frac{\color{blue}{1000}}{k}\right)} \]

       5. distribute-neg-frac 13.5%

          \[\leadsto \frac{a}{-99 + \color{blue}{\frac{-1000}{k}}} \]

       6. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \frac{\color{blue}{-1000}}{k}} \]

   12. Simplified 13.5%

       \[\leadsto \frac{a}{\color{blue}{-99 + \frac{-1000}{k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.82:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \]

Alternative 11: 60.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + k \cdot \left(k + 10\right)\\ \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{t_0 + -1}\\ \mathbf{elif}\;m \leq 1.7:\\ \;\;\;\;\frac{a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* k (+ k 10.0)))))
   (if (<= m -4.1e+15)
     (/ a (+ t_0 -1.0))
     (if (<= m 1.7) (/ a t_0) (/ a (+ -99.0 (/ -1000.0 k)))))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 + (k * (k + 10.0));
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (t_0 + -1.0);
	} else if (m <= 1.7) {
		tmp = a / t_0;
	} else {
		tmp = a / (-99.0 + (-1000.0 / 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 = 1.0d0 + (k * (k + 10.0d0))
    if (m <= (-4.1d+15)) then
        tmp = a / (t_0 + (-1.0d0))
    else if (m <= 1.7d0) then
        tmp = a / t_0
    else
        tmp = a / ((-99.0d0) + ((-1000.0d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = 1.0 + (k * (k + 10.0));
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (t_0 + -1.0);
	} else if (m <= 1.7) {
		tmp = a / t_0;
	} else {
		tmp = a / (-99.0 + (-1000.0 / k));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = 1.0 + (k * (k + 10.0))
	tmp = 0
	if m <= -4.1e+15:
		tmp = a / (t_0 + -1.0)
	elif m <= 1.7:
		tmp = a / t_0
	else:
		tmp = a / (-99.0 + (-1000.0 / k))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(1.0 + Float64(k * Float64(k + 10.0)))
	tmp = 0.0
	if (m <= -4.1e+15)
		tmp = Float64(a / Float64(t_0 + -1.0));
	elseif (m <= 1.7)
		tmp = Float64(a / t_0);
	else
		tmp = Float64(a / Float64(-99.0 + Float64(-1000.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = 1.0 + (k * (k + 10.0));
	tmp = 0.0;
	if (m <= -4.1e+15)
		tmp = a / (t_0 + -1.0);
	elseif (m <= 1.7)
		tmp = a / t_0;
	else
		tmp = a / (-99.0 + (-1000.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 + N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.1e+15], N[(a / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.7], N[(a / t$95$0), $MachinePrecision], N[(a / N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.1e15

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0 30.4%

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

   3. Taylor expanded in k around inf 35.0%

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

      1. unpow2 35.0%

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

      2. +-commutative 35.0%

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

      3. distribute-rgt-in 35.0%

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

   5. Simplified 35.0%

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

      1. expm1-log1p-u 35.0%

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

      2. expm1-udef 75.5%

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

      3. log1p-udef 75.5%

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

      4. add-exp-log 75.5%

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

   7. Applied egg-rr 75.5%

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

   if -4.1e15 < m < 1.69999999999999996

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

      3. sqr-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. +-commutative 91.4%

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

      6. sqr-neg 91.4%

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

      7. distribute-rgt-out 91.4%

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

      8. fma-def 91.4%

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

      9. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in m around 0 90.3%

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

   if 1.69999999999999996 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. +-commutative 2.9%

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

      4. flip-+ 2.9%

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

      5. associate-*l/ 2.9%

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

      6. metadata-eval 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in k around 0 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 13.5%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + 1000 \cdot \frac{1}{k}\right)}} \]
   11. Step-by-step derivation

       1. distribute-neg-in 13.5%

          \[\leadsto \frac{a}{\color{blue}{\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)}} \]

       2. metadata-eval 13.5%

          \[\leadsto \frac{a}{\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)} \]

       3. associate-*r/ 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)} \]

       4. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\frac{\color{blue}{1000}}{k}\right)} \]

       5. distribute-neg-frac 13.5%

          \[\leadsto \frac{a}{-99 + \color{blue}{\frac{-1000}{k}}} \]

       6. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \frac{\color{blue}{-1000}}{k}} \]

   12. Simplified 13.5%

       \[\leadsto \frac{a}{\color{blue}{-99 + \frac{-1000}{k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{\left(1 + k \cdot \left(k + 10\right)\right) + -1}\\ \mathbf{elif}\;m \leq 1.7:\\ \;\;\;\;\frac{a}{1 + k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \]

Alternative 12: 46.3% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.76:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.2e-17)
   (/ a (* k k))
   (if (<= m 0.76) (/ a (+ 1.0 (* k 10.0))) (/ a (+ -99.0 (/ -1000.0 k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.2e-17) {
		tmp = a / (k * k);
	} else if (m <= 0.76) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (-99.0 + (-1000.0 / 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) :: tmp
    if (m <= (-1.2d-17)) then
        tmp = a / (k * k)
    else if (m <= 0.76d0) then
        tmp = a / (1.0d0 + (k * 10.0d0))
    else
        tmp = a / ((-99.0d0) + ((-1000.0d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.2e-17) {
		tmp = a / (k * k);
	} else if (m <= 0.76) {
		tmp = a / (1.0 + (k * 10.0));
	} else {
		tmp = a / (-99.0 + (-1000.0 / k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -1.2e-17:
		tmp = a / (k * k)
	elif m <= 0.76:
		tmp = a / (1.0 + (k * 10.0))
	else:
		tmp = a / (-99.0 + (-1000.0 / k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.2e-17)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.76)
		tmp = Float64(a / Float64(1.0 + Float64(k * 10.0)));
	else
		tmp = Float64(a / Float64(-99.0 + Float64(-1000.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -1.2e-17)
		tmp = a / (k * k);
	elseif (m <= 0.76)
		tmp = a / (1.0 + (k * 10.0));
	else
		tmp = a / (-99.0 + (-1000.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.2e-17], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.76], N[(a / N[(1.0 + N[(k * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.19999999999999993e-17

   1. Initial program 98.7%

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

      1. associate-*r/ 98.7%

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

      2. *-commutative 98.7%

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

      3. sqr-neg 98.7%

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

      4. associate-+l+ 98.7%

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

      5. +-commutative 98.7%

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

      6. sqr-neg 98.7%

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

      7. distribute-rgt-out 98.7%

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

      8. fma-def 98.7%

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

      9. +-commutative 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in m around 0 36.4%

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

   5. Taylor expanded in k around inf 60.7%

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

      1. unpow2 60.7%

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

   7. Simplified 60.7%

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

   if -1.19999999999999993e-17 < m < 0.76000000000000001

   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. *-commutative 91.6%

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

      3. sqr-neg 91.6%

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

      4. associate-+l+ 91.6%

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

      5. +-commutative 91.6%

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

      6. sqr-neg 91.6%

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

      7. distribute-rgt-out 91.6%

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

      8. fma-def 91.6%

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

      9. +-commutative 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in m around 0 91.6%

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

   5. Taylor expanded in k around 0 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   if 0.76000000000000001 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. +-commutative 2.9%

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

      4. flip-+ 2.9%

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

      5. associate-*l/ 2.9%

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

      6. metadata-eval 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in k around 0 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 13.5%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + 1000 \cdot \frac{1}{k}\right)}} \]
   11. Step-by-step derivation

       1. distribute-neg-in 13.5%

          \[\leadsto \frac{a}{\color{blue}{\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)}} \]

       2. metadata-eval 13.5%

          \[\leadsto \frac{a}{\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)} \]

       3. associate-*r/ 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)} \]

       4. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\frac{\color{blue}{1000}}{k}\right)} \]

       5. distribute-neg-frac 13.5%

          \[\leadsto \frac{a}{-99 + \color{blue}{\frac{-1000}{k}}} \]

       6. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \frac{\color{blue}{-1000}}{k}} \]

   12. Simplified 13.5%

       \[\leadsto \frac{a}{\color{blue}{-99 + \frac{-1000}{k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.76:\\ \;\;\;\;\frac{a}{1 + k \cdot 10}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \]

Alternative 13: 55.4% accurate, 10.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.16:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.1e+15)
   (/ a (* k k))
   (if (<= m 1.16) (/ a (+ 1.0 (* k k))) (/ a (+ -99.0 (/ -1000.0 k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (k * k);
	} else if (m <= 1.16) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a / (-99.0 + (-1000.0 / 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) :: tmp
    if (m <= (-4.1d+15)) then
        tmp = a / (k * k)
    else if (m <= 1.16d0) then
        tmp = a / (1.0d0 + (k * k))
    else
        tmp = a / ((-99.0d0) + ((-1000.0d0) / k))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1e+15) {
		tmp = a / (k * k);
	} else if (m <= 1.16) {
		tmp = a / (1.0 + (k * k));
	} else {
		tmp = a / (-99.0 + (-1000.0 / k));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -4.1e+15:
		tmp = a / (k * k)
	elif m <= 1.16:
		tmp = a / (1.0 + (k * k))
	else:
		tmp = a / (-99.0 + (-1000.0 / k))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.1e+15)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.16)
		tmp = Float64(a / Float64(1.0 + Float64(k * k)));
	else
		tmp = Float64(a / Float64(-99.0 + Float64(-1000.0 / k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -4.1e+15)
		tmp = a / (k * k);
	elseif (m <= 1.16)
		tmp = a / (1.0 + (k * k));
	else
		tmp = a / (-99.0 + (-1000.0 / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.1e+15], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.16], N[(a / N[(1.0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(-99.0 + N[(-1000.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.1e15

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in m around 0 30.4%

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

   5. Taylor expanded in k around inf 58.5%

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

      1. unpow2 58.5%

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

   7. Simplified 58.5%

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

   if -4.1e15 < m < 1.15999999999999992

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. *-commutative 91.4%

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

      3. sqr-neg 91.4%

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

      4. associate-+l+ 91.4%

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

      5. +-commutative 91.4%

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

      6. sqr-neg 91.4%

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

      7. distribute-rgt-out 91.4%

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

      8. fma-def 91.4%

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

      9. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in m around 0 90.3%

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

   5. Taylor expanded in k around inf 87.6%

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

      1. unpow2 87.6%

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

   7. Simplified 87.6%

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

   if 1.15999999999999992 < m

   1. Initial program 82.5%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. sqr-neg 82.5%

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

      4. associate-+l+ 82.5%

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

      5. +-commutative 82.5%

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

      6. sqr-neg 82.5%

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

      7. distribute-rgt-out 82.5%

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

      8. fma-def 82.5%

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

      9. +-commutative 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in m around 0 2.9%

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

      1. +-commutative 2.9%

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

      2. *-commutative 2.9%

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

      3. +-commutative 2.9%

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

      4. flip-+ 2.9%

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

      5. associate-*l/ 2.9%

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

      6. metadata-eval 2.9%

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

   6. Applied egg-rr 2.9%

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

   7. Taylor expanded in k around 0 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{100 \cdot k}}{10 - k}} \]
   8. Step-by-step derivation

      1. *-commutative 2.3%

         \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   9. Simplified 2.3%

      \[\leadsto \frac{a}{1 + \frac{\color{blue}{k \cdot 100}}{10 - k}} \]

   10. Taylor expanded in k around inf 13.5%

       \[\leadsto \frac{a}{\color{blue}{-\left(99 + 1000 \cdot \frac{1}{k}\right)}} \]
   11. Step-by-step derivation

       1. distribute-neg-in 13.5%

          \[\leadsto \frac{a}{\color{blue}{\left(-99\right) + \left(-1000 \cdot \frac{1}{k}\right)}} \]

       2. metadata-eval 13.5%

          \[\leadsto \frac{a}{\color{blue}{-99} + \left(-1000 \cdot \frac{1}{k}\right)} \]

       3. associate-*r/ 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\color{blue}{\frac{1000 \cdot 1}{k}}\right)} \]

       4. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \left(-\frac{\color{blue}{1000}}{k}\right)} \]

       5. distribute-neg-frac 13.5%

          \[\leadsto \frac{a}{-99 + \color{blue}{\frac{-1000}{k}}} \]

       6. metadata-eval 13.5%

          \[\leadsto \frac{a}{-99 + \frac{\color{blue}{-1000}}{k}} \]

   12. Simplified 13.5%

       \[\leadsto \frac{a}{\color{blue}{-99 + \frac{-1000}{k}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -4.1 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.16:\\ \;\;\;\;\frac{a}{1 + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-99 + \frac{-1000}{k}}\\ \end{array} \]

Alternative 14: 44.1% accurate, 12.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{-233} \lor \neg \left(k \leq 1.4 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= k -2.2e-233) (not (<= k 1.4e+30))) (/ a (* k k)) a))

C

double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.2e-233) || !(k <= 1.4e+30)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if ((k <= (-2.2d-233)) .or. (.not. (k <= 1.4d+30))) then
        tmp = a / (k * k)
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if ((k <= -2.2e-233) || !(k <= 1.4e+30)) {
		tmp = a / (k * k);
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if (k <= -2.2e-233) or not (k <= 1.4e+30):
		tmp = a / (k * k)
	else:
		tmp = a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if ((k <= -2.2e-233) || !(k <= 1.4e+30))
		tmp = Float64(a / Float64(k * k));
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if ((k <= -2.2e-233) || ~((k <= 1.4e+30)))
		tmp = a / (k * k);
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[k, -2.2e-233], N[Not[LessEqual[k, 1.4e+30]], $MachinePrecision]], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], a]

Derivation

1. Split input into 2 regimes

2. if k < -2.2e-233 or 1.39999999999999992e30 < k

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. *-commutative 83.1%

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

      3. sqr-neg 83.1%

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

      4. associate-+l+ 83.1%

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

      5. +-commutative 83.1%

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

      6. sqr-neg 83.1%

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

      7. distribute-rgt-out 83.1%

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

      8. fma-def 83.1%

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

      9. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   4. Taylor expanded in m around 0 40.1%

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

   5. Taylor expanded in k around inf 41.8%

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

      1. unpow2 41.8%

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

   7. Simplified 41.8%

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

   if -2.2e-233 < k < 1.39999999999999992e30

   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. *-commutative 100.0%

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

      3. sqr-neg 100.0%

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

      4. associate-+l+ 100.0%

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

      5. +-commutative 100.0%

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

      6. sqr-neg 100.0%

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

      7. distribute-rgt-out 100.0%

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

      8. fma-def 100.0%

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

      9. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in k around 0 97.5%

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

   5. Taylor expanded in m around 0 43.9%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{-233} \lor \neg \left(k \leq 1.4 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 15: 41.0% accurate, 12.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if k < 0.10000000000000001

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-+l+ 95.7%

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

      5. +-commutative 95.7%

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

      6. sqr-neg 95.7%

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

      7. distribute-rgt-out 95.7%

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

      8. fma-def 95.7%

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

      9. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 35.8%

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

   5. Taylor expanded in k around 0 34.5%

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

   if 0.10000000000000001 < k

   1. Initial program 80.1%

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

      1. associate-*r/ 80.1%

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

      2. *-commutative 80.1%

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

      3. sqr-neg 80.1%

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

      4. associate-+l+ 80.1%

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

      5. +-commutative 80.1%

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

      6. sqr-neg 80.1%

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

      7. distribute-rgt-out 80.1%

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

      8. fma-def 80.1%

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

      9. +-commutative 80.1%

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

   3. Simplified 80.1%

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

   4. Taylor expanded in m around 0 54.9%

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

   5. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

   7. Simplified 54.9%

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

4. Final simplification 41.8%

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

Alternative 16: 41.2% accurate, 12.6× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 0.075) (+ a (* -10.0 (* k a))) (/ a (* k (+ k 10.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 0.075) {
		tmp = a + (-10.0 * (k * a));
	} else {
		tmp = a / (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 (k <= 0.075d0) then
        tmp = a + ((-10.0d0) * (k * a))
    else
        tmp = a / (k * (k + 10.0d0))
    end if
    code = tmp
end function

Java

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

Python

def code(a, k, m):
	tmp = 0
	if k <= 0.075:
		tmp = a + (-10.0 * (k * a))
	else:
		tmp = a / (k * (k + 10.0))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 0.075)
		tmp = a + (-10.0 * (k * a));
	else
		tmp = a / (k * (k + 10.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 0.075], N[(a + N[(-10.0 * N[(k * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 0.0749999999999999972

   1. Initial program 95.7%

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

      1. associate-*r/ 95.7%

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

      2. *-commutative 95.7%

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

      3. sqr-neg 95.7%

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

      4. associate-+l+ 95.7%

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

      5. +-commutative 95.7%

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

      6. sqr-neg 95.7%

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

      7. distribute-rgt-out 95.7%

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

      8. fma-def 95.7%

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

      9. +-commutative 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in m around 0 35.8%

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

   5. Taylor expanded in k around 0 34.5%

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

   if 0.0749999999999999972 < k

   1. Initial program 80.1%

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

   2. Taylor expanded in m around 0 54.9%

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

   3. Taylor expanded in k around inf 54.9%

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

      1. unpow2 54.9%

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

      2. +-commutative 54.9%

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

      3. distribute-rgt-in 54.9%

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

   5. Simplified 54.9%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.075:\\ \;\;\;\;a + -10 \cdot \left(k \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \end{array} \]

Alternative 17: 26.5% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= k 2.05e+30) a (/ a (* k 10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.05e+30) {
		tmp = a;
	} else {
		tmp = a / (k * 10.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 2.05d+30) then
        tmp = a
    else
        tmp = a / (k * 10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.05e+30) {
		tmp = a;
	} else {
		tmp = a / (k * 10.0);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.05e+30:
		tmp = a
	else:
		tmp = a / (k * 10.0)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.05e+30)
		tmp = a;
	else
		tmp = Float64(a / Float64(k * 10.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.05e+30)
		tmp = a;
	else
		tmp = a / (k * 10.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.05e+30], a, N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.05000000000000003e30

   1. Initial program 95.9%

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

      1. associate-*r/ 95.9%

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

      2. *-commutative 95.9%

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

      3. sqr-neg 95.9%

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

      4. associate-+l+ 95.9%

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

      5. +-commutative 95.9%

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

      6. sqr-neg 95.9%

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

      7. distribute-rgt-out 95.9%

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

      8. fma-def 95.9%

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

      9. +-commutative 95.9%

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

   3. Simplified 95.9%

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

   4. Taylor expanded in k around 0 98.4%

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

   5. Taylor expanded in m around 0 28.6%

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

   if 2.05000000000000003e30 < k

   1. Initial program 78.5%

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

      1. associate-*r/ 78.4%

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

      2. *-commutative 78.4%

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

      3. sqr-neg 78.4%

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

      4. associate-+l+ 78.4%

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

      5. +-commutative 78.4%

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

      6. sqr-neg 78.4%

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

      7. distribute-rgt-out 78.4%

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

      8. fma-def 78.4%

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

      9. +-commutative 78.4%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in m around 0 59.1%

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

   5. Taylor expanded in k around 0 32.8%

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

      1. *-commutative 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in k around inf 32.8%

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

      1. *-commutative 32.8%

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

   10. Simplified 32.8%

       \[\leadsto \frac{a}{\color{blue}{k \cdot 10}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.05 \cdot 10^{+30}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \end{array} \]

Alternative 18: 20.5% 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 90.1%

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

   1. associate-*r/ 90.1%

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

   2. *-commutative 90.1%

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

   3. sqr-neg 90.1%

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

   4. associate-+l+ 90.1%

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

   5. +-commutative 90.1%

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

   6. sqr-neg 90.1%

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

   7. distribute-rgt-out 90.1%

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

   8. fma-def 90.1%

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

   9. +-commutative 90.1%

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

3. Simplified 90.1%

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

4. Taylor expanded in k around 0 81.8%

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

5. Taylor expanded in m around 0 20.7%

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

6. Final simplification 20.7%

   \[\leadsto a \]

Reproduce

herbie shell --seed 2023271 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))