Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 98.6%

Time: 11.6s

Alternatives: 16

Speedup: 1.2×

• Could not converge on a ground truth

  1. a = 3.1051161665316605e+178
  2. k = -2.5048446669720267e+157
  3. m = -1.3385900517361169e+231

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.6% 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: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;k \leq 7 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t\_0}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m)))) (if (<= k 7e-15) t_0 (/ (/ t_0 k) k))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (k <= 7e-15) {
		tmp = t_0;
	} else {
		tmp = (t_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 = a * (k ** m)
    if (k <= 7d-15) then
        tmp = t_0
    else
        tmp = (t_0 / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a * Math.pow(k, m);
	double tmp;
	if (k <= 7e-15) {
		tmp = t_0;
	} else {
		tmp = (t_0 / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a * math.pow(k, m)
	tmp = 0
	if k <= 7e-15:
		tmp = t_0
	else:
		tmp = (t_0 / k) / k
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (k <= 7e-15)
		tmp = t_0;
	else
		tmp = Float64(Float64(t_0 / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a * (k ^ m);
	tmp = 0.0;
	if (k <= 7e-15)
		tmp = t_0;
	else
		tmp = (t_0 / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 7e-15], t$95$0, N[(N[(t$95$0 / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000001e-15

   1. Initial program 95.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

   if 7.0000000000000001e-15 < k

   1. Initial program 81.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   7. Applied egg-rr 99.6%

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

Alternative 2: 59.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \left(\frac{100}{k} + \frac{1000}{k \cdot k}\right)}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{fma}\left(k, k + 10, -1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/
      a
      (fma
       k
       (*
        (fma k k -100.0)
        (/ (+ (/ (+ 10.0 (+ (/ 100.0 k) (/ 1000.0 (* k k)))) k) 1.0) k))
       1.0))
     (if (<= t_0 2e+264)
       (/ a (fma k (+ k 10.0) 1.0))
       (* a (+ (* (* k (+ k 10.0)) (fma k (+ k 10.0) -1.0)) 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * ((((10.0 + ((100.0 / k) + (1000.0 / (k * k)))) / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 2e+264) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (((k * (k + 10.0)) * fma(k, (k + 10.0), -1.0)) + 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(Float64(10.0 + Float64(Float64(100.0 / k) + Float64(1000.0 / Float64(k * k)))) / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 2e+264)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(Float64(k * Float64(k + 10.0)) * fma(k, Float64(k + 10.0), -1.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(N[(10.0 + N[(N[(100.0 / k), $MachinePrecision] + N[(1000.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+264], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k + 10.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 53.2

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

   5. Simplified 53.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 53.2

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

   7. Applied egg-rr 53.2%

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

   8. Taylor expanded in k around -inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}{k}\right)}, 1\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}{k}\right)\right)}, 1\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}{\mathsf{neg}\left(k\right)}}, 1\right)} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}{\color{blue}{-1 \cdot k}}, 1\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1 \cdot \frac{10 + \left(100 \cdot \frac{1}{k} + \frac{1000}{{k}^{2}}\right)}{k} - 1}{-1 \cdot k}}, 1\right)} \]

   10. Simplified 60.0%

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{-1 - \frac{10 + \left(\frac{100}{k} + \frac{1000}{k \cdot k}\right)}{k}}{-k}}, 1\right)} \]

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000009e264

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 98.1

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

   5. Simplified 98.1%

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

   if 2.00000000000000009e264 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 53.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Simplified 2.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip3-+ N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   7. Applied egg-rr 1.4%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 72.3%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10 + \left(\frac{100}{k} + \frac{1000}{k \cdot k}\right)}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{fma}\left(k, k + 10, -1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{fma}\left(k, k + 10, -1\right) + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ (* k 10.0) 1.0) (* k k)))))
   (if (<= t_0 0.0)
     (/ a (fma k (* (fma k k -100.0) (/ (+ (/ 10.0 k) 1.0) k)) 1.0))
     (if (<= t_0 2e+264)
       (/ a (fma k (+ k 10.0) 1.0))
       (* a (+ (* (* k (+ k 10.0)) (fma k (+ k 10.0) -1.0)) 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / (((k * 10.0) + 1.0) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = a / fma(k, (fma(k, k, -100.0) * (((10.0 / k) + 1.0) / k)), 1.0);
	} else if (t_0 <= 2e+264) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = a * (((k * (k + 10.0)) * fma(k, (k + 10.0), -1.0)) + 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(Float64(k * 10.0) + 1.0) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(a / fma(k, Float64(fma(k, k, -100.0) * Float64(Float64(Float64(10.0 / k) + 1.0) / k)), 1.0));
	elseif (t_0 <= 2e+264)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(a * Float64(Float64(Float64(k * Float64(k + 10.0)) * fma(k, Float64(k + 10.0), -1.0)) + 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * 10.0), $MachinePrecision] + 1.0), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(a / N[(k * N[(N[(k * k + -100.0), $MachinePrecision] * N[(N[(N[(10.0 / k), $MachinePrecision] + 1.0), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+264], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision] * N[(k * N[(k + 10.0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0

   1. Initial program 96.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 53.2

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

   5. Simplified 53.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 53.2

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

   7. Applied egg-rr 53.2%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \color{blue}{\frac{1 + 10 \cdot \frac{1}{k}}{k}}, 1\right)} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 51.6

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

   10. Simplified 51.6%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000009e264

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 98.1

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

   5. Simplified 98.1%

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

   if 2.00000000000000009e264 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 53.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 2.6

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

   5. Simplified 2.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip3-+ N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

   7. Applied egg-rr 1.4%

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

   8. Taylor expanded in k around 0

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

   10. Simplified 72.3%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 0:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, k, -100\right) \cdot \frac{\frac{10}{k} + 1}{k}, 1\right)}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(k \cdot 10 + 1\right) + k \cdot k} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(k \cdot \left(k + 10\right)\right) \cdot \mathsf{fma}\left(k, k + 10, -1\right) + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-15}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7e-15) (* a (pow k m)) (/ (* a (pow k (+ m -1.0))) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-15) {
		tmp = a * pow(k, m);
	} else {
		tmp = (a * pow(k, (m + -1.0))) / k;
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 7d-15) then
        tmp = a * (k ** m)
    else
        tmp = (a * (k ** (m + (-1.0d0)))) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-15) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = (a * Math.pow(k, (m + -1.0))) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7e-15:
		tmp = a * math.pow(k, m)
	else:
		tmp = (a * math.pow(k, (m + -1.0))) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7e-15)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(Float64(a * (k ^ Float64(m + -1.0))) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7e-15)
		tmp = a * (k ^ m);
	else
		tmp = (a * (k ^ (m + -1.0))) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7e-15], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[k, N[(m + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000001e-15

   1. Initial program 95.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

   if 7.0000000000000001e-15 < k

   1. Initial program 81.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   7. Applied egg-rr 99.6%

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

      1. lift-pow.f64 N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-inv N/A

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

      6. lift-pow.f64 N/A

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

      7. inv-pow N/A

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

      8. pow-prod-up N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-+.f64 99.2

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

   9. Applied egg-rr 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-15}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{\left(m + -1\right)}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -30000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.0058:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -30000.0)
     t_0
     (if (<= m 0.0058) (/ a (fma k (+ k 10.0) 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -30000.0) {
		tmp = t_0;
	} else if (m <= 0.0058) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -30000.0)
		tmp = t_0;
	elseif (m <= 0.0058)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -30000.0], t$95$0, If[LessEqual[m, 0.0058], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -3e4 or 0.0058 < m

   1. Initial program 87.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 100.0

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

   5. Simplified 100.0%

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

   if -3e4 < m < 0.0058

   1. Initial program 94.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 93.4

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

   5. Simplified 93.4%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -30000:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 0.0058:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-15}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 7e-15) (* a (pow k m)) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-15) {
		tmp = a * pow(k, m);
	} else {
		tmp = a * pow(k, (m + -2.0));
	}
	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 <= 7d-15) then
        tmp = a * (k ** m)
    else
        tmp = a * (k ** (m + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 7e-15) {
		tmp = a * Math.pow(k, m);
	} else {
		tmp = a * Math.pow(k, (m + -2.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 7e-15:
		tmp = a * math.pow(k, m)
	else:
		tmp = a * math.pow(k, (m + -2.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 7e-15)
		tmp = Float64(a * (k ^ m));
	else
		tmp = Float64(a * (k ^ Float64(m + -2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 7e-15)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 7e-15], N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision], N[(a * N[Power[k, N[(m + -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.0000000000000001e-15

   1. Initial program 95.3%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 99.9

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

   5. Simplified 99.9%

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

   if 7.0000000000000001e-15 < k

   1. Initial program 81.4%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.0

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

   5. Simplified 81.0%

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

      1. lift-pow.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-div N/A

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

      10. lower-pow.f64 N/A

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

      11. sub-neg N/A

          \[\leadsto {k}^{\color{blue}{\left(m + \left(\mathsf{neg}\left(2\right)\right)\right)}} \cdot a \]

      12. lower-+.f64 N/A

          \[\leadsto {k}^{\color{blue}{\left(m + \left(\mathsf{neg}\left(2\right)\right)\right)}} \cdot a \]

      13. metadata-eval 94.2

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

   7. Applied egg-rr 94.2%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-15}:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -30000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, -10 + \frac{99}{k}, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -30000.0)
   (/ (fma (/ a k) (+ -10.0 (/ 99.0 k)) a) (* k k))
   (if (<= m 0.92) (/ a (fma k (+ k 10.0) 1.0)) (* k (* k (* a 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -30000.0) {
		tmp = fma((a / k), (-10.0 + (99.0 / k)), a) / (k * k);
	} else if (m <= 0.92) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -30000.0)
		tmp = Float64(fma(Float64(a / k), Float64(-10.0 + Float64(99.0 / k)), a) / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -30000.0], N[(N[(N[(a / k), $MachinePrecision] * N[(-10.0 + N[(99.0 / k), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3e4

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 51.3

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

   5. Simplified 51.3%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. lower-+.f64 51.3

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

   7. Applied egg-rr 51.3%

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

   8. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\frac{\left(a + -1 \cdot \frac{a + -100 \cdot a}{{k}^{2}}\right) - 10 \cdot \frac{a}{k}}{{k}^{2}}} \]
   9. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \frac{\color{blue}{a + \left(-1 \cdot \frac{a + -100 \cdot a}{{k}^{2}} - 10 \cdot \frac{a}{k}\right)}}{{k}^{2}} \]

      2. cancel-sign-sub-inv N/A

         \[\leadsto \frac{a + \color{blue}{\left(-1 \cdot \frac{a + -100 \cdot a}{{k}^{2}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}}{{k}^{2}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{a + \left(\color{blue}{\left(\mathsf{neg}\left(\frac{a + -100 \cdot a}{{k}^{2}}\right)\right)} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      4. distribute-neg-frac N/A

         \[\leadsto \frac{a + \left(\color{blue}{\frac{\mathsf{neg}\left(\left(a + -100 \cdot a\right)\right)}{{k}^{2}}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      5. distribute-rgt1-in N/A

         \[\leadsto \frac{a + \left(\frac{\mathsf{neg}\left(\color{blue}{\left(-100 + 1\right) \cdot a}\right)}{{k}^{2}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{a + \left(\frac{\mathsf{neg}\left(\color{blue}{-99} \cdot a\right)}{{k}^{2}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{a + \left(\frac{\color{blue}{\left(\mathsf{neg}\left(-99\right)\right) \cdot a}}{{k}^{2}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{a + \left(\frac{\color{blue}{99} \cdot a}{{k}^{2}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{a + \left(\color{blue}{99 \cdot \frac{a}{{k}^{2}}} + \left(\mathsf{neg}\left(10\right)\right) \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{a + \left(99 \cdot \frac{a}{{k}^{2}} + \color{blue}{-10} \cdot \frac{a}{k}\right)}{{k}^{2}} \]

      11. +-commutative N/A

          \[\leadsto \frac{a + \color{blue}{\left(-10 \cdot \frac{a}{k} + 99 \cdot \frac{a}{{k}^{2}}\right)}}{{k}^{2}} \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + \left(-10 \cdot \frac{a}{k} + 99 \cdot \frac{a}{{k}^{2}}\right)}{{k}^{2}}} \]

   10. Simplified 72.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{a}{k}, -10 + \frac{99}{k}, a\right)}{k \cdot k}} \]

   if -3e4 < m < 0.92000000000000004

   1. Initial program 94.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 93.4

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

   5. Simplified 93.4%

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

   if 0.92000000000000004 < m

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Simplified 3.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 51.4

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

   11. Simplified 51.4%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -30000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{k}, -10 + \frac{99}{k}, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{k \cdot \left(k + 10\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.75e-40)
   (/ a (* k k))
   (if (<= m 2.1e-183)
     (/ a (fma k 10.0 1.0))
     (if (<= m 0.92) (/ a (* k (+ k 10.0))) (* k (* k (* a 99.0)))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.75e-40) {
		tmp = a / (k * k);
	} else if (m <= 2.1e-183) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (m <= 0.92) {
		tmp = a / (k * (k + 10.0));
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.75e-40)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.1e-183)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (m <= 0.92)
		tmp = Float64(a / Float64(k * Float64(k + 10.0)));
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.75e-40], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.1e-183], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(k * N[(k + 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -1.7500000000000001e-40

   1. Initial program 97.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 51.6

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

   5. Simplified 51.6%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.3

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

   8. Simplified 65.3%

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

   if -1.7500000000000001e-40 < m < 2.1000000000000002e-183

   1. Initial program 98.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 71.7

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

   8. Simplified 71.7%

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

   if 2.1000000000000002e-183 < m < 0.92000000000000004

   1. Initial program 91.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. associate-*l* N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l* N/A

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

      7. lft-mult-inverse N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 67.6

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

   8. Simplified 67.6%

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

   if 0.92000000000000004 < m

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Simplified 3.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 51.4

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

   11. Simplified 51.4%

       \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 57.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.75 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.1 \cdot 10^{-183}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.75e-40)
     t_0
     (if (<= m 2.1e-183)
       (/ a (fma k 10.0 1.0))
       (if (<= m 0.92) t_0 (* k (* k (* a 99.0))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.75e-40) {
		tmp = t_0;
	} else if (m <= 2.1e-183) {
		tmp = a / fma(k, 10.0, 1.0);
	} else if (m <= 0.92) {
		tmp = t_0;
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.75e-40)
		tmp = t_0;
	elseif (m <= 2.1e-183)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	elseif (m <= 0.92)
		tmp = t_0;
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.75e-40], t$95$0, If[LessEqual[m, 2.1e-183], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], t$95$0, N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.7500000000000001e-40 or 2.1000000000000002e-183 < m < 0.92000000000000004

   1. Initial program 95.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 62.9

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

   5. Simplified 62.9%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   8. Simplified 65.9%

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

   if -1.7500000000000001e-40 < m < 2.1000000000000002e-183

   1. Initial program 98.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 98.4

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

   5. Simplified 98.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 71.7

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

   8. Simplified 71.7%

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

   if 0.92000000000000004 < m

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Simplified 3.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 51.4

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

   11. Simplified 51.4%

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

Alternative 10: 54.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -2.7 \cdot 10^{-277}:\\ \;\;\;\;a\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -3.4e-49)
     t_0
     (if (<= m -2.7e-277) a (if (<= m 0.92) t_0 (* k (* k (* a 99.0))))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -3.4e-49) {
		tmp = t_0;
	} else if (m <= -2.7e-277) {
		tmp = a;
	} else if (m <= 0.92) {
		tmp = t_0;
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-3.4d-49)) then
        tmp = t_0
    else if (m <= (-2.7d-277)) then
        tmp = a
    else if (m <= 0.92d0) then
        tmp = t_0
    else
        tmp = k * (k * (a * 99.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -3.4e-49) {
		tmp = t_0;
	} else if (m <= -2.7e-277) {
		tmp = a;
	} else if (m <= 0.92) {
		tmp = t_0;
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -3.4e-49:
		tmp = t_0
	elif m <= -2.7e-277:
		tmp = a
	elif m <= 0.92:
		tmp = t_0
	else:
		tmp = k * (k * (a * 99.0))
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -3.4e-49)
		tmp = t_0;
	elseif (m <= -2.7e-277)
		tmp = a;
	elseif (m <= 0.92)
		tmp = t_0;
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -3.4e-49)
		tmp = t_0;
	elseif (m <= -2.7e-277)
		tmp = a;
	elseif (m <= 0.92)
		tmp = t_0;
	else
		tmp = k * (k * (a * 99.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -3.4e-49], t$95$0, If[LessEqual[m, -2.7e-277], a, If[LessEqual[m, 0.92], t$95$0, N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.40000000000000005e-49 or -2.69999999999999975e-277 < m < 0.92000000000000004

   1. Initial program 96.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 69.8

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

   5. Simplified 69.8%

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

   6. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 63.4

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

   8. Simplified 63.4%

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

   if -3.40000000000000005e-49 < m < -2.69999999999999975e-277

   1. Initial program 97.1%

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

   3. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 74.9

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

   5. Simplified 74.9%

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

   6. Taylor expanded in m around 0

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

   8. Simplified 74.9%

      \[\leadsto a \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 74.9

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

   10. Applied egg-rr 74.9%

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

   if 0.92000000000000004 < m

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Simplified 3.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

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

      11. associate-*l* N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

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

       2. associate-*r* N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 51.4

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

   11. Simplified 51.4%

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

Alternative 11: 68.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-34}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -5e-34)
   (* a (/ 1.0 (* k k)))
   (if (<= m 0.92) (/ a (fma k (+ k 10.0) 1.0)) (* k (* k (* a 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -5e-34) {
		tmp = a * (1.0 / (k * k));
	} else if (m <= 0.92) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -5e-34)
		tmp = Float64(a * Float64(1.0 / Float64(k * k)));
	elseif (m <= 0.92)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -5e-34], N[(a * N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.0000000000000003e-34

   1. Initial program 98.8%

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

   3. Taylor expanded in k around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in m around 0

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

      1. lower-/.f64 58.6

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

   10. Simplified 58.6%

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

       1. associate-/l/ N/A

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

       2. lift-*.f64 N/A

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

       3. div-inv N/A

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

       4. remove-double-div N/A

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

       5. lift-/.f64 N/A

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

       6. inv-pow N/A

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

       7. inv-pow N/A

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

       8. pow-prod-down N/A

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

       9. *-commutative N/A

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

       10. pow-prod-down N/A

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

       11. inv-pow N/A

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

       12. inv-pow N/A

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

       13. lift-/.f64 N/A

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

       14. remove-double-div N/A

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

       15. lower-*.f64 N/A

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

       16. lower-/.f64 68.6

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

   12. Applied egg-rr 68.6%

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

   if -5.0000000000000003e-34 < m < 0.92000000000000004

   1. Initial program 94.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 94.5

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

   5. Simplified 94.5%

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

   if 0.92000000000000004 < m

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. lft-mult-inverse N/A

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

      7. associate-*l* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-rgt-in N/A

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

      16. associate-*l* N/A

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

      17. lft-mult-inverse N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. lower-+.f64 3.0

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

   5. Simplified 3.0%

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

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a, a\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}, a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10} + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right), a\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(a, -10, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)\right)}, a\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} \cdot k\right)\right), a\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(-100 + 1\right) \cdot \left(a \cdot k\right)}\right)\right), a\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto k \cdot \color{blue}{\left(\left(99 \cdot a\right) \cdot k\right)} \]

       7. *-commutative N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

       10. lower-*.f64 51.4

           \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

   11. Simplified 51.4%

       \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{-34}:\\ \;\;\;\;a \cdot \frac{1}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -30000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -30000.0)
   (/ a (* k k))
   (if (<= m 0.92) (/ a (fma k (+ k 10.0) 1.0)) (* k (* k (* a 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -30000.0) {
		tmp = a / (k * k);
	} else if (m <= 0.92) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -30000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.92)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -30000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.92], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3e4

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 51.3

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

      2. lower-*.f64 65.9

         \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   8. Simplified 65.9%

      \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]

   if -3e4 < m < 0.92000000000000004

   1. Initial program 94.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 93.4

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 93.4%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   if 0.92000000000000004 < m

   1. Initial program 76.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.0

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.0%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a, a\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}, a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10} + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right), a\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(a, -10, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)\right)}, a\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} \cdot k\right)\right), a\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(-100 + 1\right) \cdot \left(a \cdot k\right)}\right)\right), a\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto k \cdot \color{blue}{\left(\left(99 \cdot a\right) \cdot k\right)} \]

       7. *-commutative N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

       10. lower-*.f64 51.4

           \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

   11. Simplified 51.4%

       \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -30000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.92:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.37:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(k \cdot \left(a \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.37) a (* k (* k (* a 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.37) {
		tmp = a;
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.37d0) then
        tmp = a
    else
        tmp = k * (k * (a * 99.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.37) {
		tmp = a;
	} else {
		tmp = k * (k * (a * 99.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.37:
		tmp = a
	else:
		tmp = k * (k * (a * 99.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.37)
		tmp = a;
	else
		tmp = Float64(k * Float64(k * Float64(a * 99.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.37)
		tmp = a;
	else
		tmp = k * (k * (a * 99.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.37], a, N[(k * N[(k * N[(a * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.37

   1. Initial program 96.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

      2. lower-pow.f64 71.9

         \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   6. Taylor expanded in m around 0

      \[\leadsto a \cdot \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 28.8%

      \[\leadsto a \cdot \color{blue}{1} \]
   9. Step-by-step derivation

      1. *-rgt-identity 28.8

         \[\leadsto \color{blue}{a} \]

   10. Applied egg-rr 28.8%

       \[\leadsto \color{blue}{a} \]

   if 0.37 < m

   1. Initial program 76.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   2. Add Preprocessing

   3. Taylor expanded in m around 0

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

      3. distribute-rgt-in N/A

         \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

      6. lft-mult-inverse N/A

         \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

      7. associate-*l* N/A

         \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

      8. *-lft-identity N/A

         \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

      9. distribute-rgt-in N/A

         \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

      10. +-commutative N/A

          \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

      11. *-commutative N/A

          \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

      15. distribute-rgt-in N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

      16. associate-*l* N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

      17. lft-mult-inverse N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

      18. metadata-eval N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

      19. *-lft-identity N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

      20. lower-+.f64 3.0

          \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

   5. Simplified 3.0%

      \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

   6. Taylor expanded in k around 0

      \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

      3. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a, a\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}, a\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10} + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right), a\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(a, -10, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)\right)}, a\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

      10. distribute-rgt1-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} \cdot k\right)\right), a\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(-100 + 1\right) \cdot \left(a \cdot k\right)}\right)\right), a\right) \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

      16. lower-*.f64 30.0

          \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

   8. Simplified 30.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

   9. Taylor expanded in k around inf

      \[\leadsto \color{blue}{99 \cdot \left(a \cdot {k}^{2}\right)} \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto 99 \cdot \left(a \cdot \color{blue}{\left(k \cdot k\right)}\right) \]

       2. associate-*r* N/A

          \[\leadsto 99 \cdot \color{blue}{\left(\left(a \cdot k\right) \cdot k\right)} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(99 \cdot \left(a \cdot k\right)\right) \cdot k} \]

       4. *-commutative N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto \color{blue}{k \cdot \left(99 \cdot \left(a \cdot k\right)\right)} \]

       6. associate-*r* N/A

          \[\leadsto k \cdot \color{blue}{\left(\left(99 \cdot a\right) \cdot k\right)} \]

       7. *-commutative N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto k \cdot \color{blue}{\left(k \cdot \left(99 \cdot a\right)\right)} \]

       9. *-commutative N/A

          \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

       10. lower-*.f64 51.4

           \[\leadsto k \cdot \left(k \cdot \color{blue}{\left(a \cdot 99\right)}\right) \]

   11. Simplified 51.4%

       \[\leadsto \color{blue}{k \cdot \left(k \cdot \left(a \cdot 99\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 20.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(k, a \cdot -10, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma k (* a -10.0) a))

C

double code(double a, double k, double m) {
	return fma(k, (a * -10.0), a);
}

Julia

function code(a, k, m)
	return fma(k, Float64(a * -10.0), a)
end

Wolfram

code[a_, k_, m_] := N[(k * N[(a * -10.0), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 89.6%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 49.6

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Simplified 49.6%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a + k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, a\right)} \]

   3. cancel-sign-sub-inv N/A

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \left(\mathsf{neg}\left(10\right)\right) \cdot a}, a\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(k, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) + \color{blue}{-10} \cdot a, a\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)}, a\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10} + -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right), a\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(k, \color{blue}{\mathsf{fma}\left(a, -10, -1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right)\right)}, a\right) \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)}\right), a\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(a + -100 \cdot a\right) \cdot k}\right)\right), a\right) \]

   10. distribute-rgt1-in N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(\left(-100 + 1\right) \cdot a\right)} \cdot k\right)\right), a\right) \]

   11. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \mathsf{neg}\left(\color{blue}{\left(-100 + 1\right) \cdot \left(a \cdot k\right)}\right)\right), a\right) \]

   12. distribute-lft-neg-in N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{\left(\mathsf{neg}\left(\left(-100 + 1\right)\right)\right) \cdot \left(a \cdot k\right)}\right), a\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \left(\mathsf{neg}\left(\color{blue}{-99}\right)\right) \cdot \left(a \cdot k\right)\right), a\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, \color{blue}{99} \cdot \left(a \cdot k\right)\right), a\right) \]

   16. lower-*.f64 28.6

       \[\leadsto \mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \color{blue}{\left(a \cdot k\right)}\right), a\right) \]

8. Simplified 28.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(a, -10, 99 \cdot \left(a \cdot k\right)\right), a\right)} \]

9. Taylor expanded in k around 0

   \[\leadsto \mathsf{fma}\left(k, \color{blue}{-10 \cdot a}, a\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10}, a\right) \]

    2. lower-*.f64 21.5

       \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10}, a\right) \]

11. Simplified 21.5%

    \[\leadsto \mathsf{fma}\left(k, \color{blue}{a \cdot -10}, a\right) \]
12. Add Preprocessing

Alternative 15: 20.7% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, k \cdot -10, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma a (* k -10.0) a))

C

double code(double a, double k, double m) {
	return fma(a, (k * -10.0), a);
}

Julia

function code(a, k, m)
	return fma(a, Float64(k * -10.0), a)
end

Wolfram

code[a_, k_, m_] := N[(a * N[(k * -10.0), $MachinePrecision] + a), $MachinePrecision]

Derivation

1. Initial program 89.6%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in m around 0

   \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a}{1 + \left(10 \cdot k + {k}^{2}\right)}} \]

   2. unpow2 N/A

      \[\leadsto \frac{a}{1 + \left(10 \cdot k + \color{blue}{k \cdot k}\right)} \]

   3. distribute-rgt-in N/A

      \[\leadsto \frac{a}{1 + \color{blue}{k \cdot \left(10 + k\right)}} \]

   4. +-commutative N/A

      \[\leadsto \frac{a}{\color{blue}{k \cdot \left(10 + k\right) + 1}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{10 \cdot 1} + k\right) + 1} \]

   6. lft-mult-inverse N/A

      \[\leadsto \frac{a}{k \cdot \left(10 \cdot \color{blue}{\left(\frac{1}{k} \cdot k\right)} + k\right) + 1} \]

   7. associate-*l* N/A

      \[\leadsto \frac{a}{k \cdot \left(\color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k} + k\right) + 1} \]

   8. *-lft-identity N/A

      \[\leadsto \frac{a}{k \cdot \left(\left(10 \cdot \frac{1}{k}\right) \cdot k + \color{blue}{1 \cdot k}\right) + 1} \]

   9. distribute-rgt-in N/A

      \[\leadsto \frac{a}{k \cdot \color{blue}{\left(k \cdot \left(10 \cdot \frac{1}{k} + 1\right)\right)} + 1} \]

   10. +-commutative N/A

       \[\leadsto \frac{a}{k \cdot \left(k \cdot \color{blue}{\left(1 + 10 \cdot \frac{1}{k}\right)}\right) + 1} \]

   11. *-commutative N/A

       \[\leadsto \frac{a}{k \cdot \color{blue}{\left(\left(1 + 10 \cdot \frac{1}{k}\right) \cdot k\right)} + 1} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(k, \left(1 + 10 \cdot \frac{1}{k}\right) \cdot k, 1\right)}} \]

   13. *-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{k \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}, 1\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, k \cdot \color{blue}{\left(10 \cdot \frac{1}{k} + 1\right)}, 1\right)} \]

   15. distribute-rgt-in N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{\left(10 \cdot \frac{1}{k}\right) \cdot k + 1 \cdot k}, 1\right)} \]

   16. associate-*l* N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 \cdot \left(\frac{1}{k} \cdot k\right)} + 1 \cdot k, 1\right)} \]

   17. lft-mult-inverse N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 \cdot \color{blue}{1} + 1 \cdot k, 1\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10} + 1 \cdot k, 1\right)} \]

   19. *-lft-identity N/A

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, 10 + \color{blue}{k}, 1\right)} \]

   20. lower-+.f64 49.6

       \[\leadsto \frac{a}{\mathsf{fma}\left(k, \color{blue}{10 + k}, 1\right)} \]

5. Simplified 49.6%

   \[\leadsto \color{blue}{\frac{a}{\mathsf{fma}\left(k, 10 + k, 1\right)}} \]

6. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a + -10 \cdot \left(a \cdot k\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot k\right) + a} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(a \cdot k\right) \cdot -10} + a \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{a \cdot \left(k \cdot -10\right)} + a \]

   4. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(-10 \cdot k\right)} + a \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, -10 \cdot k, a\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

   7. lower-*.f64 21.5

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{k \cdot -10}, a\right) \]

8. Simplified 21.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, k \cdot -10, a\right)} \]
9. Add Preprocessing

Alternative 16: 19.8% accurate, 134.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 a)

C

double code(double a, double k, double m) {
	return a;
}

Fortran

real(8) function code(a, k, m)
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a
end function

Java

public static double code(double a, double k, double m) {
	return a;
}

Python

def code(a, k, m):
	return a

Julia

function code(a, k, m)
	return a
end

MATLAB

function tmp = code(a, k, m)
	tmp = a;
end

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 89.6%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

   2. lower-pow.f64 81.8

      \[\leadsto a \cdot \color{blue}{{k}^{m}} \]

5. Simplified 81.8%

   \[\leadsto \color{blue}{a \cdot {k}^{m}} \]

6. Taylor expanded in m around 0

   \[\leadsto a \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 20.0%

   \[\leadsto a \cdot \color{blue}{1} \]
9. Step-by-step derivation

   1. *-rgt-identity 20.0

      \[\leadsto \color{blue}{a} \]

10. Applied egg-rr 20.0%

    \[\leadsto \color{blue}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))