Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 99.1%

Time: 10.2s

Alternatives: 11

Speedup: 1.2×

• 22.9% 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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		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, 1.0], 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 < 1

   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 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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1 < k

   1. Initial program 79.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 79.2

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

   5. Applied rewrites 79.2%

      \[\leadsto \frac{a \cdot {k}^{m}}{\color{blue}{k \cdot k}} \]
   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. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-prod-up N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-+.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -0.0195:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -0.0195)
     t_0
     (if (<= m 4.3e-6) (/ 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 <= -0.0195) {
		tmp = t_0;
	} else if (m <= 4.3e-6) {
		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 <= -0.0195)
		tmp = t_0;
	elseif (m <= 4.3e-6)
		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, -0.0195], t$95$0, If[LessEqual[m, 4.3e-6], 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 < -0.0195 or 4.30000000000000033e-6 < m

   1. Initial program 84.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. Applied rewrites 100.0%

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

   if -0.0195 < m < 4.30000000000000033e-6

   1. Initial program 94.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0195:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{elif}\;m \leq 4.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{m}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.0) (* a (pow k m)) (* a (pow k (+ m -2.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.0) {
		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 <= 1.0d0) 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 <= 1.0) {
		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 <= 1.0:
		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 <= 1.0)
		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 <= 1.0)
		tmp = a * (k ^ m);
	else
		tmp = a * (k ^ (m + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.0], 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 < 1

   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 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 99.5

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

   5. Applied rewrites 99.5%

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

   if 1 < k

   1. Initial program 79.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 79.2

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

   5. Applied rewrites 79.2%

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

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

      3. associate-/l* N/A

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

      4. *-commutative 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 94.5

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

   7. Applied rewrites 94.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1:\\ \;\;\;\;a \cdot {k}^{m}\\ \mathbf{else}:\\ \;\;\;\;a \cdot {k}^{\left(m + -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.045:\\ \;\;\;\;\frac{a \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.045)
   (/ (* a 99.0) (* (* k k) (* k k)))
   (if (<= m 0.95) (/ a (fma k (+ k 10.0) 1.0)) (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.045) {
		tmp = (a * 99.0) / ((k * k) * (k * k));
	} else if (m <= 0.95) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.045)
		tmp = Float64(Float64(a * 99.0) / Float64(Float64(k * k) * Float64(k * k)));
	elseif (m <= 0.95)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.045], N[(N[(a * 99.0), $MachinePrecision] / N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.044999999999999998

   1. Initial program 98.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.6%

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

   9. Taylor expanded in k around 0

      \[\leadsto \color{blue}{99 \cdot \frac{a}{{k}^{4}}} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{99 \cdot a}{{k}^{4}}} \]

       3. *-commutative N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{a \cdot 99}}{{k}^{4}} \]

       5. metadata-eval N/A

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

       6. pow-sqr N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       7. lower-*.f64 N/A

          \[\leadsto \frac{a \cdot 99}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]

       8. unpow2 N/A

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

       9. lower-*.f64 N/A

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

       10. unpow2 N/A

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

       11. lower-*.f64 78.1

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

   11. Applied rewrites 78.1%

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

   if -0.044999999999999998 < m < 0.94999999999999996

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.94999999999999996 < m

   1. Initial program 71.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.6

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

   14. Applied rewrites 52.6%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.045:\\ \;\;\;\;\frac{a \cdot 99}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.039:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.039)
   (/ a (* k k))
   (if (<= m 0.95) (/ a (fma k (+ k 10.0) 1.0)) (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.039) {
		tmp = a / (k * k);
	} else if (m <= 0.95) {
		tmp = a / fma(k, (k + 10.0), 1.0);
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.039)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.95)
		tmp = Float64(a / fma(k, Float64(k + 10.0), 1.0));
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.039], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.95], N[(a / N[(k * N[(k + 10.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0389999999999999999

   1. Initial program 98.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 38.8

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

   5. Applied rewrites 38.8%

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

   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 63.8

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

   8. Applied rewrites 63.8%

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

   if -0.0389999999999999999 < m < 0.94999999999999996

   1. Initial program 94.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if 0.94999999999999996 < m

   1. Initial program 71.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.6

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

   14. Applied rewrites 52.6%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.039:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.95:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k + 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.85 \cdot 10^{-96}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.85e-96)
   (/ a (* k k))
   (if (<= m 0.26) (/ a (fma k 10.0 1.0)) (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.85e-96) {
		tmp = a / (k * k);
	} else if (m <= 0.26) {
		tmp = a / fma(k, 10.0, 1.0);
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.85e-96)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.26)
		tmp = Float64(a / fma(k, 10.0, 1.0));
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.85e-96], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.26], N[(a / N[(k * 10.0 + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.84999999999999993e-96

   1. Initial program 98.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 49.7

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

   5. Applied rewrites 49.7%

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

   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 63.9

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

   8. Applied rewrites 63.9%

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

   if -1.84999999999999993e-96 < m < 0.26000000000000001

   1. Initial program 94.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   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 63.8

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

   8. Applied rewrites 63.8%

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

   if 0.26000000000000001 < m

   1. Initial program 71.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.6

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

   14. Applied rewrites 52.6%

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

Alternative 7: 41.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{a}{k \cdot 10}\\ \mathbf{elif}\;m \leq 0.24:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.8e-21)
   (/ a (* k 10.0))
   (if (<= m 0.24) a (* k (* a (* k 99.0))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.8e-21) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.24) {
		tmp = a;
	} else {
		tmp = k * (a * (k * 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 <= (-3.8d-21)) then
        tmp = a / (k * 10.0d0)
    else if (m <= 0.24d0) then
        tmp = a
    else
        tmp = k * (a * (k * 99.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -3.8e-21) {
		tmp = a / (k * 10.0);
	} else if (m <= 0.24) {
		tmp = a;
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -3.8e-21:
		tmp = a / (k * 10.0)
	elif m <= 0.24:
		tmp = a
	else:
		tmp = k * (a * (k * 99.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -3.8e-21)
		tmp = Float64(a / Float64(k * 10.0));
	elseif (m <= 0.24)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -3.8e-21)
		tmp = a / (k * 10.0);
	elseif (m <= 0.24)
		tmp = a;
	else
		tmp = k * (a * (k * 99.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -3.8e-21], N[(a / N[(k * 10.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.24], a, N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.7999999999999998e-21

   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 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 40.8

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

   5. Applied rewrites 40.8%

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

   6. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2} \cdot \left(1 + 10 \cdot \frac{1}{k}\right)}} \]
   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. associate-*l* N/A

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

      6. lft-mult-inverse N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      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 48.9

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

   8. Applied rewrites 48.9%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 25.5

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

   11. Applied rewrites 25.5%

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

   if -3.7999999999999998e-21 < m < 0.23999999999999999

   1. Initial program 94.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 47.3%

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

      1. /-rgt-identity 47.3

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

   10. Applied rewrites 47.3%

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

   if 0.23999999999999999 < m

   1. Initial program 71.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.6

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

   14. Applied rewrites 52.6%

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

Alternative 8: 51.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.00125:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.00125) (/ a (* k k)) (* k (* a (* k 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00125) {
		tmp = a / (k * k);
	} else {
		tmp = k * (a * (k * 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 <= 0.00125d0) then
        tmp = a / (k * k)
    else
        tmp = k * (a * (k * 99.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00125) {
		tmp = a / (k * k);
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.00125:
		tmp = a / (k * k)
	else:
		tmp = k * (a * (k * 99.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.00125)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.00125)
		tmp = a / (k * k);
	else
		tmp = k * (a * (k * 99.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.00125], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.00125000000000000003

   1. Initial program 96.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

   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 57.5

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

   8. Applied rewrites 57.5%

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

   if 0.00125000000000000003 < m

   1. Initial program 72.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.3

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

   11. Applied rewrites 32.3%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.1

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

   14. Applied rewrites 52.1%

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

Alternative 9: 35.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.24:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(a \cdot \left(k \cdot 99\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.24) a (* k (* a (* k 99.0)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.24) {
		tmp = a;
	} else {
		tmp = k * (a * (k * 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 <= 0.24d0) then
        tmp = a
    else
        tmp = k * (a * (k * 99.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.24) {
		tmp = a;
	} else {
		tmp = k * (a * (k * 99.0));
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.24:
		tmp = a
	else:
		tmp = k * (a * (k * 99.0))
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.24)
		tmp = a;
	else
		tmp = Float64(k * Float64(a * Float64(k * 99.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.24)
		tmp = a;
	else
		tmp = k * (a * (k * 99.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.24], a, N[(k * N[(a * N[(k * 99.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.23999999999999999

   1. Initial program 96.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 27.4%

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

      1. /-rgt-identity 27.4

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

   10. Applied rewrites 27.4%

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

   if 0.23999999999999999 < m

   1. Initial program 71.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 14.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(k, \mathsf{fma}\left(k, \mathsf{fma}\left(a, 99, k \cdot \left(10 \cdot \left(-98 \cdot a\right)\right)\right), a \cdot -10\right), a\right)} \]

   9. Taylor expanded in k around 0

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

       1. +-commutative N/A

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

       2. +-commutative N/A

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

       3. distribute-rgt-in N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. distribute-rgt-out N/A

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

       9. lower-fma.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-fma.f64 32.4

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

   11. Applied rewrites 32.4%

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

   12. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 52.6

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

   14. Applied rewrites 52.6%

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

Alternative 10: 20.6% 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 88.5%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 48.3

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

5. Applied rewrites 48.3%

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

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 20.8

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

8. Applied rewrites 20.8%

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

Alternative 11: 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 88.5%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 48.3

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

5. Applied rewrites 48.3%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 20.0%

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

   1. /-rgt-identity 20.0

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

10. Applied rewrites 20.0%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))