Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 97.7%

Time: 8.5s

Alternatives: 14

Speedup: 1.0×

• Could not converge on a ground truth

  1. a = 1.496573486953673e-175
  2. k = 6.964512473137672e-157
  3. m = -3.251951087984515e+231

• 21.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    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: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{{\left({k}^{\left(-m\right)}\right)}^{-1}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.5)
   (* (/ (pow (pow k (- m)) -1.0) (fma (+ k 10.0) k 1.0)) a)
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.5) {
		tmp = (pow(pow(k, -m), -1.0) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.5)
		tmp = Float64(Float64(((k ^ Float64(-m)) ^ -1.0) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.5], N[(N[(N[Power[N[Power[k, (-m)], $MachinePrecision], -1.0], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.7

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

   4. Applied rewrites 97.7%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. log-rec N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. distribute-lft-neg-out N/A

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

      7. *-commutative N/A

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

      8. exp-to-pow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-neg.f64 97.7

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

   9. Applied rewrites 97.7%

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

   11. Applied rewrites 97.7%

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

   if 3.5 < m

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 75.9

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{{\left({k}^{\left(-m\right)}\right)}^{-1}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 17.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
   (* (* a k) -10.0)
   (* (fma -10.0 k 1.0) a)))

C

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

Julia

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

Wolfram

code[a_, k_, m_] := If[LessEqual[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], 0.0], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.4

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 14.6%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 9.3%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
   12. Step-by-step derivation

   13. Applied rewrites 9.3%

       \[\leadsto \left(a \cdot k\right) \cdot -10 \]

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

   1. Initial program 74.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 25.7%

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

4. Final simplification 14.5%

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

Alternative 3: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(\frac{{k}^{-1}}{k} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -68.0)
   (/ (/ (* (/ (/ a k) k) 99.0) k) k)
   (if (<= m 1.3e-12)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (/ a (* (* (/ (pow k -1.0) k) k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -68.0) {
		tmp = ((((a / k) / k) * 99.0) / k) / k;
	} else if (m <= 1.3e-12) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = a / (((pow(k, -1.0) / k) * k) * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -68.0)
		tmp = Float64(Float64(Float64(Float64(Float64(a / k) / k) * 99.0) / k) / k);
	elseif (m <= 1.3e-12)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(a / Float64(Float64(Float64((k ^ -1.0) / k) * k) * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -68.0], N[(N[(N[(N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision] * 99.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[m, 1.3e-12], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] / k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -68

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 3.3%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 71.2%

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

   12. Taylor expanded in k around 0

       \[\leadsto \frac{\frac{99 \cdot \frac{a}{{k}^{2}}}{k}}{k} \]
   13. Step-by-step derivation

   14. Applied rewrites 81.4%

       \[\leadsto \frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k} \]

   if -68 < m < 1.29999999999999991e-12

   1. Initial program 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if 1.29999999999999991e-12 < m

   1. Initial program 76.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 55.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{\frac{\frac{\frac{a}{k}}{k} \cdot 99}{k}}{k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(\frac{{k}^{-1}}{k} \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(\frac{{k}^{-1}}{k} \cdot k\right) \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -68.0)
   (/ a (* k k))
   (if (<= m 1.3e-12)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (/ a (* (* (/ (pow k -1.0) k) k) k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -68.0) {
		tmp = a / (k * k);
	} else if (m <= 1.3e-12) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = a / (((pow(k, -1.0) / k) * k) * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -68.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.3e-12)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(a / Float64(Float64(Float64((k ^ -1.0) / k) * k) * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -68.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.3e-12], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(N[(N[Power[k, -1.0], $MachinePrecision] / k), $MachinePrecision] * k), $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -68

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.7%

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

   if -68 < m < 1.29999999999999991e-12

   1. Initial program 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if 1.29999999999999991e-12 < m

   1. Initial program 76.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.6

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 32.9%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 55.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\left(\frac{{k}^{-1}}{k} \cdot k\right) \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.5:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.5) (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.5) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.5)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.5], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 97.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.7

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 97.7

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

   4. Applied rewrites 97.7%

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

   if 3.5 < m

   1. Initial program 75.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 75.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 75.9

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

   4. Applied rewrites 75.9%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 75.9

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

   6. Applied rewrites 75.9%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 100.0

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

   9. Applied rewrites 100.0%

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

Alternative 6: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -68 \lor \neg \left(m \leq 1.3 \cdot 10^{-12}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -68.0) (not (<= m 1.3e-12)))
   (* (pow k m) a)
   (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -68.0) || !(m <= 1.3e-12)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -68.0) || !(m <= 1.3e-12))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -68.0], N[Not[LessEqual[m, 1.3e-12]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -68 or 1.29999999999999991e-12 < m

   1. Initial program 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 87.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. rem-square-sqrt N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 87.8

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

   6. Applied rewrites 87.8%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 99.5

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

   9. Applied rewrites 99.5%

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

   if -68 < m < 1.29999999999999991e-12

   1. Initial program 95.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 95.1

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

   7. Applied rewrites 95.1%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -68 \lor \neg \left(m \leq 1.3 \cdot 10^{-12}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -68.0)
   (/ a (* k k))
   (if (<= m 1.6)
     (* (pow (fma (+ 10.0 k) k 1.0) -1.0) a)
     (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -68.0) {
		tmp = a / (k * k);
	} else if (m <= 1.6) {
		tmp = pow(fma((10.0 + k), k, 1.0), -1.0) * a;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -68.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.6)
		tmp = Float64((fma(Float64(10.0 + k), k, 1.0) ^ -1.0) * a);
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -68.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.6], N[(N[Power[N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -68

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.7%

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

   if -68 < m < 1.6000000000000001

   1. Initial program 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 95.5

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-+r+ N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. distribute-rgt-out N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 93.7

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

   7. Applied rewrites 93.7%

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

   if 1.6000000000000001 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;{\left(\mathsf{fma}\left(10 + k, k, 1\right)\right)}^{-1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -2.45 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -2.45e-36)
     t_0
     (if (<= m 2.3e-129)
       (/ a 1.0)
       (if (<= m 1.6) t_0 (* (* (* a k) k) 99.0))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -2.45e-36) {
		tmp = t_0;
	} else if (m <= 2.3e-129) {
		tmp = a / 1.0;
	} else if (m <= 1.6) {
		tmp = t_0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (m <= (-2.45d-36)) then
        tmp = t_0
    else if (m <= 2.3d-129) then
        tmp = a / 1.0d0
    else if (m <= 1.6d0) then
        tmp = t_0
    else
        tmp = ((a * k) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -2.45e-36) {
		tmp = t_0;
	} else if (m <= 2.3e-129) {
		tmp = a / 1.0;
	} else if (m <= 1.6) {
		tmp = t_0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if m <= -2.45e-36:
		tmp = t_0
	elif m <= 2.3e-129:
		tmp = a / 1.0
	elif m <= 1.6:
		tmp = t_0
	else:
		tmp = ((a * k) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -2.45e-36)
		tmp = t_0;
	elseif (m <= 2.3e-129)
		tmp = Float64(a / 1.0);
	elseif (m <= 1.6)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (m <= -2.45e-36)
		tmp = t_0;
	elseif (m <= 2.3e-129)
		tmp = a / 1.0;
	elseif (m <= 1.6)
		tmp = t_0;
	else
		tmp = ((a * k) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2.45e-36], t$95$0, If[LessEqual[m, 2.3e-129], N[(a / 1.0), $MachinePrecision], If[LessEqual[m, 1.6], t$95$0, N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.4499999999999998e-36 or 2.3e-129 < m < 1.6000000000000001

   1. Initial program 99.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 43.9

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

   5. Applied rewrites 43.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 67.2%

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

   if -2.4499999999999998e-36 < m < 2.3e-129

   1. Initial program 95.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 55.9%

      \[\leadsto \frac{a}{1} \]

   if 1.6000000000000001 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.8%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.45 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -68.0)
   (/ a (* k k))
   (if (<= m 1.6) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -68.0) {
		tmp = a / (k * k);
	} else if (m <= 1.6) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -68.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.6)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -68.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.6], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -68

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.7%

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

   if -68 < m < 1.6000000000000001

   1. Initial program 95.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.7

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

   5. Applied rewrites 93.7%

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

   if 1.6000000000000001 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.8%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -68:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.5e-36)
   (/ a (* k k))
   (if (<= m 1.6) (/ a (fma 10.0 k 1.0)) (* (* (* a k) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.5e-36) {
		tmp = a / (k * k);
	} else if (m <= 1.6) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.5e-36)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1.6)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.5e-36], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.6], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.50000000000000002e-36

   1. Initial program 100.0%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 36.4

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

   5. Applied rewrites 36.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 66.9%

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

   if -2.50000000000000002e-36 < m < 1.6000000000000001

   1. Initial program 95.3%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.5

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

   5. Applied rewrites 93.5%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.3%

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

   if 1.6000000000000001 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.8%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1.6:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 1.35) (/ a 1.0) (* (* (* a k) k) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / 1.0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 1.35d0) then
        tmp = a / 1.0d0
    else
        tmp = ((a * k) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 1.35) {
		tmp = a / 1.0;
	} else {
		tmp = ((a * k) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 1.35:
		tmp = a / 1.0
	else:
		tmp = ((a * k) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 1.35)
		tmp = Float64(a / 1.0);
	else
		tmp = Float64(Float64(Float64(a * k) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 1.35)
		tmp = a / 1.0;
	else
		tmp = ((a * k) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 1.35], N[(a / 1.0), $MachinePrecision], N[(N[(N[(a * k), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.3500000000000001

   1. Initial program 97.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 26.3%

      \[\leadsto \frac{a}{1} \]

   if 1.3500000000000001 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 23.9%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 53.8%

       \[\leadsto \left(\left(a \cdot k\right) \cdot k\right) \cdot 99 \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 1.35:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot k\right) \cdot k\right) \cdot 99\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 135000:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 135000.0) (/ a 1.0) (* (* -10.0 k) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 135000.0) {
		tmp = a / 1.0;
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 135000.0d0) then
        tmp = a / 1.0d0
    else
        tmp = ((-10.0d0) * k) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 135000.0) {
		tmp = a / 1.0;
	} else {
		tmp = (-10.0 * k) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 135000.0:
		tmp = a / 1.0
	else:
		tmp = (-10.0 * k) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 135000.0)
		tmp = Float64(a / 1.0);
	else
		tmp = Float64(Float64(-10.0 * k) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 135000.0)
		tmp = a / 1.0;
	else
		tmp = (-10.0 * k) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 135000.0], N[(a / 1.0), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 135000

   1. Initial program 97.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 26.0%

      \[\leadsto \frac{a}{1} \]

   if 135000 < m

   1. Initial program 76.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 4.0%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 17.3%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
   12. Step-by-step derivation

   13. Applied rewrites 17.3%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 135000:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot k\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 8.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 43.5

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

5. Applied rewrites 43.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 18.1%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 7.4%

    \[\leadsto \left(-10 \cdot a\right) \cdot k \]
12. Step-by-step derivation

13. Applied rewrites 7.4%

    \[\leadsto \left(a \cdot k\right) \cdot -10 \]

14. Final simplification 7.4%

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

Alternative 14: 8.8% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 43.5

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

5. Applied rewrites 43.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 18.1%

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

9. Taylor expanded in k around inf

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

11. Applied rewrites 7.4%

    \[\leadsto \left(-10 \cdot a\right) \cdot k \]

12. Final simplification 7.4%

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

Reproduce

herbie shell --seed 2024356 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))