Falkner and Boettcher, Appendix A

Percentage Accurate: 90.6% → 97.8%

Time: 8.2s

Alternatives: 11

Speedup: 1.1×

• 21.5% 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: 90.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.00049:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)} \cdot 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.00049)
   (* (/ (pow k m) (fma (+ k 10.0) k 1.0)) a)
   (/ a (* (pow k (- m)) 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.00049) {
		tmp = (pow(k, m) / fma((k + 10.0), k, 1.0)) * a;
	} else {
		tmp = a / (pow(k, -m) * 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.00049)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k + 10.0), k, 1.0)) * a);
	else
		tmp = Float64(a / Float64((k ^ Float64(-m)) * 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.00049], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k + 10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[Power[k, (-m)], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.8999999999999998e-4

   1. Initial program 98.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.3

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

   if 4.8999999999999998e-4 < m

   1. Initial program 69.6%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(\cosh \left(\log k \cdot m\right) + \sinh \left(\log k \cdot m\right)\right)}}{1} \]

      6. lift-cosh.f64 N/A

         \[\leadsto \frac{a \cdot \left(\color{blue}{\cosh \left(\log k \cdot m\right)} + \sinh \left(\log k \cdot m\right)\right)}{1} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \frac{a \cdot \left(\cosh \left(\log k \cdot m\right) + \color{blue}{\sinh \left(\log k \cdot m\right)}\right)}{1} \]

      8. flip-+ N/A

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

      9. lift-cosh.f64 N/A

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

      10. lift-sinh.f64 N/A

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

      11. sinh---cosh-rev N/A

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq 0.00049:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k + 10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)} \cdot 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)} \cdot 1}\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.5e-9)
   (/ a (* (pow k (- m)) 1.0))
   (if (<= m 1.75e-14) (/ a (fma (+ 10.0 k) k 1.0)) (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.5e-9) {
		tmp = a / (pow(k, -m) * 1.0);
	} else if (m <= 1.75e-14) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.5e-9)
		tmp = Float64(a / Float64((k ^ Float64(-m)) * 1.0));
	elseif (m <= 1.75e-14)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.5e-9], N[(a / N[(N[Power[k, (-m)], $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.75e-14], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.49999999999999933e-9

   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 k around 0

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

   5. Applied rewrites 100.0%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lift-log.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sinh-+-cosh-rev N/A

         \[\leadsto \frac{a \cdot \color{blue}{\left(\cosh \left(\log k \cdot m\right) + \sinh \left(\log k \cdot m\right)\right)}}{1} \]

      6. lift-cosh.f64 N/A

         \[\leadsto \frac{a \cdot \left(\color{blue}{\cosh \left(\log k \cdot m\right)} + \sinh \left(\log k \cdot m\right)\right)}{1} \]

      7. lift-sinh.f64 N/A

         \[\leadsto \frac{a \cdot \left(\cosh \left(\log k \cdot m\right) + \color{blue}{\sinh \left(\log k \cdot m\right)}\right)}{1} \]

      8. flip-+ N/A

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

      9. lift-cosh.f64 N/A

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

      10. lift-sinh.f64 N/A

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

      11. sinh---cosh-rev N/A

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

   7. Applied rewrites 100.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 100.0

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

   9. Applied rewrites 100.0%

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

   if -7.49999999999999933e-9 < m < 1.7500000000000001e-14

   1. Initial program 96.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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 96.3

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

   5. Applied rewrites 96.3%

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

   if 1.7500000000000001e-14 < m

   1. Initial program 70.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 70.0

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

      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 70.0

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

   4. Applied rewrites 70.0%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{{k}^{\left(-m\right)} \cdot 1}\\ \mathbf{elif}\;m \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-9} \lor \neg \left(m \leq 1.75 \cdot 10^{-14}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -7.5e-9) (not (<= m 1.75e-14)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -7.5e-9) || !(m <= 1.75e-14)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -7.5e-9) || !(m <= 1.75e-14))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -7.5e-9], N[Not[LessEqual[m, 1.75e-14]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -7.49999999999999933e-9 or 1.7500000000000001e-14 < m

   1. Initial program 85.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 85.1

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

      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 85.1

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

   4. Applied rewrites 85.1%

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

   5. Taylor expanded in k around 0

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

      1. lower-pow.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if -7.49999999999999933e-9 < m < 1.7500000000000001e-14

   1. Initial program 96.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7.5 \cdot 10^{-9} \lor \neg \left(m \leq 1.75 \cdot 10^{-14}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;{\left(k \cdot k\right)}^{-1} \cdot a\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.21)
   (* (pow (* k k) -1.0) a)
   (if (<= m 0.39) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k a) k) 101.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.21) {
		tmp = pow((k * k), -1.0) * a;
	} else if (m <= 0.39) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.21)
		tmp = Float64((Float64(k * k) ^ -1.0) * a);
	elseif (m <= 0.39)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.21], N[(N[Power[N[(k * k), $MachinePrecision], -1.0], $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.39], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.209999999999999992

   1. Initial program 100.0%

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

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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. distribute-rgt-in N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 33.2

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

   7. Applied rewrites 33.2%

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

   8. Taylor expanded in k around inf

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

   10. Applied rewrites 60.2%

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

   if -0.209999999999999992 < m < 0.39000000000000001

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;{\left(k \cdot k\right)}^{-1} \cdot a\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{\frac{\frac{99}{k} + -10}{k} + 1}{k \cdot k} \cdot a\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.21)
   (* (/ (+ (/ (+ (/ 99.0 k) -10.0) k) 1.0) (* k k)) a)
   (if (<= m 0.39) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k a) k) 101.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.21) {
		tmp = (((((99.0 / k) + -10.0) / k) + 1.0) / (k * k)) * a;
	} else if (m <= 0.39) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.21)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(99.0 / k) + -10.0) / k) + 1.0) / Float64(k * k)) * a);
	elseif (m <= 0.39)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.21], N[(N[(N[(N[(N[(N[(99.0 / k), $MachinePrecision] + -10.0), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 0.39], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.209999999999999992

   1. Initial program 100.0%

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

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

      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 100.0

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

   4. Applied rewrites 100.0%

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

   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. distribute-rgt-in N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 33.2

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

   7. Applied rewrites 33.2%

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

   8. Taylor expanded in k around inf

      \[\leadsto \frac{\left(1 + \frac{99}{{k}^{2}}\right) - 10 \cdot \frac{1}{k}}{\color{blue}{{k}^{2}}} \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 67.7%

       \[\leadsto \frac{\frac{\frac{99}{k} + -10}{k} + 1}{\color{blue}{k \cdot k}} \cdot a \]

   if -0.209999999999999992 < m < 0.39000000000000001

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.21:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.21)
   (/ a (* k k))
   (if (<= m 0.39) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k a) k) 101.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.21) {
		tmp = a / (k * k);
	} else if (m <= 0.39) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.21)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.39)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.21], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.39], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.209999999999999992

   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 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 59.1%

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

   if -0.209999999999999992 < m < 0.39000000000000001

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 58.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -7.2e-39)
   (/ a (* k k))
   (if (<= m 0.39) (/ a (fma 10.0 k 1.0)) (* (* (* k a) k) 101.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -7.2e-39) {
		tmp = a / (k * k);
	} else if (m <= 0.39) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -7.2e-39)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.39)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -7.2e-39], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.39], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -7.2000000000000001e-39

   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 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.2%

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

   if -7.2000000000000001e-39 < m < 0.39000000000000001

   1. Initial program 97.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.1

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

   5. Applied rewrites 96.1%

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

   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 67.2%

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

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.39:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -6.8e-39)
   (/ a (* k k))
   (if (<= m 0.39) (* 1.0 a) (* (* (* k a) k) 101.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e-39) {
		tmp = a / (k * k);
	} else if (m <= 0.39) {
		tmp = 1.0 * a;
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	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 <= (-6.8d-39)) then
        tmp = a / (k * k)
    else if (m <= 0.39d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((k * a) * k) * 101.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= -6.8e-39) {
		tmp = a / (k * k);
	} else if (m <= 0.39) {
		tmp = 1.0 * a;
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= -6.8e-39:
		tmp = a / (k * k)
	elif m <= 0.39:
		tmp = 1.0 * a
	else:
		tmp = ((k * a) * k) * 101.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -6.8e-39)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.39)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= -6.8e-39)
		tmp = a / (k * k);
	elseif (m <= 0.39)
		tmp = 1.0 * a;
	else
		tmp = ((k * a) * k) * 101.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -6.8e-39], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.39], N[(1.0 * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -6.7999999999999998e-39

   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 34.3

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.2%

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

   if -6.7999999999999998e-39 < m < 0.39000000000000001

   1. Initial program 97.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.4

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

      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 97.5

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

   4. Applied rewrites 97.5%

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

   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. distribute-rgt-in N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 96.1

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

   7. Applied rewrites 96.1%

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

   8. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 60.1%

       \[\leadsto 1 \cdot a \]

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 36.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.39:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 101\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.39) (* 1.0 a) (* (* (* k a) k) 101.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.39) {
		tmp = 1.0 * a;
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	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 <= 0.39d0) then
        tmp = 1.0d0 * a
    else
        tmp = ((k * a) * k) * 101.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.39) {
		tmp = 1.0 * a;
	} else {
		tmp = ((k * a) * k) * 101.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.39:
		tmp = 1.0 * a
	else:
		tmp = ((k * a) * k) * 101.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.39)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 101.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.39)
		tmp = 1.0 * a;
	else
		tmp = ((k * a) * k) * 101.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.39], N[(1.0 * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 101.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.39000000000000001

   1. Initial program 98.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.3

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

      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 98.3

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

   4. Applied rewrites 98.3%

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

   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. distribute-rgt-in N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 61.5

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

   7. Applied rewrites 61.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 28.8%

       \[\leadsto 1 \cdot a \]

   if 0.39000000000000001 < m

   1. Initial program 69.3%

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

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 1.3%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 33.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-k, a \cdot -101, -10 \cdot a\right), \color{blue}{k}, a\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto 101 \cdot \left(a \cdot \color{blue}{{k}^{2}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 53.8%

       \[\leadsto \left(\left(k \cdot a\right) \cdot k\right) \cdot 101 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 25.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2850000000000:\\ \;\;\;\;1 \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot k\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2850000000000.0) (* 1.0 a) (* (* a k) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2850000000000.0) {
		tmp = 1.0 * a;
	} else {
		tmp = (a * k) * -10.0;
	}
	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 <= 2850000000000.0d0) then
        tmp = 1.0d0 * a
    else
        tmp = (a * k) * (-10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2850000000000.0) {
		tmp = 1.0 * a;
	} else {
		tmp = (a * k) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2850000000000.0:
		tmp = 1.0 * a
	else:
		tmp = (a * k) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2850000000000.0)
		tmp = Float64(1.0 * a);
	else
		tmp = Float64(Float64(a * k) * -10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2850000000000.0)
		tmp = 1.0 * a;
	else
		tmp = (a * k) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2850000000000.0], N[(1.0 * a), $MachinePrecision], N[(N[(a * k), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.85e12

   1. Initial program 97.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 97.8

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

      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 97.8

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

   4. Applied rewrites 97.8%

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

   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. distribute-rgt-in N/A

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

      2. unpow2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 60.5

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

   7. Applied rewrites 60.5%

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

   8. Taylor expanded in k around 0

      \[\leadsto 1 \cdot a \]
   9. Step-by-step derivation

   10. Applied rewrites 28.4%

       \[\leadsto 1 \cdot a \]

   if 2.85e12 < m

   1. Initial program 69.4%

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

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

   5. Applied rewrites 2.7%

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

   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 13.2%

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

   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 23.9%

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

Alternative 11: 20.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ 1 \cdot a \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (* 1.0 a))

C

double code(double a, double k, double m) {
	return 1.0 * a;
}

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 = 1.0d0 * a
end function

Java

public static double code(double a, double k, double m) {
	return 1.0 * a;
}

Python

def code(a, k, m):
	return 1.0 * a

Julia

function code(a, k, m)
	return Float64(1.0 * a)
end

MATLAB

function tmp = code(a, k, m)
	tmp = 1.0 * a;
end

Wolfram

code[a_, k_, m_] := N[(1.0 * a), $MachinePrecision]

Derivation

1. Initial program 88.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 88.3

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

   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 88.3

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

4. Applied rewrites 88.3%

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

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. distribute-rgt-in N/A

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

   2. unpow2 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-in N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 41.3

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

7. Applied rewrites 41.3%

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

8. Taylor expanded in k around 0

   \[\leadsto 1 \cdot a \]
9. Step-by-step derivation

10. Applied rewrites 20.2%

    \[\leadsto 1 \cdot a \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024360 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))