Falkner and Boettcher, Appendix A

Percentage Accurate: 91.0% → 97.6%

Time: 7.7s

Alternatives: 14

Speedup: 1.1×

• 21.2% 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: 91.0% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.4e-17)
   (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.4e-17) {
		tmp = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 4.4d-17) then
        tmp = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 4.4e-17) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 4.4e-17:
		tmp = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 4.4e-17)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 4.4e-17)
		tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 4.4e-17], 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], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4.4e-17

   1. Initial program 96.8%

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

   if 4.4e-17 < m

   1. Initial program 74.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. log-rec N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 41.1

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 100.0

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

   11. Applied rewrites 100.0%

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

Alternative 2: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-18} \lor \neg \left(m \leq 4.4 \cdot 10^{-17}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -5.4e-18) (not (<= m 4.4e-17)))
   (* (pow k m) a)
   (/ a (fma (- k -10.0) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -5.4e-18) || !(m <= 4.4e-17)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((k - -10.0), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -5.4e-18) || !(m <= 4.4e-17))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -5.4e-18], N[Not[LessEqual[m, 4.4e-17]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.39999999999999977e-18 or 4.4e-17 < m

   1. Initial program 87.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. exp-prod N/A

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

      8. lower-pow.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. log-rec N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-/.f64 49.5

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

   8. Applied rewrites 49.5%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 100.0

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

   11. Applied rewrites 100.0%

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

   if -5.39999999999999977e-18 < m < 4.4e-17

   1. Initial program 93.0%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 93.0

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

   5. Applied rewrites 93.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.4 \cdot 10^{-18} \lor \neg \left(m \leq 4.4 \cdot 10^{-17}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.44:\\ \;\;\;\;\frac{\frac{99}{k \cdot k} \cdot a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.44)
   (/ (* (/ 99.0 (* k k)) a) (* k k))
   (if (<= m 0.48) (/ a (fma (- k -10.0) k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.44) {
		tmp = ((99.0 / (k * k)) * a) / (k * k);
	} else if (m <= 0.48) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.44)
		tmp = Float64(Float64(Float64(99.0 / Float64(k * k)) * a) / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.44], N[(N[(N[(99.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.440000000000000002

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around -inf

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.4%

       \[\leadsto \frac{\left(\frac{\frac{99}{k} - 10}{k} + 1\right) \cdot a}{k \cdot k} \]

   12. Taylor expanded in k around 0

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

   14. Applied rewrites 74.6%

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

   if -0.440000000000000002 < m < 0.47999999999999998

   1. Initial program 93.3%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 0.47999999999999998 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 4: 68.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.44:\\ \;\;\;\;\frac{\frac{a}{k} \cdot -10}{\left(-k\right) \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.44)
   (/ (* (/ a k) -10.0) (* (- k) k))
   (if (<= m 0.48) (/ a (fma (- k -10.0) k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.44) {
		tmp = ((a / k) * -10.0) / (-k * k);
	} else if (m <= 0.48) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.44)
		tmp = Float64(Float64(Float64(a / k) * -10.0) / Float64(Float64(-k) * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.44], N[(N[(N[(a / k), $MachinePrecision] * -10.0), $MachinePrecision] / N[((-k) * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.440000000000000002

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 45.0%

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

   10. Applied rewrites 52.1%

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

   11. Taylor expanded in k around 0

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

   13. Applied rewrites 59.4%

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

   if -0.440000000000000002 < m < 0.47999999999999998

   1. Initial program 93.3%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 0.47999999999999998 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 5: 54.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;m \leq -1.7 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.4 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot a\\ \mathbf{elif}\;m \leq 4.4 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= m -1.7e-178)
     t_0
     (if (<= m 4.4e-78)
       (* (fma -10.0 k 1.0) a)
       (if (<= m 4.4e-17) t_0 (* (* (* k a) k) 99.0))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (m <= -1.7e-178) {
		tmp = t_0;
	} else if (m <= 4.4e-78) {
		tmp = fma(-10.0, k, 1.0) * a;
	} else if (m <= 4.4e-17) {
		tmp = t_0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (m <= -1.7e-178)
		tmp = t_0;
	elseif (m <= 4.4e-78)
		tmp = Float64(fma(-10.0, k, 1.0) * a);
	elseif (m <= 4.4e-17)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.7e-178], t$95$0, If[LessEqual[m, 4.4e-78], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[m, 4.4e-17], t$95$0, N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.69999999999999986e-178 or 4.3999999999999998e-78 < m < 4.4e-17

   1. Initial program 99.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 51.5

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

   5. Applied rewrites 51.5%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 58.7%

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

   if -1.69999999999999986e-178 < m < 4.3999999999999998e-78

   1. Initial program 90.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   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 63.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 63.7%

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

   if 4.4e-17 < m

   1. Initial program 74.4%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 4.3

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

   5. Applied rewrites 4.3%

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

   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 8.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.9%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 53.9%

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

Alternative 6: 70.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.44:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.44)
   (/ a (* k k))
   (if (<= m 0.48) (/ a (fma (- k -10.0) k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.44) {
		tmp = a / (k * k);
	} else if (m <= 0.48) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.44)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.44], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.440000000000000002

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 59.0%

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

   if -0.440000000000000002 < m < 0.47999999999999998

   1. Initial program 93.3%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if 0.47999999999999998 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 7: 60.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.00027:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.48:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.00027)
   (/ a (* k k))
   (if (<= m 0.48) (/ a (fma 10.0 k 1.0)) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.00027) {
		tmp = a / (k * k);
	} else if (m <= 0.48) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.00027)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.48)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.00027], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.48], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.70000000000000003e-4

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 35.9

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

   5. Applied rewrites 35.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 59.0%

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

   if -2.70000000000000003e-4 < m < 0.47999999999999998

   1. Initial program 93.3%

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

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   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 67.9%

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

   if 0.47999999999999998 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 8: 54.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(99, k, -10\right), k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e-178)
   (/ a (* k k))
   (if (<= m 0.26) (fma (* a (fma 99.0 k -10.0)) k a) (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e-178) {
		tmp = a / (k * k);
	} else if (m <= 0.26) {
		tmp = fma((a * fma(99.0, k, -10.0)), k, a);
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e-178)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.26)
		tmp = fma(Float64(a * fma(99.0, k, -10.0)), k, a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e-178], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.26], N[(N[(a * N[(99.0 * k + -10.0), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.69999999999999986e-178

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.7%

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

   if -1.69999999999999986e-178 < m < 0.26000000000000001

   1. Initial program 90.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   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 56.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 57.1%

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

   13. Applied rewrites 57.1%

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

   if 0.26000000000000001 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 9: 54.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.26:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.7e-178)
   (/ a (* k k))
   (if (<= m 0.26)
     (* (fma (fma 99.0 k -10.0) k 1.0) a)
     (* (* (* k a) k) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.7e-178) {
		tmp = a / (k * k);
	} else if (m <= 0.26) {
		tmp = fma(fma(99.0, k, -10.0), k, 1.0) * a;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.7e-178)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.26)
		tmp = Float64(fma(fma(99.0, k, -10.0), k, 1.0) * a);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.7e-178], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.26], N[(N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] * a), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * k), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.69999999999999986e-178

   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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 48.0

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

   5. Applied rewrites 48.0%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 57.7%

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

   if -1.69999999999999986e-178 < m < 0.26000000000000001

   1. Initial program 90.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   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 56.7%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 57.1%

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

   12. Taylor expanded in a around 0

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

   14. Applied rewrites 57.1%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right) \cdot a \]

   if 0.26000000000000001 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 10: 37.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.26:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot a\right) \cdot k\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26) (/ a 1.0) (* (* (* k a) k) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / 1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.26d0) then
        tmp = a / 1.0d0
    else
        tmp = ((k * a) * k) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = a / 1.0;
	} else {
		tmp = ((k * a) * k) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.26:
		tmp = a / 1.0
	else:
		tmp = ((k * a) * k) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.26)
		tmp = Float64(a / 1.0);
	else
		tmp = Float64(Float64(Float64(k * a) * k) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.26)
		tmp = a / 1.0;
	else
		tmp = ((k * a) * k) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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 42.0%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 27.6%

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

   if 0.26000000000000001 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 30.3%

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

   12. Taylor expanded in k around inf

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

   14. Applied rewrites 54.5%

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

Alternative 11: 26.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 5.5e+15) (/ a 1.0) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.5e+15) {
		tmp = a / 1.0;
	} else {
		tmp = (-10.0 * a) * k;
	}
	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 <= 5.5d+15) then
        tmp = a / 1.0d0
    else
        tmp = ((-10.0d0) * a) * k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 5.5e+15) {
		tmp = a / 1.0;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 5.5e+15:
		tmp = a / 1.0
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 5.5e+15)
		tmp = Float64(a / 1.0);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 5.5e+15)
		tmp = a / 1.0;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 5.5e+15], N[(a / 1.0), $MachinePrecision], N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 5.5e15

   1. Initial program 95.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 61.8

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

   5. Applied rewrites 61.8%

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

   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 41.5%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 27.3%

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

   if 5.5e15 < m

   1. Initial program 75.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.4

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

   5. Applied rewrites 3.4%

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

   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 7.7%

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

   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 21.3%

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

   13. Applied rewrites 21.3%

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

Alternative 12: 26.0% accurate, 7.4× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26) (fma (* k a) -10.0 a) (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.26) {
		tmp = fma((k * a), -10.0, a);
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.26)
		tmp = fma(Float64(k * a), -10.0, a);
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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 27.1%

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

   if 0.26000000000000001 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   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 20.8%

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

   13. Applied rewrites 20.8%

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

Alternative 13: 25.9% accurate, 7.4× speedup

Math

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

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 0.26) (* (fma -10.0 k 1.0) a) (* (* -10.0 a) k)))

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 0.26000000000000001

   1. Initial program 96.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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 27.1%

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

   9. Taylor expanded in k around 0

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

   11. Applied rewrites 27.1%

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

   if 0.26000000000000001 < m

   1. Initial program 74.2%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-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. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. fp-cancel-sign-sub-inv N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 3.3

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

   5. Applied rewrites 3.3%

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

   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 7.6%

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

   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 20.8%

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

   13. Applied rewrites 20.8%

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

Alternative 14: 8.9% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.9%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. metadata-eval N/A

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

   9. fp-cancel-sign-sub-inv N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 41.9

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

5. Applied rewrites 41.9%

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

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 20.3%

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

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 8.8%

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

13. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))