Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.9%

Time: 5.0s

Alternatives: 14

Speedup: 1.3×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% 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: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.9:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, \frac{k}{\mathsf{fma}\left(10, k, 1\right)}, 1\right)} \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.9)
   (* (/ a (fma k (/ k (fma 10.0 k 1.0)) 1.0)) (/ (pow k m) (fma 10.0 k 1.0)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.9) {
		tmp = (a / fma(k, (k / fma(10.0, k, 1.0)), 1.0)) * (pow(k, m) / fma(10.0, k, 1.0));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.9)
		tmp = Float64(Float64(a / fma(k, Float64(k / fma(10.0, k, 1.0)), 1.0)) * Float64((k ^ m) / fma(10.0, k, 1.0)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.9], N[(N[(a / N[(k * N[(k / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.89999999999999991

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. sum-to-mult N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 96.2

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

      18. lift-+.f64 N/A

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

   3. Applied rewrites 96.2%

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

   if 3.89999999999999991 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 3.9:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 3.9)
   (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ (pow k m) (fma 10.0 k 1.0)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 3.9) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (pow(k, m) / fma(10.0, k, 1.0));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 3.9)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64((k ^ m) / fma(10.0, k, 1.0)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 3.9], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 3.89999999999999991

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   if 3.89999999999999991 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.095:\\ \;\;\;\;\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \frac{1}{1 + 10 \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.2e-11)
     t_0
     (if (<= m 0.095)
       (* (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (/ 1.0 (+ 1.0 (* 10.0 k))))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -1.2e-11) {
		tmp = t_0;
	} else if (m <= 0.095) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / (1.0 + (10.0 * k)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (k ** m) * a
    if (m <= (-1.2d-11)) then
        tmp = t_0
    else if (m <= 0.095d0) then
        tmp = (a / ((k / (10.0d0 - ((-1.0d0) / k))) - (-1.0d0))) * (1.0d0 / (1.0d0 + (10.0d0 * k)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.pow(k, m) * a;
	double tmp;
	if (m <= -1.2e-11) {
		tmp = t_0;
	} else if (m <= 0.095) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / (1.0 + (10.0 * k)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = math.pow(k, m) * a
	tmp = 0
	if m <= -1.2e-11:
		tmp = t_0
	elif m <= 0.095:
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / (1.0 + (10.0 * k)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.2e-11)
		tmp = t_0;
	elseif (m <= 0.095)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) * Float64(1.0 / Float64(1.0 + Float64(10.0 * k))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (k ^ m) * a;
	tmp = 0.0;
	if (m <= -1.2e-11)
		tmp = t_0;
	elseif (m <= 0.095)
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) * (1.0 / (1.0 + (10.0 * k)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.2e-11], t$95$0, If[LessEqual[m, 0.095], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.2000000000000001e-11 or 0.095000000000000001 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

   if -1.2000000000000001e-11 < m < 0.095000000000000001

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 44.3

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

   8. Applied rewrites 44.3%

      \[\leadsto \frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1} \cdot \color{blue}{\frac{1}{1 + 10 \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.095:\\ \;\;\;\;\frac{\frac{a}{\frac{k}{10 - \frac{-1}{k}} - -1}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.2e-11)
     t_0
     (if (<= m 0.095)
       (/ (/ a (- (/ k (- 10.0 (/ -1.0 k))) -1.0)) (fma 10.0 k 1.0))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -1.2e-11) {
		tmp = t_0;
	} else if (m <= 0.095) {
		tmp = (a / ((k / (10.0 - (-1.0 / k))) - -1.0)) / fma(10.0, k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.2e-11)
		tmp = t_0;
	elseif (m <= 0.095)
		tmp = Float64(Float64(a / Float64(Float64(k / Float64(10.0 - Float64(-1.0 / k))) - -1.0)) / fma(10.0, k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.2e-11], t$95$0, If[LessEqual[m, 0.095], N[(N[(a / N[(N[(k / N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.2000000000000001e-11 or 0.095000000000000001 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

   if -1.2000000000000001e-11 < m < 0.095000000000000001

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate-+l+ N/A

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

      5. sum-to-mult N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-/.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. associate-/r* N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

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

Alternative 5: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. sum-to-mult N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if m < 3.5

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.3

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-rgt-out N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-+.f64 90.3

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

   3. Applied rewrites 90.3%

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

   if 3.5 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

Alternative 7: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.2 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.095:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.2e-11)
     t_0
     (if (<= m 0.095) (/ a (fma k 10.0 (fma k k 1.0))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -1.2e-11) {
		tmp = t_0;
	} else if (m <= 0.095) {
		tmp = a / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.2e-11)
		tmp = t_0;
	elseif (m <= 0.095)
		tmp = Float64(a / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -1.2e-11], t$95$0, If[LessEqual[m, 0.095], N[(a / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.2000000000000001e-11 or 0.095000000000000001 < m

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in k around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 82.5

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

   8. Applied rewrites 82.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 82.5

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

   10. Applied rewrites 82.5%

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

   if -1.2000000000000001e-11 < m < 0.095000000000000001

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

Alternative 8: 61.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 120000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.3)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 120000.0)
     (/ a (fma k 10.0 (fma k k 1.0)))
     (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.3) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 120000.0) {
		tmp = a / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.3)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 120000.0)
		tmp = Float64(a / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.3], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 120000.0], N[(a / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.299999999999999989

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -0.299999999999999989 < m < 1.2e5

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   if 1.2e5 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. distribute-rgt-in N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. associate-/r/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 44.6

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.6

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. add-flip N/A

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

      20. lower--.f64 N/A

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

      21. metadata-eval 44.6

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

   6. Applied rewrites 44.6%

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

   7. Taylor expanded in k around 0

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 28.5

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

   9. Applied rewrites 28.5%

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 54.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, 10, \mathsf{fma}\left(k, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.3)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 1.25e+16)
     (/ a (fma k 10.0 (fma k k 1.0)))
     (fma (* -10.0 k) a a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.3) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 1.25e+16) {
		tmp = a / fma(k, 10.0, fma(k, k, 1.0));
	} else {
		tmp = fma((-10.0 * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.3)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 1.25e+16)
		tmp = Float64(a / fma(k, 10.0, fma(k, k, 1.0)));
	else
		tmp = fma(Float64(-10.0 * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.3], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.25e+16], N[(a / N[(k * 10.0 + N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.299999999999999989

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -0.299999999999999989 < m < 1.25e16

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   if 1.25e16 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(10\right)\right) \cdot k\right) \cdot a + a \]

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

         \[\leadsto \left(\mathsf{neg}\left(10 \cdot k\right)\right) \cdot a + a \]

      9. lower-fma.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 20.6

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

   9. Applied rewrites 20.6%

      \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 54.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.3:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.3)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 1.25e+16) (/ a (fma (- k -10.0) k 1.0)) (fma (* -10.0 k) a a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.3) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 1.25e+16) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma((-10.0 * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.3)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 1.25e+16)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = fma(Float64(-10.0 * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.3], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.25e+16], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.299999999999999989

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if -0.299999999999999989 < m < 1.25e16

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 44.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. add-flip N/A

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

      13. lower--.f64 N/A

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

      14. metadata-eval 44.6

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

   6. Applied rewrites 44.6%

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

   if 1.25e16 < m

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.6

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

   7. Applied rewrites 20.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(10\right)\right) \cdot k\right) \cdot a + a \]

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

         \[\leadsto \left(\mathsf{neg}\left(10 \cdot k\right)\right) \cdot a + a \]

      9. lower-fma.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 20.6

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

   9. Applied rewrites 20.6%

      \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 40.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := 1 + 10 \cdot k\\ t_2 := \frac{a \cdot {k}^{m}}{t\_1 + k \cdot k}\\ \mathbf{if}\;t\_2 \leq 10^{-317}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{a}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (+ 1.0 (* 10.0 k)))
        (t_2 (/ (* a (pow k m)) (+ t_1 (* k k)))))
   (if (<= t_2 1e-317) t_0 (if (<= t_2 4e+291) (/ a t_1) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = 1.0 + (10.0 * k);
	double t_2 = (a * pow(k, m)) / (t_1 + (k * k));
	double tmp;
	if (t_2 <= 1e-317) {
		tmp = t_0;
	} else if (t_2 <= 4e+291) {
		tmp = a / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = a / (k * k)
    t_1 = 1.0d0 + (10.0d0 * k)
    t_2 = (a * (k ** m)) / (t_1 + (k * k))
    if (t_2 <= 1d-317) then
        tmp = t_0
    else if (t_2 <= 4d+291) then
        tmp = a / t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = 1.0 + (10.0 * k);
	double t_2 = (a * Math.pow(k, m)) / (t_1 + (k * k));
	double tmp;
	if (t_2 <= 1e-317) {
		tmp = t_0;
	} else if (t_2 <= 4e+291) {
		tmp = a / t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	t_1 = 1.0 + (10.0 * k)
	t_2 = (a * math.pow(k, m)) / (t_1 + (k * k))
	tmp = 0
	if t_2 <= 1e-317:
		tmp = t_0
	elif t_2 <= 4e+291:
		tmp = a / t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(1.0 + Float64(10.0 * k))
	t_2 = Float64(Float64(a * (k ^ m)) / Float64(t_1 + Float64(k * k)))
	tmp = 0.0
	if (t_2 <= 1e-317)
		tmp = t_0;
	elseif (t_2 <= 4e+291)
		tmp = Float64(a / t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	t_1 = 1.0 + (10.0 * k);
	t_2 = (a * (k ^ m)) / (t_1 + (k * k));
	tmp = 0.0;
	if (t_2 <= 1e-317)
		tmp = t_0;
	elseif (t_2 <= 4e+291)
		tmp = a / t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 1e-317], t$95$0, If[LessEqual[t$95$2, 4e+291], N[(a / t$95$1), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000023e-317 or 3.9999999999999998e291 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. pow2 N/A

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

      3. lift-*.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if 1.00000023e-317 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.9999999999999998e291

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 27.9

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

   7. Applied rewrites 27.9%

      \[\leadsto \frac{a}{1 + 10 \cdot \color{blue}{k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 10 \cdot k\\ t_1 := \frac{a \cdot {k}^{m}}{t\_0 + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 10^{-317}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{a}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{k \cdot k} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 10.0 k))) (t_1 (/ (* a (pow k m)) (+ t_0 (* k k)))))
   (if (<= t_1 1e-317)
     (/ a (* k k))
     (if (<= t_1 4e+291) (/ a t_0) (* (/ 1.0 (* k k)) a)))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 + (10.0 * k);
	double t_1 = (a * pow(k, m)) / (t_0 + (k * k));
	double tmp;
	if (t_1 <= 1e-317) {
		tmp = a / (k * k);
	} else if (t_1 <= 4e+291) {
		tmp = a / t_0;
	} else {
		tmp = (1.0 / (k * k)) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (10.0d0 * k)
    t_1 = (a * (k ** m)) / (t_0 + (k * k))
    if (t_1 <= 1d-317) then
        tmp = a / (k * k)
    else if (t_1 <= 4d+291) then
        tmp = a / t_0
    else
        tmp = (1.0d0 / (k * k)) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = 1.0 + (10.0 * k);
	double t_1 = (a * Math.pow(k, m)) / (t_0 + (k * k));
	double tmp;
	if (t_1 <= 1e-317) {
		tmp = a / (k * k);
	} else if (t_1 <= 4e+291) {
		tmp = a / t_0;
	} else {
		tmp = (1.0 / (k * k)) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = 1.0 + (10.0 * k)
	t_1 = (a * math.pow(k, m)) / (t_0 + (k * k))
	tmp = 0
	if t_1 <= 1e-317:
		tmp = a / (k * k)
	elif t_1 <= 4e+291:
		tmp = a / t_0
	else:
		tmp = (1.0 / (k * k)) * a
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(1.0 + Float64(10.0 * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(t_0 + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 1e-317)
		tmp = Float64(a / Float64(k * k));
	elseif (t_1 <= 4e+291)
		tmp = Float64(a / t_0);
	else
		tmp = Float64(Float64(1.0 / Float64(k * k)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = 1.0 + (10.0 * k);
	t_1 = (a * (k ^ m)) / (t_0 + (k * k));
	tmp = 0.0;
	if (t_1 <= 1e-317)
		tmp = a / (k * k);
	elseif (t_1 <= 4e+291)
		tmp = a / t_0;
	else
		tmp = (1.0 / (k * k)) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-317], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+291], N[(a / t$95$0), $MachinePrecision], N[(N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000023e-317

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. pow2 N/A

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

      3. lift-*.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if 1.00000023e-317 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.9999999999999998e291

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around 0

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

      1. lower-*.f64 27.9

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

   7. Applied rewrites 27.9%

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

   if 3.9999999999999998e291 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. div-flip N/A

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

      6. associate-/r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{-1 \cdot -1}{k \cdot k} \cdot a \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot -1}{k \cdot k} \cdot a \]

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. frac-times N/A

          \[\leadsto \frac{-1 \cdot -1}{k \cdot k} \cdot a \]

      16. metadata-eval N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 35.9

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

   9. Applied rewrites 35.9%

      \[\leadsto \frac{1}{k \cdot k} \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 40.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 10^{-317}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+291}:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_1 1e-317) t_0 (if (<= t_1 4e+291) (/ a 1.0) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 1e-317) {
		tmp = t_0;
	} else if (t_1 <= 4e+291) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = a / (k * k)
    t_1 = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    if (t_1 <= 1d-317) then
        tmp = t_0
    else if (t_1 <= 4d+291) then
        tmp = a / 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 1e-317) {
		tmp = t_0;
	} else if (t_1 <= 4e+291) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	t_1 = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_1 <= 1e-317:
		tmp = t_0
	elif t_1 <= 4e+291:
		tmp = a / 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 1e-317)
		tmp = t_0;
	elseif (t_1 <= 4e+291)
		tmp = Float64(a / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	t_1 = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_1 <= 1e-317)
		tmp = t_0;
	elseif (t_1 <= 4e+291)
		tmp = a / 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, 1e-317], t$95$0, If[LessEqual[t$95$1, 4e+291], N[(a / 1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.00000023e-317 or 3.9999999999999998e291 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.8

         \[\leadsto \frac{a}{{k}^{2}} \]

   7. Applied rewrites 35.8%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{a}{{k}^{2}} \]

      2. pow2 N/A

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

      3. lift-*.f64 35.8

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

   9. Applied rewrites 35.8%

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

   if 1.00000023e-317 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 3.9999999999999998e291

   1. Initial program 90.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.6

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

   4. Applied rewrites 44.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+l+ N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-fma.f64 44.6

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

   6. Applied rewrites 44.6%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 19.6%

      \[\leadsto \frac{a}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 19.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.3%

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

2. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 44.6

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

4. Applied rewrites 44.6%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. associate-+l+ N/A

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

   4. +-commutative N/A

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

   5. +-commutative N/A

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

   6. associate-+l+ N/A

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

   7. +-commutative N/A

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

   8. associate-+l+ N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lower-fma.f64 44.6

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

6. Applied rewrites 44.6%

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

7. Taylor expanded in k around 0

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

9. Applied rewrites 19.6%

   \[\leadsto \frac{a}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))