Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 99.7%

Time: 4.7s

Alternatives: 13

Speedup: 1.3×

• 22.4% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{10}{t\_0} + \frac{k}{t\_0}, k, \frac{1}{t\_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 1e-83)
     t_0
     (/ 1.0 (fma (+ (/ 10.0 t_0) (/ k t_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 (k <= 1e-83) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(((10.0 / t_0) + (k / t_0)), k, (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 1e-83)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(Float64(Float64(10.0 / t_0) + Float64(k / t_0)), k, Float64(1.0 / 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[k, 1e-83], t$95$0, N[(1.0 / N[(N[(N[(10.0 / t$95$0), $MachinePrecision] + N[(k / t$95$0), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1e-83

   1. Initial program 94.4%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 1e-83 < k

   1. Initial program 85.1%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 57.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 57.7

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

      17. *-commutative 57.7

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

   6. Applied rewrites 57.7%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 99.4%

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

Alternative 2: 98.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -150:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.00026:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{10}{a} + \frac{k}{a}, k, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -150.0)
     t_0
     (if (<= m 0.00026)
       (/ 1.0 (fma (+ (/ 10.0 a) (/ k a)) k (/ 1.0 a)))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -150.0) {
		tmp = t_0;
	} else if (m <= 0.00026) {
		tmp = 1.0 / fma(((10.0 / a) + (k / a)), k, (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -150.0)
		tmp = t_0;
	elseif (m <= 0.00026)
		tmp = Float64(1.0 / fma(Float64(Float64(10.0 / a) + Float64(k / a)), k, Float64(1.0 / a)));
	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, -150.0], t$95$0, If[LessEqual[m, 0.00026], N[(1.0 / N[(N[(N[(10.0 / a), $MachinePrecision] + N[(k / a), $MachinePrecision]), $MachinePrecision] * k + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -150 or 2.59999999999999977e-4 < m

   1. Initial program 88.5%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if -150 < m < 2.59999999999999977e-4

   1. Initial program 93.6%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 91.3

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

      17. *-commutative 91.3

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

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   9. Applied rewrites 99.2%

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

   10. Taylor expanded in m around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. mult-flip-rev N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-/.f64 97.2

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

   12. Applied rewrites 97.2%

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

Alternative 3: 97.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -150:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -150.0)
     t_0
     (if (<= m 4.5e-6) (* a (/ 1.0 (fma (+ 10.0 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 <= -150.0) {
		tmp = t_0;
	} else if (m <= 4.5e-6) {
		tmp = a * (1.0 / fma((10.0 + 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 <= -150.0)
		tmp = t_0;
	elseif (m <= 4.5e-6)
		tmp = Float64(a * Float64(1.0 / fma(Float64(10.0 + 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, -150.0], t$95$0, If[LessEqual[m, 4.5e-6], N[(a * N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -150 or 4.50000000000000011e-6 < m

   1. Initial program 88.5%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -150 < m < 4.50000000000000011e-6

   1. Initial program 93.6%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 91.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. mult-flip N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

          \[\leadsto a \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{1}{1 + k \cdot \left(10 + k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite=>}\left(+-commutative, \left(k \cdot \left(10 + k\right) + 1\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(*-commutative, \left(\left(10 + k\right) \cdot k\right)\right) + 1} \]

      16. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(10 + k, k, 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(10 + k\right)\right), k, 1\right)} \]

   6. Applied rewrites 91.9%

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

Alternative 4: 60.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -150:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;a \cdot \frac{1}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+188}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -150.0)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 2.0)
     (* a (/ 1.0 (fma (+ 10.0 k) k 1.0)))
     (if (<= m 6.2e+188)
       (* a (fma (fma 99.0 k -10.0) k 1.0))
       (* a (* -10.0 k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -150.0) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 2.0) {
		tmp = a * (1.0 / fma((10.0 + k), k, 1.0));
	} else if (m <= 6.2e+188) {
		tmp = a * fma(fma(99.0, k, -10.0), k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -150.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 2.0)
		tmp = Float64(a * Float64(1.0 / fma(Float64(10.0 + k), k, 1.0)));
	elseif (m <= 6.2e+188)
		tmp = Float64(a * fma(fma(99.0, k, -10.0), k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -150.0], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a * N[(1.0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+188], N[(a * N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -150

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 34.4

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

      17. *-commutative 34.4

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 60.9

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

   9. Applied rewrites 60.9%

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

   if -150 < m < 2

   1. Initial program 93.6%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. mult-flip N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

          \[\leadsto a \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{1}{1 + k \cdot \left(10 + k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite=>}\left(+-commutative, \left(k \cdot \left(10 + k\right) + 1\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(*-commutative, \left(\left(10 + k\right) \cdot k\right)\right) + 1} \]

      16. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(10 + k, k, 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(10 + k\right)\right), k, 1\right)} \]

   6. Applied rewrites 91.5%

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

   if 2 < m < 6.2000000000000004e188

   1. Initial program 77.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. mult-flip N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

          \[\leadsto a \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{1}{1 + k \cdot \left(10 + k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite=>}\left(+-commutative, \left(k \cdot \left(10 + k\right) + 1\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(*-commutative, \left(\left(10 + k\right) \cdot k\right)\right) + 1} \]

      16. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(10 + k, k, 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(10 + k\right)\right), k, 1\right)} \]

   6. Applied rewrites 3.1%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 32.1

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

   9. Applied rewrites 32.1%

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

   if 6.2000000000000004e188 < m

   1. Initial program 78.3%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 6.2

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

   7. Applied rewrites 6.2%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 18.3

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

   10. Applied rewrites 18.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 18.3

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

   12. Applied rewrites 18.3%

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

Alternative 5: 60.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -150:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 2:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{elif}\;m \leq 6.2 \cdot 10^{+188}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(\mathsf{fma}\left(99, k, -10\right), k, 1\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -150.0)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 2.0)
     (/ a (fma (+ 10.0 k) k 1.0))
     (if (<= m 6.2e+188)
       (* a (fma (fma 99.0 k -10.0) k 1.0))
       (* a (* -10.0 k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -150.0) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 2.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else if (m <= 6.2e+188) {
		tmp = a * fma(fma(99.0, k, -10.0), k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -150.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 2.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	elseif (m <= 6.2e+188)
		tmp = Float64(a * fma(fma(99.0, k, -10.0), k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -150.0], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 6.2e+188], N[(a * N[(N[(99.0 * k + -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if m < -150

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 34.4

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

      17. *-commutative 34.4

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 60.9

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

   9. Applied rewrites 60.9%

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

   if -150 < m < 2

   1. Initial program 93.6%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 2 < m < 6.2000000000000004e188

   1. Initial program 77.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. mult-flip N/A

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

      7. pow2 N/A

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

      8. associate-+l+ N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

          \[\leadsto a \cdot \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{1}{1 + k \cdot \left(10 + k\right)}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite=>}\left(+-commutative, \left(k \cdot \left(10 + k\right) + 1\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(*-commutative, \left(\left(10 + k\right) \cdot k\right)\right) + 1} \]

      16. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{Rewrite<=}\left(lift-fma.f64, \left(\mathsf{fma}\left(10 + k, k, 1\right)\right)\right)} \]

      17. *-commutative N/A

          \[\leadsto a \cdot \frac{1}{\mathsf{fma}\left(\mathsf{Rewrite<=}\left(lift-+.f64, \left(10 + k\right)\right), k, 1\right)} \]

   6. Applied rewrites 3.1%

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

   7. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-flip N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 32.1

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

   9. Applied rewrites 32.1%

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

   if 6.2000000000000004e188 < m

   1. Initial program 78.3%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 6.2

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

   7. Applied rewrites 6.2%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 18.3

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

   10. Applied rewrites 18.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 18.3

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

   12. Applied rewrites 18.3%

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

Alternative 6: 58.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -150:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 23000000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -150.0)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 23000000000.0) (/ a (fma (+ 10.0 k) k 1.0)) (* a (* -10.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -150.0) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 23000000000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -150.0)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 23000000000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -150.0], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 23000000000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -150

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 34.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 34.4

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

      17. *-commutative 34.4

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

   6. Applied rewrites 34.4%

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

   7. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 60.9

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

   9. Applied rewrites 60.9%

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

   if -150 < m < 2.3e10

   1. Initial program 93.3%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 90.2

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

   4. Applied rewrites 90.2%

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

   if 2.3e10 < m

   1. Initial program 77.5%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 7.7

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.1

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

   10. Applied rewrites 20.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 20.1

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

   12. Applied rewrites 20.1%

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

Alternative 7: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{\frac{k \cdot k}{a}}\\ \mathbf{elif}\;m \leq 1050000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+17)
   (/ 1.0 (/ (* k k) a))
   (if (<= m 1050000.0) (/ a (fma k k 1.0)) (* a (* -10.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+17) {
		tmp = 1.0 / ((k * k) / a);
	} else if (m <= 1050000.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+17)
		tmp = Float64(1.0 / Float64(Float64(k * k) / a));
	elseif (m <= 1050000.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+17], N[(1.0 / N[(N[(k * k), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1050000.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e17

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

   4. Applied rewrites 34.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. division-flip N/A

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

      7. lower-/.f64 N/A

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. pow2 N/A

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

      11. distribute-rgt-in N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. *-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-+.f64 34.7

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

      17. *-commutative 34.7

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

   6. Applied rewrites 34.7%

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

   7. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 61.1

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

   9. Applied rewrites 61.1%

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

   if -1.05e17 < m < 1.05e6

   1. Initial program 93.7%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in k around inf

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

   7. Applied rewrites 86.7%

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

   if 1.05e6 < m

   1. Initial program 77.4%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 7.6

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 20.1

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

   12. Applied rewrites 20.1%

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

Alternative 8: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.05 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 1050000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.05e+17)
   (/ a (* k k))
   (if (<= m 1050000.0) (/ a (fma k k 1.0)) (* a (* -10.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.05e+17) {
		tmp = a / (k * k);
	} else if (m <= 1050000.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.05e+17)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 1050000.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.05e+17], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1050000.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.05e17

   1. Initial program 100.0%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.5

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

   4. Applied rewrites 34.5%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.9

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

   7. Applied rewrites 60.9%

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

   if -1.05e17 < m < 1.05e6

   1. Initial program 93.7%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.8

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

   4. Applied rewrites 88.8%

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

   5. Taylor expanded in k around inf

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

   7. Applied rewrites 86.7%

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

   if 1.05e6 < m

   1. Initial program 77.4%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 7.6

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

   7. Applied rewrites 7.6%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.0

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

   10. Applied rewrites 20.0%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 20.1

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

   12. Applied rewrites 20.1%

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

Alternative 9: 47.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -150:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2 \cdot 10^{+28}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -150.0)
   (/ a (* k k))
   (if (<= m 2e+28) (/ a (fma 10.0 k 1.0)) (* a (* -10.0 k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -150.0) {
		tmp = a / (k * k);
	} else if (m <= 2e+28) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -150.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2e+28)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -150.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2e+28], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -150

   1. Initial program 100.0%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 34.2

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.7

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

   7. Applied rewrites 60.7%

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

   if -150 < m < 1.99999999999999992e28

   1. Initial program 92.4%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 85.8

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

   4. Applied rewrites 85.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 59.2%

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

   if 1.99999999999999992e28 < m

   1. Initial program 77.7%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 7.7

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.1

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

   10. Applied rewrites 20.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 20.1

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

   12. Applied rewrites 20.1%

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

Alternative 10: 42.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \mathbf{elif}\;k \leq 0.112:\\ \;\;\;\;\mathsf{fma}\left(k \cdot a, -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5e-310)
   (* a (* -10.0 k))
   (if (<= k 0.112) (fma (* k a) -10.0 a) (/ a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-310) {
		tmp = a * (-10.0 * k);
	} else if (k <= 0.112) {
		tmp = fma((k * a), -10.0, a);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5e-310)
		tmp = Float64(a * Float64(-10.0 * k));
	elseif (k <= 0.112)
		tmp = fma(Float64(k * a), -10.0, a);
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5e-310], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.112], N[(N[(k * a), $MachinePrecision] * -10.0 + a), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.999999999999985e-310

   1. Initial program 89.4%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 18.6

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

   4. Applied rewrites 18.6%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 8.7

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

   7. Applied rewrites 8.7%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 15.5

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

   10. Applied rewrites 15.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 15.6

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

   12. Applied rewrites 15.6%

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

   if 4.999999999999985e-310 < k < 0.112000000000000002

   1. Initial program 100.0%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 49.6

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

   4. Applied rewrites 49.6%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 49.3

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

   7. Applied rewrites 49.3%

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

   if 0.112000000000000002 < k

   1. Initial program 81.1%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 58.7

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

   7. Applied rewrites 58.7%

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

Alternative 11: 42.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-310}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \mathbf{elif}\;k \leq 820:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 5e-310) (* a (* -10.0 k)) (if (<= k 820.0) a (/ a (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-310) {
		tmp = a * (-10.0 * k);
	} else if (k <= 820.0) {
		tmp = a;
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 5d-310) then
        tmp = a * ((-10.0d0) * k)
    else if (k <= 820.0d0) then
        tmp = a
    else
        tmp = a / (k * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 5e-310) {
		tmp = a * (-10.0 * k);
	} else if (k <= 820.0) {
		tmp = a;
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 5e-310:
		tmp = a * (-10.0 * k)
	elif k <= 820.0:
		tmp = a
	else:
		tmp = a / (k * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 5e-310)
		tmp = Float64(a * Float64(-10.0 * k));
	elseif (k <= 820.0)
		tmp = a;
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 5e-310)
		tmp = a * (-10.0 * k);
	elseif (k <= 820.0)
		tmp = a;
	else
		tmp = a / (k * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 5e-310], N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 820.0], a, N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 4.999999999999985e-310

   1. Initial program 89.4%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 18.6

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

   4. Applied rewrites 18.6%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 8.7

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

   7. Applied rewrites 8.7%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 15.5

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

   10. Applied rewrites 15.5%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 15.6

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

   12. Applied rewrites 15.6%

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

   if 4.999999999999985e-310 < k < 820

   1. Initial program 100.0%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 49.7

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

   4. Applied rewrites 49.7%

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

   5. Taylor expanded in k around 0

      \[\leadsto a \]
   6. Step-by-step derivation

   7. Applied rewrites 48.3%

      \[\leadsto a \]

   if 820 < k

   1. Initial program 80.8%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 60.1

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

   4. Applied rewrites 60.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 59.3

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

   7. Applied rewrites 59.3%

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

Alternative 12: 25.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2 \cdot 10^{+28}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(-10 \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 2e+28) a (* a (* -10.0 k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2e+28) {
		tmp = a;
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 2d+28) then
        tmp = a
    else
        tmp = a * ((-10.0d0) * k)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 2e+28) {
		tmp = a;
	} else {
		tmp = a * (-10.0 * k);
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 2e+28:
		tmp = a
	else:
		tmp = a * (-10.0 * k)
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2e+28)
		tmp = a;
	else
		tmp = Float64(a * Float64(-10.0 * k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 2e+28)
		tmp = a;
	else
		tmp = a * (-10.0 * k);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2e+28], a, N[(a * N[(-10.0 * k), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 1.99999999999999992e28

   1. Initial program 95.8%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 62.3

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

   4. Applied rewrites 62.3%

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

   5. Taylor expanded in k around 0

      \[\leadsto a \]
   6. Step-by-step derivation

   7. Applied rewrites 27.4%

      \[\leadsto a \]

   if 1.99999999999999992e28 < m

   1. Initial program 77.7%

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

   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. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 7.7

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

   7. Applied rewrites 7.7%

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

   8. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 20.1

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

   10. Applied rewrites 20.1%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 20.1

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

   12. Applied rewrites 20.1%

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

Alternative 13: 20.2% accurate, 34.5× speedup

Math

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

FPCore

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

C

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

Fortran

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
end function

Java

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

Python

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

Julia

function code(a, k, m)
	return a
end

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 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. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 44.2

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

4. Applied rewrites 44.2%

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

5. Taylor expanded in k around 0

   \[\leadsto a \]
6. Step-by-step derivation

7. Applied rewrites 20.2%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))