Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 97.5%

Time: 4.4s

Alternatives: 14

Speedup: 1.3×

• 22.7% 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: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m 4e-12) (/ t_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 <= 4e-12) {
		tmp = t_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 <= 4e-12)
		tmp = Float64(t_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, 4e-12], N[(t$95$0 / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if m < 3.99999999999999992e-12

   1. Initial program 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 96.9

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. pow2 N/A

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

      11. associate-+r+ N/A

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

      12. pow2 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 97.0

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

   3. Applied rewrites 97.0%

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

   if 3.99999999999999992e-12 < m

   1. Initial program 77.8%

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

   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 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 2: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.75 \cdot 10^{-14}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.00021:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -1.75e-14)
   (/ (* a (pow k m)) (* k k))
   (if (<= m 0.00021)
     (/ (fma (* (log k) m) a a) (fma (+ 10.0 k) k 1.0))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -1.75e-14) {
		tmp = (a * pow(k, m)) / (k * k);
	} else if (m <= 0.00021) {
		tmp = fma((log(k) * m), a, a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -1.75e-14)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(k * k));
	elseif (m <= 0.00021)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -1.75e-14], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.00021], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.7500000000000001e-14

   1. Initial program 99.7%

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

   2. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -1.7500000000000001e-14 < m < 2.1000000000000001e-4

   1. Initial program 94.1%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 93.9

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

   4. Applied rewrites 93.9%

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

   if 2.1000000000000001e-4 < m

   1. Initial program 77.5%

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

   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 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 3: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -8400000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 0.00021:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -8400000000.0)
     t_0
     (if (<= m 0.00021)
       (/ (fma (* (log k) m) a a) (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 <= -8400000000.0) {
		tmp = t_0;
	} else if (m <= 0.00021) {
		tmp = fma((log(k) * m), a, a) / 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 <= -8400000000.0)
		tmp = t_0;
	elseif (m <= 0.00021)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / 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, -8400000000.0], t$95$0, If[LessEqual[m, 0.00021], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -8.4e9 or 2.1000000000000001e-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 99.7

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

   4. Applied rewrites 99.7%

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

   if -8.4e9 < m < 2.1000000000000001e-4

   1. Initial program 94.1%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

Alternative 4: 96.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -8400000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -8400000000.0)
     t_0
     (if (<= m 4e-12) (/ a (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 <= -8400000000.0) {
		tmp = t_0;
	} else if (m <= 4e-12) {
		tmp = a / 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 <= -8400000000.0)
		tmp = t_0;
	elseif (m <= 4e-12)
		tmp = Float64(a / 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, -8400000000.0], t$95$0, If[LessEqual[m, 4e-12], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -8.4e9 or 3.99999999999999992e-12 < m

   1. Initial program 88.6%

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

   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 99.2

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

   4. Applied rewrites 99.2%

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

   if -8.4e9 < m < 3.99999999999999992e-12

   1. Initial program 94.1%

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

   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 91.7

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

   4. Applied rewrites 91.7%

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

Alternative 5: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8400000000.0)
   (/ a (* k k))
   (if (<= m 5.5e+15)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (- (- (* (* -99.0 a) k)) (* 10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 5.5e+15) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((-((-99.0 * a) * k) - (10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 5.5e+15)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(Float64(-Float64(Float64(-99.0 * a) * k)) - Float64(10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5.5e+15], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(-99.0 * a), $MachinePrecision] * k), $MachinePrecision]) - N[(10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.4e9

   1. Initial program 99.9%

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

   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 35.8

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.0

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

   7. Applied rewrites 62.0%

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

   if -8.4e9 < m < 5.5e15

   1. Initial program 93.8%

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

   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 88.7

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

   4. Applied rewrites 88.7%

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

   if 5.5e15 < 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

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

      1. +-commutative N/A

         \[\leadsto k \cdot \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) + a \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a\right) \cdot k + a \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \left(k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(k \cdot \left(a + -100 \cdot a\right)\right)\right) - 10 \cdot a, k, a\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-k \cdot \left(a + -100 \cdot a\right)\right) - 10 \cdot a, k, a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(a + -100 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(-\left(\left(-100 + 1\right) \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right) \]

      11. metadata-eval N/A

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

      12. lower-*.f64 28.4

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

   7. Applied rewrites 28.4%

      \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, \color{blue}{k}, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 54.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log k \cdot m, a, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8400000000.0)
   (/ a (* k k))
   (if (<= m 4e-12) (/ a (fma (+ 10.0 k) k 1.0)) (fma (* (log k) m) a a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 4e-12) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((log(k) * m), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 4e-12)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(log(k) * m), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4e-12], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.4e9

   1. Initial program 99.9%

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

   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 35.8

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.0

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

   7. Applied rewrites 62.0%

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

   if -8.4e9 < m < 3.99999999999999992e-12

   1. Initial program 94.1%

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

   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 91.7

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

   4. Applied rewrites 91.7%

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

   if 3.99999999999999992e-12 < m

   1. Initial program 77.8%

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

   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 98.6

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in m around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

         \[\leadsto a \cdot \left(m \cdot \log k\right) + a \]

      3. *-commutative N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a + a \]

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 9.4

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

   7. Applied rewrites 9.4%

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

Alternative 7: 54.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -8400000000:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\log k \cdot m\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -8400000000.0)
   (/ a (* k k))
   (if (<= m 7.2e+25) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (log k) m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -8400000000.0) {
		tmp = a / (k * k);
	} else if (m <= 7.2e+25) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (log(k) * m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -8400000000.0)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 7.2e+25)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(log(k) * m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -8400000000.0], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 7.2e+25], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8.4e9

   1. Initial program 99.9%

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

   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 35.8

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.0

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

   7. Applied rewrites 62.0%

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

   if -8.4e9 < m < 7.20000000000000031e25

   1. Initial program 93.2%

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

   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 86.2

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

   4. Applied rewrites 86.2%

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

   if 7.20000000000000031e25 < m

   1. Initial program 77.5%

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

   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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in m around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

         \[\leadsto a \cdot \left(m \cdot \log k\right) + a \]

      3. *-commutative N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a + a \]

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-log.f64 9.3

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

   7. Applied rewrites 9.3%

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

   8. Taylor expanded in m around inf

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

      1. *-commutative N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

      4. lift-log.f64 N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

      5. lift-*.f64 9.3

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

   10. Applied rewrites 9.3%

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

Alternative 8: 47.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 0.0)
     (/ (/ a k) k)
     (if (<= t_0 2e+305) (/ a (fma (+ 10.0 k) k 1.0)) (/ a (* k k))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (a / k) / k;
	} else if (t_0 <= 2e+305) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(a / k) / k);
	elseif (t_0 <= 2e+305)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$0, 2e+305], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 97.0%

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

   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 46.9

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 15.5%

      \[\leadsto a \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   9. 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-+r+ N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 40.8

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

   10. Applied rewrites 40.8%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 39.5

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

   12. Applied rewrites 39.5%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

   if 1.9999999999999999e305 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 61.4%

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

   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 2.9

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

   4. Applied rewrites 2.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

Alternative 9: 47.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_0 \leq 10^{-255}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_0 1e-255)
     (/ (/ a k) k)
     (if (<= t_0 2e+305) (/ a (fma k k 1.0)) (/ a (* k k))))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 1e-255) {
		tmp = (a / k) / k;
	} else if (t_0 <= 2e+305) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = a / (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 1e-255)
		tmp = Float64(Float64(a / k) / k);
	elseif (t_0 <= 2e+305)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(a / Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-255], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$0, 2e+305], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1e-255

   1. Initial program 97.1%

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

   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 47.9

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

   4. Applied rewrites 47.9%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 16.3%

      \[\leadsto a \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   9. 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-+r+ N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 40.7

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

   10. Applied rewrites 40.7%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 39.5

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

   12. Applied rewrites 39.5%

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

   if 1e-255 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.9999999999999999e305

   1. Initial program 99.8%

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

   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 96.6

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

   4. Applied rewrites 96.6%

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

   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 93.2%

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

   if 1.9999999999999999e305 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 61.4%

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

   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 2.9

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

   4. Applied rewrites 2.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

Alternative 10: 47.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 10:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.32e-307)
   (/ a (* k k))
   (if (<= k 10.0) (/ a (fma 10.0 k 1.0)) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.32e-307) {
		tmp = a / (k * k);
	} else if (k <= 10.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.32e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 10.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.32e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 10.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3199999999999999e-307

   1. Initial program 89.5%

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

   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 20.4

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

   4. Applied rewrites 20.4%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

   if 1.3199999999999999e-307 < k < 10

   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

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

   7. Applied rewrites 49.0%

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

   if 10 < k

   1. Initial program 81.7%

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

   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 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 4.3%

      \[\leadsto a \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   9. 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-+r+ N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 58.7

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

   10. Applied rewrites 58.7%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 61.5

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

   12. Applied rewrites 61.5%

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

Alternative 11: 46.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.32e-307)
   (/ a (* k k))
   (if (<= k 0.1) (fma (* a k) -10.0 a) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.32e-307) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = fma((a * k), -10.0, a);
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.32e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = fma(Float64(a * k), -10.0, a);
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.32e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3199999999999999e-307

   1. Initial program 89.5%

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

   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 20.4

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

   4. Applied rewrites 20.4%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

   if 1.3199999999999999e-307 < k < 0.10000000000000001

   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. lower-*.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if 0.10000000000000001 < k

   1. Initial program 81.8%

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

   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 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 4.4%

      \[\leadsto a \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   9. 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-+r+ N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 61.2

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

   12. Applied rewrites 61.2%

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

Alternative 12: 43.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.32 \cdot 10^{-307}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 1.32e-307) (/ a (* k k)) (if (<= k 1.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 1.32e-307) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		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 <= 1.32d-307) then
        tmp = a / (k * k)
    else if (k <= 1.0d0) 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 <= 1.32e-307) {
		tmp = a / (k * k);
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 1.32e-307:
		tmp = a / (k * k)
	elif k <= 1.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 1.32e-307)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 1.32e-307)
		tmp = a / (k * k);
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 1.32e-307], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.3199999999999999e-307

   1. Initial program 89.5%

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

   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 20.4

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

   4. Applied rewrites 20.4%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.8

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

   7. Applied rewrites 29.8%

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

   if 1.3199999999999999e-307 < k < 1

   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 48.4%

      \[\leadsto a \]

   if 1 < k

   1. Initial program 81.8%

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

   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 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 4.3%

      \[\leadsto a \]

   8. Taylor expanded in k around inf

      \[\leadsto \frac{a}{\color{blue}{{k}^{2}}} \]
   9. 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-+r+ N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-/.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 58.6

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

   10. Applied rewrites 58.6%

       \[\leadsto \frac{a}{\color{blue}{k \cdot k}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

       5. lower-/.f64 61.4

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

   12. Applied rewrites 61.4%

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

Alternative 13: 42.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 1.32 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 1:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))) (if (<= k 1.32e-307) t_0 (if (<= k 1.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.32e-307) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 1.32d-307) then
        tmp = t_0
    else if (k <= 1.0d0) then
        tmp = a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 1.32e-307) {
		tmp = t_0;
	} else if (k <= 1.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 1.32e-307:
		tmp = t_0
	elif k <= 1.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 1.32e-307)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 1.32e-307)
		tmp = t_0;
	elseif (k <= 1.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.32e-307], t$95$0, If[LessEqual[k, 1.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3199999999999999e-307 or 1 < k

   1. Initial program 85.4%

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

   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 41.4

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

   4. Applied rewrites 41.4%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.2

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

   7. Applied rewrites 45.2%

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

   if 1.3199999999999999e-307 < k < 1

   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 48.4%

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

Alternative 14: 19.7% 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 90.5%

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

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 44.3

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

4. Applied rewrites 44.3%

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

5. Taylor expanded in k around 0

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

7. Applied rewrites 19.7%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))