Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 99.9%

Time: 5.3s

Alternatives: 19

Speedup: 1.1×

• 21.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 90.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := 1 + 10 \cdot k\\ t_1 := \left|a\right| \cdot {k}^{m}\\ t_2 := \frac{t\_1}{t\_0 + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{\left|a\right|}{10 + k}}{k}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{t\_1}{t\_0 + \frac{{k}^{1}}{{k}^{-1}}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 10.0 k)))
        (t_1 (* (fabs a) (pow k m)))
        (t_2 (/ t_1 (+ t_0 (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_2 0.0)
      (/ (* (pow k m) (/ (fabs a) (+ 10.0 k))) k)
      (if (<= t_2 INFINITY)
        (/ t_1 (+ t_0 (/ (pow k 1.0) (pow k -1.0))))
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = 1.0 + (10.0 * k);
	double t_1 = fabs(a) * pow(k, m);
	double t_2 = t_1 / (t_0 + (k * k));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (pow(k, m) * (fabs(a) / (10.0 + k))) / k;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1 / (t_0 + (pow(k, 1.0) / pow(k, -1.0)));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = 1.0 + (10.0 * k);
	double t_1 = Math.abs(a) * Math.pow(k, m);
	double t_2 = t_1 / (t_0 + (k * k));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (Math.pow(k, m) * (Math.abs(a) / (10.0 + k))) / k;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_0 + (Math.pow(k, 1.0) / Math.pow(k, -1.0)));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * Math.abs(a);
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = 1.0 + (10.0 * k)
	t_1 = math.fabs(a) * math.pow(k, m)
	t_2 = t_1 / (t_0 + (k * k))
	tmp = 0
	if t_2 <= 0.0:
		tmp = (math.pow(k, m) * (math.fabs(a) / (10.0 + k))) / k
	elif t_2 <= math.inf:
		tmp = t_1 / (t_0 + (math.pow(k, 1.0) / math.pow(k, -1.0)))
	else:
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * math.fabs(a)
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	t_0 = Float64(1.0 + Float64(10.0 * k))
	t_1 = Float64(abs(a) * (k ^ m))
	t_2 = Float64(t_1 / Float64(t_0 + Float64(k * k)))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64((k ^ m) * Float64(abs(a) / Float64(10.0 + k))) / k);
	elseif (t_2 <= Inf)
		tmp = Float64(t_1 / Float64(t_0 + Float64((k ^ 1.0) / (k ^ -1.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = 1.0 + (10.0 * k);
	t_1 = abs(a) * (k ^ m);
	t_2 = t_1 / (t_0 + (k * k));
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((k ^ m) * (abs(a) / (10.0 + k))) / k;
	elseif (t_2 <= Inf)
		tmp = t_1 / (t_0 + ((k ^ 1.0) / (k ^ -1.0)));
	else
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * abs(a);
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[(t$95$0 + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(t$95$1 / N[(t$95$0 + N[(N[Power[k, 1.0], $MachinePrecision] / N[Power[k, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 73.3%

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

   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))) < +inf.0

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 2: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{\left|a\right|}{10 + k}}{k}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 0.0)
      (/ (* (pow k m) (/ (fabs a) (+ 10.0 k))) k)
      (if (<= t_0 INFINITY)
        t_0
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (pow(k, m) * (fabs(a) / (10.0 + k))) / k;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = (Math.abs(a) * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (Math.pow(k, m) * (Math.abs(a) / (10.0 + k))) / k;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * Math.abs(a);
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = (math.fabs(a) * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (math.pow(k, m) * (math.fabs(a) / (10.0 + k))) / k
	elif t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * math.fabs(a)
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(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((k ^ m) * Float64(abs(a) / Float64(10.0 + k))) / k);
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = (abs(a) * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = ((k ^ m) * (abs(a) / (10.0 + k))) / k;
	elseif (t_0 <= Inf)
		tmp = t_0;
	else
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * abs(a);
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$0, N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 73.3%

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

   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))) < +inf.0

   1. Initial program 90.4%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 3: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{\left|a\right|}{10 + k}}{k}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{{k}^{m}}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot \left|a\right|\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 0.0)
      (/ (* (pow k m) (/ (fabs a) (+ 10.0 k))) k)
      (if (<= t_0 INFINITY)
        (* (/ (pow k m) (fma (- k -10.0) k 1.0)) (fabs a))
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (pow(k, m) * (fabs(a) / (10.0 + k))) / k;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (pow(k, m) / fma((k - -10.0), k, 1.0)) * fabs(a);
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(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((k ^ m) * Float64(abs(a) / Float64(10.0 + k))) / k);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64((k ^ m) / fma(Float64(k - -10.0), k, 1.0)) * abs(a));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[Power[k, m], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 73.3%

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

   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))) < +inf.0

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 4: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot {k}^{m}\\ t_1 := \left(1 + 10 \cdot k\right) + k \cdot k\\ t_2 := \frac{t\_0}{t\_1}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{\left|a\right|}{10 + k}}{k}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left|a\right| + \left|a\right| \cdot \left(m \cdot \log k\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (fabs a) (pow k m)))
        (t_1 (+ (+ 1.0 (* 10.0 k)) (* k k)))
        (t_2 (/ t_0 t_1)))
   (*
    (copysign 1.0 a)
    (if (<= t_2 0.0)
      (/ (* (pow k m) (/ (fabs a) (+ 10.0 k))) k)
      (if (<= t_2 2e+301)
        (/ (+ (fabs a) (* (fabs a) (* m (log k)))) t_1)
        (/ t_0 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = fabs(a) * pow(k, m);
	double t_1 = (1.0 + (10.0 * k)) + (k * k);
	double t_2 = t_0 / t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (pow(k, m) * (fabs(a) / (10.0 + k))) / k;
	} else if (t_2 <= 2e+301) {
		tmp = (fabs(a) + (fabs(a) * (m * log(k)))) / t_1;
	} else {
		tmp = t_0 / 1.0;
	}
	return copysign(1.0, a) * tmp;
}

Java

public static double code(double a, double k, double m) {
	double t_0 = Math.abs(a) * Math.pow(k, m);
	double t_1 = (1.0 + (10.0 * k)) + (k * k);
	double t_2 = t_0 / t_1;
	double tmp;
	if (t_2 <= 0.0) {
		tmp = (Math.pow(k, m) * (Math.abs(a) / (10.0 + k))) / k;
	} else if (t_2 <= 2e+301) {
		tmp = (Math.abs(a) + (Math.abs(a) * (m * Math.log(k)))) / t_1;
	} else {
		tmp = t_0 / 1.0;
	}
	return Math.copySign(1.0, a) * tmp;
}

Python

def code(a, k, m):
	t_0 = math.fabs(a) * math.pow(k, m)
	t_1 = (1.0 + (10.0 * k)) + (k * k)
	t_2 = t_0 / t_1
	tmp = 0
	if t_2 <= 0.0:
		tmp = (math.pow(k, m) * (math.fabs(a) / (10.0 + k))) / k
	elif t_2 <= 2e+301:
		tmp = (math.fabs(a) + (math.fabs(a) * (m * math.log(k)))) / t_1
	else:
		tmp = t_0 / 1.0
	return math.copysign(1.0, a) * tmp

Julia

function code(a, k, m)
	t_0 = Float64(abs(a) * (k ^ m))
	t_1 = Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))
	t_2 = Float64(t_0 / t_1)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64((k ^ m) * Float64(abs(a) / Float64(10.0 + k))) / k);
	elseif (t_2 <= 2e+301)
		tmp = Float64(Float64(abs(a) + Float64(abs(a) * Float64(m * log(k)))) / t_1);
	else
		tmp = Float64(t_0 / 1.0);
	end
	return Float64(copysign(1.0, a) * tmp)
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = abs(a) * (k ^ m);
	t_1 = (1.0 + (10.0 * k)) + (k * k);
	t_2 = t_0 / t_1;
	tmp = 0.0;
	if (t_2 <= 0.0)
		tmp = ((k ^ m) * (abs(a) / (10.0 + k))) / k;
	elseif (t_2 <= 2e+301)
		tmp = (abs(a) + (abs(a) * (m * log(k)))) / t_1;
	else
		tmp = t_0 / 1.0;
	end
	tmp_2 = (sign(a) * abs(1.0)) * tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$2, 2e+301], N[(N[(N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] * N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(t$95$0 / 1.0), $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 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 73.3%

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

   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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 41.1

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

   4. Applied rewrites 41.1%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

Alternative 5: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot {k}^{m}\\ t_1 := \frac{t\_0}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{{k}^{m} \cdot \frac{\left|a\right|}{10 + k}}{k}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1 + m \cdot \log k}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot \left|a\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (fabs a) (pow k m)))
        (t_1 (/ t_0 (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_1 0.0)
      (/ (* (pow k m) (/ (fabs a) (+ 10.0 k))) k)
      (if (<= t_1 2e+301)
        (* (/ (+ 1.0 (* m (log k))) (fma (- k -10.0) k 1.0)) (fabs a))
        (/ t_0 1.0))))))

C

double code(double a, double k, double m) {
	double t_0 = fabs(a) * pow(k, m);
	double t_1 = t_0 / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (pow(k, m) * (fabs(a) / (10.0 + k))) / k;
	} else if (t_1 <= 2e+301) {
		tmp = ((1.0 + (m * log(k))) / fma((k - -10.0), k, 1.0)) * fabs(a);
	} else {
		tmp = t_0 / 1.0;
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(abs(a) * (k ^ m))
	t_1 = Float64(t_0 / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64((k ^ m) * Float64(abs(a) / Float64(10.0 + k))) / k);
	elseif (t_1 <= 2e+301)
		tmp = Float64(Float64(Float64(1.0 + Float64(m * log(k))) / fma(Float64(k - -10.0), k, 1.0)) * abs(a));
	else
		tmp = Float64(t_0 / 1.0);
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, 0.0], N[(N[(N[Power[k, m], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(10.0 + k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], If[LessEqual[t$95$1, 2e+301], N[(N[(N[(1.0 + N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 1.0), $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 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 73.3%

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

   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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.6

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

   6. Applied rewrites 41.6%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

Alternative 6: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -1.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1 + m \cdot \log k}{\mathsf{fma}\left(k - -10, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -1.5)
     t_0
     (if (<= m 3.4e-17)
       (* (/ (+ 1.0 (* m (log k))) (fma (- k -10.0) k 1.0)) a)
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -1.5) {
		tmp = t_0;
	} else if (m <= 3.4e-17) {
		tmp = ((1.0 + (m * log(k))) / fma((k - -10.0), k, 1.0)) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -1.5)
		tmp = t_0;
	elseif (m <= 3.4e-17)
		tmp = Float64(Float64(Float64(1.0 + Float64(m * log(k))) / fma(Float64(k - -10.0), k, 1.0)) * a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -1.5], t$95$0, If[LessEqual[m, 3.4e-17], N[(N[(N[(1.0 + N[(m * N[Log[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.5 or 3.3999999999999998e-17 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

   if -1.5 < m < 3.3999999999999998e-17

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 41.6

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

   6. Applied rewrites 41.6%

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

Alternative 7: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -6.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 3.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{1 + \mathsf{fma}\left(10, k, \frac{k}{\frac{1}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -6.4)
     t_0
     (if (<= m 3.4e-17) (/ a (+ 1.0 (fma 10.0 k (/ k (/ 1.0 k))))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -6.4) {
		tmp = t_0;
	} else if (m <= 3.4e-17) {
		tmp = a / (1.0 + fma(10.0, k, (k / (1.0 / k))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -6.4)
		tmp = t_0;
	elseif (m <= 3.4e-17)
		tmp = Float64(a / Float64(1.0 + fma(10.0, k, Float64(k / Float64(1.0 / k)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -6.4], t$95$0, If[LessEqual[m, 3.4e-17], N[(a / N[(1.0 + N[(10.0 * k + N[(k / N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -6.4000000000000004 or 3.3999999999999998e-17 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

   if -6.4000000000000004 < m < 3.3999999999999998e-17

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-div N/A

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

      4. unpow1 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow-1 N/A

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

      7. lower-/.f64 45.3

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

   8. Applied rewrites 45.3%

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

Alternative 8: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq -4.1:\\ \;\;\;\;\frac{a}{{k}^{2}}\\ \mathbf{elif}\;m \leq 1.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{a}{1 + \mathsf{fma}\left(10, k, \frac{k}{\frac{1}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -4.1)
   (/ a (pow k 2.0))
   (if (<= m 1.4e-23)
     (/ a (+ 1.0 (fma 10.0 k (/ k (/ 1.0 k)))))
     (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -4.1) {
		tmp = a / pow(k, 2.0);
	} else if (m <= 1.4e-23) {
		tmp = a / (1.0 + fma(10.0, k, (k / (1.0 / k))));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -4.1)
		tmp = Float64(a / (k ^ 2.0));
	elseif (m <= 1.4e-23)
		tmp = Float64(a / Float64(1.0 + fma(10.0, k, Float64(k / Float64(1.0 / k)))));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -4.1], N[(a / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.4e-23], N[(a / N[(1.0 + N[(10.0 * k + N[(k / N[(1.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.0999999999999996

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. lower-pow.f64 35.6

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

   9. Applied rewrites 35.6%

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

   if -4.0999999999999996 < m < 1.3999999999999999e-23

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-div N/A

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

      4. unpow1 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow-1 N/A

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

      7. lower-/.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 1.3999999999999999e-23 < m

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 9: 58.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-279}:\\ \;\;\;\;\frac{\left|a\right|}{\left(10 - \frac{-1}{k}\right) + k} \cdot \frac{1}{k}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left|a\right|}{1 + \mathsf{fma}\left(10, k, k \cdot k\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, \left|a\right|, \left|a\right|\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e-279)
      (* (/ (fabs a) (+ (- 10.0 (/ -1.0 k)) k)) (/ 1.0 k))
      (if (<= t_0 2e+301)
        (/ (fabs a) (+ 1.0 (fma 10.0 k (* k k))))
        (if (<= t_0 INFINITY)
          (/ (fma (* (log k) m) (fabs a) (fabs a)) 1.0)
          (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a))))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e-279) {
		tmp = (fabs(a) / ((10.0 - (-1.0 / k)) + k)) * (1.0 / k);
	} else if (t_0 <= 2e+301) {
		tmp = fabs(a) / (1.0 + fma(10.0, k, (k * k)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((log(k) * m), fabs(a), fabs(a)) / 1.0;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e-279)
		tmp = Float64(Float64(abs(a) / Float64(Float64(10.0 - Float64(-1.0 / k)) + k)) * Float64(1.0 / k));
	elseif (t_0 <= 2e+301)
		tmp = Float64(abs(a) / Float64(1.0 + fma(10.0, k, Float64(k * k))));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(log(k) * m), abs(a), abs(a)) / 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e-279], N[(N[(N[Abs[a], $MachinePrecision] / N[(N[(10.0 - N[(-1.0 / k), $MachinePrecision]), $MachinePrecision] + k), $MachinePrecision]), $MachinePrecision] * N[(1.0 / k), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+301], N[(N[Abs[a], $MachinePrecision] / N[(1.0 + N[(10.0 * k + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 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))) < 2.00000000000000011e-279

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-div N/A

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

      10. unpow1 N/A

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

      11. add-to-fraction N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 44.7%

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

   if 2.00000000000000011e-279 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 24.2

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

   7. Applied rewrites 24.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 24.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 10: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{\left|a\right|}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, \left|a\right|, \left|a\right|\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e+301)
      (/ 1.0 (/ (fma (- k -10.0) k 1.0) (fabs a)))
      (if (<= t_0 INFINITY)
        (/ (fma (* (log k) m) (fabs a) (fabs a)) 1.0)
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = 1.0 / (fma((k - -10.0), k, 1.0) / fabs(a));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((log(k) * m), fabs(a), fabs(a)) / 1.0;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = Float64(1.0 / Float64(fma(Float64(k - -10.0), k, 1.0) / abs(a)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(log(k) * m), abs(a), abs(a)) / 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e+301], N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * N[Abs[a], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-unsound-/.f64 N/A

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

      18. lower-/.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 24.2

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

   7. Applied rewrites 24.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 24.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 24.2

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

   9. Applied rewrites 24.2%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 11: 56.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{\left|a\right|}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left|a\right| \cdot m, \log k, \left|a\right|\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e+301)
      (/ 1.0 (/ (fma (- k -10.0) k 1.0) (fabs a)))
      (if (<= t_0 INFINITY)
        (/ (fma (* (fabs a) m) (log k) (fabs a)) 1.0)
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = 1.0 / (fma((k - -10.0), k, 1.0) / fabs(a));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((fabs(a) * m), log(k), fabs(a)) / 1.0;
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = Float64(1.0 / Float64(fma(Float64(k - -10.0), k, 1.0) / abs(a)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(abs(a) * m), log(k), abs(a)) / 1.0);
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e+301], N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[Abs[a], $MachinePrecision] * m), $MachinePrecision] * N[Log[k], $MachinePrecision] + N[Abs[a], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-unsound-/.f64 N/A

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

      18. lower-/.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 24.2

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

   7. Applied rewrites 24.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 24.1

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

   9. Applied rewrites 24.1%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 12: 55.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(k - -10, k, 1\right)}{\left|a\right|}}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e+301)
      (/ 1.0 (/ (fma (- k -10.0) k 1.0) (fabs a)))
      (if (<= t_0 INFINITY)
        (* k (fma -10.0 (fabs a) (/ (fabs a) k)))
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = 1.0 / (fma((k - -10.0), k, 1.0) / fabs(a));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = Float64(1.0 / Float64(fma(Float64(k - -10.0), k, 1.0) / abs(a)));
	elseif (t_0 <= Inf)
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e+301], N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-+.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. distribute-rgt-in N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. sub-flip N/A

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

      11. lift--.f64 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-unsound-/.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lower-unsound-/.f64 N/A

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

      18. lower-/.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 20.3

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

   10. Applied rewrites 20.3%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 13: 55.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left|a\right|}{1 + \mathsf{fma}\left(10, k, k \cdot k\right)}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (*
    (copysign 1.0 a)
    (if (<= t_0 2e+301)
      (/ (fabs a) (+ 1.0 (fma 10.0 k (* k k))))
      (if (<= t_0 INFINITY)
        (* k (fma -10.0 (fabs a) (/ (fabs a) k)))
        (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))))

C

double code(double a, double k, double m) {
	double t_0 = (fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+301) {
		tmp = fabs(a) / (1.0 + fma(10.0, k, (k * k)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e+301)
		tmp = Float64(abs(a) / Float64(1.0 + fma(10.0, k, Float64(k * k))));
	elseif (t_0 <= Inf)
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$0, 2e+301], N[(N[Abs[a], $MachinePrecision] / N[(1.0 + N[(10.0 * k + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $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))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 20.3

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

   10. Applied rewrites 20.3%

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

   if +inf.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 90.4%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 45.3

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

   6. Applied rewrites 45.3%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

Alternative 14: 50.4% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left|a\right|}{1 + \mathsf{fma}\left(10, k, k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+301)
    (/ (fabs a) (+ 1.0 (fma 10.0 k (* k k))))
    (* k (fma -10.0 (fabs a) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+301) {
		tmp = fabs(a) / (1.0 + fma(10.0, k, (k * k)));
	} else {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+301)
		tmp = Float64(abs(a) / Float64(1.0 + fma(10.0, k, Float64(k * k))));
	else
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+301], N[(N[Abs[a], $MachinePrecision] / N[(1.0 + N[(10.0 * k + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 20.3

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

   10. Applied rewrites 20.3%

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

Alternative 15: 50.0% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+301)
    (/ (fabs a) (fma (- k -10.0) k 1.0))
    (* k (fma -10.0 (fabs a) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+301) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+301)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+301], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 2.00000000000000011e301

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 2.00000000000000011e301 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 64.7

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

   4. Applied rewrites 64.7%

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

   5. Taylor expanded in m around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.5

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

   7. Applied rewrites 21.5%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 20.3

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

   10. Applied rewrites 20.3%

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

Alternative 16: 50.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \left(-10 \cdot a\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4e+17) (/ a (fma (- k -10.0) k 1.0)) (* k (* -10.0 a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 4e+17) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * (-10.0 * a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 4e+17)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * Float64(-10.0 * a));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 4e+17], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 4e17

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. pow2 N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. pow-sub N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-pow.f64 N/A

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

      10. lower-unsound-pow.f64 90.3

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

   3. Applied rewrites 90.3%

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

   4. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 45.3

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

   6. Applied rewrites 45.3%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-in N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 45.3

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

   8. Applied rewrites 45.3%

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

   if 4e17 < m

   1. Initial program 90.4%

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

   2. Taylor expanded in k around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

      6. lower-pow.f64 64.7

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      3. lower-*.f64 21.5

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.5%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

      3. lower-/.f64 20.3

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   10. Applied rewrites 20.3%

       \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 7.9

          \[\leadsto k \cdot \left(-10 \cdot a\right) \]

   13. Applied rewrites 7.9%

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 25.3% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left|a\right| \cdot -10, k, \left|a\right|\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
    (* k (* -10.0 (fabs a)))
    (fma (* (fabs a) -10.0) k (fabs a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 0.0) {
		tmp = k * (-10.0 * fabs(a));
	} else {
		tmp = fma((fabs(a) * -10.0), k, fabs(a));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 0.0)
		tmp = Float64(k * Float64(-10.0 * abs(a)));
	else
		tmp = fma(Float64(abs(a) * -10.0), k, abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[a], $MachinePrecision] * -10.0), $MachinePrecision] * k + N[Abs[a], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot \left(k \cdot {k}^{m}\right)\right) + a \cdot {k}^{m}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, \color{blue}{a \cdot \left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \color{blue}{\left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot \color{blue}{{k}^{m}}\right), a \cdot {k}^{m}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{\color{blue}{m}}\right), a \cdot {k}^{m}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

      6. lower-pow.f64 64.7

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      3. lower-*.f64 21.5

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.5%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

      3. lower-/.f64 20.3

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   10. Applied rewrites 20.3%

       \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 7.9

          \[\leadsto k \cdot \left(-10 \cdot a\right) \]

   13. Applied rewrites 7.9%

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]

   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. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot \left(k \cdot {k}^{m}\right)\right) + a \cdot {k}^{m}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, \color{blue}{a \cdot \left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \color{blue}{\left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot \color{blue}{{k}^{m}}\right), a \cdot {k}^{m}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{\color{blue}{m}}\right), a \cdot {k}^{m}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

      6. lower-pow.f64 64.7

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      3. lower-*.f64 21.5

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.5%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. +-commutative N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      3. lift-*.f64 N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      4. lift-*.f64 N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      5. associate-*r* N/A

         \[\leadsto \left(-10 \cdot a\right) \cdot k + a \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10 \cdot a, k, a\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot -10, k, a\right) \]

      8. lower-*.f64 21.5

         \[\leadsto \mathsf{fma}\left(a \cdot -10, k, a\right) \]

   9. Applied rewrites 21.5%

      \[\leadsto \mathsf{fma}\left(a \cdot -10, k, a\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 25.3% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;k \cdot \left(-10 \cdot \left|a\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10, k, 1\right) \cdot \left|a\right|\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 0.0)
    (* k (* -10.0 (fabs a)))
    (* (fma -10.0 k 1.0) (fabs a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 0.0) {
		tmp = k * (-10.0 * fabs(a));
	} else {
		tmp = fma(-10.0, k, 1.0) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 0.0)
		tmp = Float64(k * Float64(-10.0 * abs(a)));
	else
		tmp = Float64(fma(-10.0, k, 1.0) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k + 1.0), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 0.0

   1. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot \left(k \cdot {k}^{m}\right)\right) + a \cdot {k}^{m}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, \color{blue}{a \cdot \left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \color{blue}{\left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot \color{blue}{{k}^{m}}\right), a \cdot {k}^{m}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{\color{blue}{m}}\right), a \cdot {k}^{m}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

      6. lower-pow.f64 64.7

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      3. lower-*.f64 21.5

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.5%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

   8. Taylor expanded in k around inf

      \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

      3. lower-/.f64 20.3

         \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   10. Applied rewrites 20.3%

       \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]

   11. Taylor expanded in k around inf

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 7.9

          \[\leadsto k \cdot \left(-10 \cdot a\right) \]

   13. Applied rewrites 7.9%

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]

   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. Initial program 90.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

   2. Taylor expanded in k around 0

      \[\leadsto \color{blue}{-10 \cdot \left(a \cdot \left(k \cdot {k}^{m}\right)\right) + a \cdot {k}^{m}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, \color{blue}{a \cdot \left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \color{blue}{\left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot \color{blue}{{k}^{m}}\right), a \cdot {k}^{m}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{\color{blue}{m}}\right), a \cdot {k}^{m}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

      6. lower-pow.f64 64.7

         \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   4. Applied rewrites 64.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

   5. Taylor expanded in m around 0

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

      3. lower-*.f64 21.5

         \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

   7. Applied rewrites 21.5%

      \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

      2. +-commutative N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      3. lift-*.f64 N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      4. lift-*.f64 N/A

         \[\leadsto -10 \cdot \left(a \cdot k\right) + a \]

      5. *-commutative N/A

         \[\leadsto -10 \cdot \left(k \cdot a\right) + a \]

      6. associate-*r* N/A

         \[\leadsto \left(-10 \cdot k\right) \cdot a + a \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(10\right)\right) \cdot k\right) \cdot a + a \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(10 \cdot k\right)\right) \cdot a + a \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(10 \cdot k\right)\right) \cdot a + a \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(10 \cdot k\right), a, a\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(10 \cdot k\right), a, a\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(10\right)\right) \cdot k, a, a\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]

      14. lower-*.f64 21.5

          \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]

   9. Applied rewrites 21.5%

      \[\leadsto \mathsf{fma}\left(-10 \cdot k, a, a\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \left(-10 \cdot k\right) \cdot a + a \]

       2. distribute-lft1-in N/A

          \[\leadsto \left(-10 \cdot k + 1\right) \cdot a \]

       3. lower-*.f64 N/A

          \[\leadsto \left(-10 \cdot k + 1\right) \cdot a \]

       4. lift-*.f64 N/A

          \[\leadsto \left(-10 \cdot k + 1\right) \cdot a \]

       5. lower-fma.f64 21.5

          \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]

   11. Applied rewrites 21.5%

       \[\leadsto \mathsf{fma}\left(-10, k, 1\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 7.9% accurate, 5.0× speedup

Math

\[k \cdot \left(-10 \cdot a\right) \]

FPCore

(FPCore (a k m) :precision binary64 (* k (* -10.0 a)))

C

double code(double a, double k, double m) {
	return k * (-10.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = k * ((-10.0d0) * a)
end function

Java

public static double code(double a, double k, double m) {
	return k * (-10.0 * a);
}

Python

def code(a, k, m):
	return k * (-10.0 * a)

Julia

function code(a, k, m)
	return Float64(k * Float64(-10.0 * a))
end

MATLAB

function tmp = code(a, k, m)
	tmp = k * (-10.0 * a);
end

Wolfram

code[a_, k_, m_] := N[(k * N[(-10.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.4%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]

2. Taylor expanded in k around 0

   \[\leadsto \color{blue}{-10 \cdot \left(a \cdot \left(k \cdot {k}^{m}\right)\right) + a \cdot {k}^{m}} \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-10, \color{blue}{a \cdot \left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-10, a \cdot \color{blue}{\left(k \cdot {k}^{m}\right)}, a \cdot {k}^{m}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot \color{blue}{{k}^{m}}\right), a \cdot {k}^{m}\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{\color{blue}{m}}\right), a \cdot {k}^{m}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

   6. lower-pow.f64 64.7

      \[\leadsto \mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right) \]

4. Applied rewrites 64.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-10, a \cdot \left(k \cdot {k}^{m}\right), a \cdot {k}^{m}\right)} \]

5. Taylor expanded in m around 0

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto a + -10 \cdot \color{blue}{\left(a \cdot k\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto a + -10 \cdot \left(a \cdot \color{blue}{k}\right) \]

   3. lower-*.f64 21.5

      \[\leadsto a + -10 \cdot \left(a \cdot k\right) \]

7. Applied rewrites 21.5%

   \[\leadsto a + \color{blue}{-10 \cdot \left(a \cdot k\right)} \]

8. Taylor expanded in k around inf

   \[\leadsto k \cdot \left(-10 \cdot a + \color{blue}{\frac{a}{k}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto k \cdot \left(-10 \cdot a + \frac{a}{\color{blue}{k}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

   3. lower-/.f64 20.3

      \[\leadsto k \cdot \mathsf{fma}\left(-10, a, \frac{a}{k}\right) \]

10. Applied rewrites 20.3%

    \[\leadsto k \cdot \mathsf{fma}\left(-10, \color{blue}{a}, \frac{a}{k}\right) \]

11. Taylor expanded in k around inf

    \[\leadsto k \cdot \left(-10 \cdot a\right) \]
12. Step-by-step derivation

    1. lower-*.f64 7.9

       \[\leadsto k \cdot \left(-10 \cdot a\right) \]

13. Applied rewrites 7.9%

    \[\leadsto k \cdot \left(-10 \cdot a\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))