Falkner and Boettcher, Appendix A

Percentage Accurate: 90.1% → 97.5%

Time: 5.2s

Alternatives: 12

Speedup: 1.1×

• 21.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+254}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (* a_s (if (<= t_0 2e+254) t_0 (* (pow k m) a_m)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+254) {
		tmp = t_0;
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a_m * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    if (t_0 <= 2d+254) then
        tmp = t_0
    else
        tmp = (k ** m) * a_m
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double t_0 = (a_m * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_0 <= 2e+254) {
		tmp = t_0;
	} else {
		tmp = Math.pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	t_0 = (a_m * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	tmp = 0
	if t_0 <= 2e+254:
		tmp = t_0
	else:
		tmp = math.pow(k, m) * a_m
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_0 <= 2e+254)
		tmp = t_0;
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	t_0 = (a_m * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	tmp = 0.0;
	if (t_0 <= 2e+254)
		tmp = t_0;
	else
		tmp = (k ^ m) * a_m;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$0, 2e+254], t$95$0, N[(N[Power[k, m], $MachinePrecision] * a$95$m), $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))) < 1.9999999999999999e254

   1. Initial program 95.4%

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

   if 1.9999999999999999e254 < (/.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 64.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 2 \cdot 10^{+254}:\\ \;\;\;\;a\_m \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 2e+254)
    (* a_m (/ (pow k m) (fma (+ 10.0 k) k 1.0)))
    (* (pow k m) a_m))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (((a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 2e+254) {
		tmp = a_m * (pow(k, m) / fma((10.0 + k), k, 1.0));
	} else {
		tmp = pow(k, m) * a_m;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 2e+254)
		tmp = Float64(a_m * Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)));
	else
		tmp = Float64((k ^ m) * a_m);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+254], N[(a$95$m * N[(N[Power[k, m], $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $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))) < 1.9999999999999999e254

   1. Initial program 95.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 95.4

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

   4. Applied rewrites 95.4%

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

   if 1.9999999999999999e254 < (/.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 64.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 3: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.0122 \lor \neg \left(m \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;{k}^{m} \cdot a\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (or (<= m -0.0122) (not (<= m 5e-8)))
    (* (pow k m) a_m)
    (/ a_m (fma (+ 10.0 k) k 1.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if ((m <= -0.0122) || !(m <= 5e-8)) {
		tmp = pow(k, m) * a_m;
	} else {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if ((m <= -0.0122) || !(m <= 5e-8))
		tmp = Float64((k ^ m) * a_m);
	else
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[Or[LessEqual[m, -0.0122], N[Not[LessEqual[m, 5e-8]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a$95$m), $MachinePrecision], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < -0.0122000000000000008 or 4.9999999999999998e-8 < m

   1. Initial program 89.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -0.0122000000000000008 < m < 4.9999999999999998e-8

   1. Initial program 89.5%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0122 \lor \neg \left(m \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.25:\\ \;\;\;\;\frac{\frac{a\_m}{k \cdot k} \cdot 99}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.245:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.25)
    (/ (* (/ a_m (* k k)) 99.0) (* k k))
    (if (<= m 0.245)
      (/ a_m (fma k k (fma 10.0 k 1.0)))
      (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.25) {
		tmp = ((a_m / (k * k)) * 99.0) / (k * k);
	} else if (m <= 0.245) {
		tmp = a_m / fma(k, k, fma(10.0, k, 1.0));
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.25)
		tmp = Float64(Float64(Float64(a_m / Float64(k * k)) * 99.0) / Float64(k * k));
	elseif (m <= 0.245)
		tmp = Float64(a_m / fma(k, k, fma(10.0, k, 1.0)));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.25], N[(N[(N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision] * 99.0), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.245], N[(a$95$m / N[(k * k + N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.25

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in k around -inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{a + -1 \cdot \frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k}}{{k}^{\color{blue}{2}}} \]

      2. +-commutative N/A

         \[\leadsto \frac{-1 \cdot \frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k} + a}{{k}^{2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k} \cdot -1 + a}{{k}^{2}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) - -10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 \cdot \frac{a}{k} + \frac{a}{k}\right) + \left(\mathsf{neg}\left(-10\right)\right) \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      7. distribute-lft1-in N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 + 1\right) \cdot \frac{a}{k} + \left(\mathsf{neg}\left(-10\right)\right) \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\left(-100 + 1\right) \cdot \frac{a}{k} + 10 \cdot a}{k}, -1, a\right)}{{k}^{2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-100 + 1, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{{k}^{2}} \]

      13. pow2 N/A

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

   8. Applied rewrites 65.7%

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

   9. Taylor expanded in k around 0

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

       1. *-commutative N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{\frac{a}{{k}^{2}} \cdot 99}{k \cdot k} \]

       4. pow2 N/A

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

       5. lift-*.f64 76.6

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

   11. Applied rewrites 76.6%

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

   if -0.25 < m < 0.245

   1. Initial program 89.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 88.4

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

   7. Applied rewrites 88.4%

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

   if 0.245 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 5: 71.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -14500:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.245:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -14500.0)
    (/ a_m (* k k))
    (if (<= m 0.245)
      (/ a_m (fma k k (fma 10.0 k 1.0)))
      (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -14500.0) {
		tmp = a_m / (k * k);
	} else if (m <= 0.245) {
		tmp = a_m / fma(k, k, fma(10.0, k, 1.0));
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -14500.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.245)
		tmp = Float64(a_m / fma(k, k, fma(10.0, k, 1.0)));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -14500.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.245], N[(a$95$m / N[(k * k + N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -14500

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   if -14500 < m < 0.245

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 87.4

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

   7. Applied rewrites 87.4%

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

   if 0.245 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 6: 55.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := \frac{a\_m}{k \cdot k}\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq -6.7 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a\_m, k, a\_m\right)\\ \mathbf{elif}\;m \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (/ a_m (* k k))))
   (*
    a_s
    (if (<= m -9.8e-14)
      t_0
      (if (<= m -6.7e-192)
        (fma (* -10.0 a_m) k a_m)
        (if (<= m 9.5e-8) t_0 (* (* (* k k) a_m) 99.0)))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = a_m / (k * k);
	double tmp;
	if (m <= -9.8e-14) {
		tmp = t_0;
	} else if (m <= -6.7e-192) {
		tmp = fma((-10.0 * a_m), k, a_m);
	} else if (m <= 9.5e-8) {
		tmp = t_0;
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(a_m / Float64(k * k))
	tmp = 0.0
	if (m <= -9.8e-14)
		tmp = t_0;
	elseif (m <= -6.7e-192)
		tmp = fma(Float64(-10.0 * a_m), k, a_m);
	elseif (m <= 9.5e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[m, -9.8e-14], t$95$0, If[LessEqual[m, -6.7e-192], N[(N[(-10.0 * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision], If[LessEqual[m, 9.5e-8], t$95$0, N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if m < -9.79999999999999989e-14 or -6.69999999999999991e-192 < m < 9.50000000000000036e-8

   1. Initial program 94.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 54.8

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 55.9

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

   8. Applied rewrites 55.9%

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

   if -9.79999999999999989e-14 < m < -6.69999999999999991e-192

   1. Initial program 93.3%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in k around 0

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

       1. lift-*.f64 73.8

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

   11. Applied rewrites 73.8%

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

   if 9.50000000000000036e-8 < m

   1. Initial program 79.1%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.8

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

   8. Applied rewrites 22.8%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 56.7

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 56.7%

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

Alternative 7: 71.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -14500:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.245:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -14500.0)
    (/ a_m (* k k))
    (if (<= m 0.245)
      (/ a_m (fma (+ 10.0 k) k 1.0))
      (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -14500.0) {
		tmp = a_m / (k * k);
	} else if (m <= 0.245) {
		tmp = a_m / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -14500.0)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.245)
		tmp = Float64(a_m / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -14500.0], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.245], N[(a$95$m / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -14500

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

   if -14500 < m < 0.245

   1. Initial program 89.8%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if 0.245 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 8: 70.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -0.31:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.245:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -0.31)
    (/ a_m (* k k))
    (if (<= m 0.245) (/ a_m (fma k k 1.0)) (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -0.31) {
		tmp = a_m / (k * k);
	} else if (m <= 0.245) {
		tmp = a_m / fma(k, k, 1.0);
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -0.31)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.245)
		tmp = Float64(a_m / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -0.31], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.245], N[(a$95$m / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -0.309999999999999998

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 57.1

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

   8. Applied rewrites 57.1%

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

   if -0.309999999999999998 < m < 0.245

   1. Initial program 89.7%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 86.4%

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

   if 0.245 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 9: 61.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq -4 \cdot 10^{-9}:\\ \;\;\;\;\frac{a\_m}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.245:\\ \;\;\;\;\frac{a\_m}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= m -4e-9)
    (/ a_m (* k k))
    (if (<= m 0.245) (/ a_m (fma 10.0 k 1.0)) (* (* (* k k) a_m) 99.0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= -4e-9) {
		tmp = a_m / (k * k);
	} else if (m <= 0.245) {
		tmp = a_m / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= -4e-9)
		tmp = Float64(a_m / Float64(k * k));
	elseif (m <= 0.245)
		tmp = Float64(a_m / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, -4e-9], N[(a$95$m / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.245], N[(a$95$m / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if m < -4.00000000000000025e-9

   1. Initial program 98.9%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in k around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 57.6

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

   8. Applied rewrites 57.6%

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

   if -4.00000000000000025e-9 < m < 0.245

   1. Initial program 89.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 88.3

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

   5. Applied rewrites 88.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 63.4%

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

   if 0.245 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 10: 38.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;m \leq 0.225:\\ \;\;\;\;a\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\_m\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (if (<= m 0.225) a_m (* (* (* k k) a_m) 99.0))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 0.225) {
		tmp = a_m;
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
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_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.225d0) then
        tmp = a_m
    else
        tmp = ((k * k) * a_m) * 99.0d0
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (m <= 0.225) {
		tmp = a_m;
	} else {
		tmp = ((k * k) * a_m) * 99.0;
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if m <= 0.225:
		tmp = a_m
	else:
		tmp = ((k * k) * a_m) * 99.0
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (m <= 0.225)
		tmp = a_m;
	else
		tmp = Float64(Float64(Float64(k * k) * a_m) * 99.0);
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (m <= 0.225)
		tmp = a_m;
	else
		tmp = ((k * k) * a_m) * 99.0;
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[m, 0.225], a$95$m, N[(N[(N[(k * k), $MachinePrecision] * a$95$m), $MachinePrecision] * 99.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if m < 0.225000000000000006

   1. Initial program 94.4%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.9%

      \[\leadsto a \]

   if 0.225000000000000006 < m

   1. Initial program 78.6%

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

   3. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 22.2

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

   8. Applied rewrites 22.2%

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

   9. Taylor expanded in k around inf

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

       1. *-commutative N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       2. lower-*.f64 N/A

          \[\leadsto \left(a \cdot {k}^{2}\right) \cdot 99 \]

       3. *-commutative N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       4. lower-*.f64 N/A

          \[\leadsto \left({k}^{2} \cdot a\right) \cdot 99 \]

       5. pow2 N/A

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

       6. lift-*.f64 57.9

          \[\leadsto \left(\left(k \cdot k\right) \cdot a\right) \cdot 99 \]

   11. Applied rewrites 57.9%

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

Alternative 11: 20.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \mathsf{fma}\left(-10 \cdot a\_m, k, a\_m\right) \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s (fma (* -10.0 a_m) k a_m)))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * fma((-10.0 * a_m), k, a_m);
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * fma(Float64(-10.0 * a_m), k, a_m))
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * N[(N[(-10.0 * a$95$m), $MachinePrecision] * k + a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

6. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

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

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. distribute-rgt1-in N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 24.1

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

8. Applied rewrites 24.1%

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

9. Taylor expanded in k around 0

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

    1. lift-*.f64 19.2

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

11. Applied rewrites 19.2%

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

Alternative 12: 19.7% accurate, 134.0× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot a\_m \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m) :precision binary64 (* a_s a_m))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Fortran

a\_m =     private
a\_s =     private
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_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * a_m
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	return a_s * a_m;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	return a_s * a_m

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	return Float64(a_s * a_m)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp = code(a_s, a_m, k, m)
	tmp = a_s * a_m;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * a$95$m), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 41.9

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

5. Applied rewrites 41.9%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 18.5%

   \[\leadsto a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))