Falkner and Boettcher, Appendix A

Percentage Accurate: 91.3% → 97.9%

Time: 4.9s

Alternatives: 12

Speedup: 1.1×

• 21.1% 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: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 18:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 18.0)
   (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))
   (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 18.0) {
		tmp = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 18.0d0) then
        tmp = (a * (k ** m)) / ((1.0d0 + (10.0d0 * k)) + (k * k))
    else
        tmp = (k ** m) * a
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 18.0) {
		tmp = (a * Math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	} else {
		tmp = Math.pow(k, m) * a;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 18.0:
		tmp = (a * math.pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))
	else:
		tmp = math.pow(k, m) * a
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 18.0)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 18.0)
		tmp = (a * (k ^ m)) / ((1.0 + (10.0 * k)) + (k * k));
	else
		tmp = (k ^ m) * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 18.0], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 18

   1. Initial program 96.9%

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

   if 18 < m

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 2: 45.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_1 5e-229)
     t_0
     (if (<= t_1 5e+280)
       (/ a (fma 10.0 k 1.0))
       (if (<= t_1 INFINITY) t_0 (fma (* (* k a) 99.0) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 5e-229) {
		tmp = t_0;
	} else if (t_1 <= 5e+280) {
		tmp = a / fma(10.0, k, 1.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(((k * a) * 99.0), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e-229)
		tmp = t_0;
	elseif (t_1 <= 5e+280)
		tmp = Float64(a / fma(10.0, k, 1.0));
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(k * a) * 99.0), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-229], t$95$0, If[LessEqual[t$95$1, 5e+280], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e-229 or 5.0000000000000002e280 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 97.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 42.4

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

   5. Applied rewrites 42.4%

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

   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. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if 5.00000000000000016e-229 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.0000000000000002e280

   1. Initial program 99.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   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 70.9%

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

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

   1. Initial program 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.1

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

   5. Applied rewrites 2.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 39.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 77.7

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

   11. Applied rewrites 77.7%

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

   12. Taylor expanded in k around inf

       \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 77.7

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

   14. Applied rewrites 77.7%

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

Alternative 3: 44.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ t_1 := \frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k)))
        (t_1 (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k)))))
   (if (<= t_1 5e-229)
     t_0
     (if (<= t_1 5e+280)
       (fma (* -10.0 a) k a)
       (if (<= t_1 INFINITY) t_0 (fma (* (* k a) 99.0) k a))))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double t_1 = (a * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double tmp;
	if (t_1 <= 5e-229) {
		tmp = t_0;
	} else if (t_1 <= 5e+280) {
		tmp = fma((-10.0 * a), k, a);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = fma(((k * a) * 99.0), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	t_1 = Float64(Float64(a * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	tmp = 0.0
	if (t_1 <= 5e-229)
		tmp = t_0;
	elseif (t_1 <= 5e+280)
		tmp = fma(Float64(-10.0 * a), k, a);
	elseif (t_1 <= Inf)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(k * a) * 99.0), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-229], t$95$0, If[LessEqual[t$95$1, 5e+280], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$0, N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000016e-229 or 5.0000000000000002e280 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < +inf.0

   1. Initial program 97.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 42.4

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

   5. Applied rewrites 42.4%

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

   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. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   if 5.00000000000000016e-229 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.0000000000000002e280

   1. Initial program 99.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 69.3%

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

   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 68.9

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

   11. Applied rewrites 68.9%

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

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

   1. Initial program 0.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.1

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

   5. Applied rewrites 2.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 39.2%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 77.7

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

   11. Applied rewrites 77.7%

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

   12. Taylor expanded in k around inf

       \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 77.7

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

   14. Applied rewrites 77.7%

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

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 18:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 18.0) (* a (/ (pow k m) (fma (+ 10.0 k) k 1.0))) (* (pow k m) a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 18.0) {
		tmp = a * (pow(k, m) / fma((10.0 + k), k, 1.0));
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 18.0)
		tmp = Float64(a * Float64((k ^ m) / fma(Float64(10.0 + k), k, 1.0)));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 18.0], N[(a * 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), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 18

   1. Initial program 96.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   if 18 < m

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

Alternative 5: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.5e-18)
   (/ (* a (pow k m)) (fma 10.0 k 1.0))
   (if (<= m 1.65e-14) (/ a (fma (+ 10.0 k) k 1.0)) (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.5e-18) {
		tmp = (a * pow(k, m)) / fma(10.0, k, 1.0);
	} else if (m <= 1.65e-14) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.5e-18)
		tmp = Float64(Float64(a * (k ^ m)) / fma(10.0, k, 1.0));
	elseif (m <= 1.65e-14)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.5e-18], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.65e-14], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.50000000000000018e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -2.50000000000000018e-18 < m < 1.6499999999999999e-14

   1. Initial program 94.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

   if 1.6499999999999999e-14 < m

   1. Initial program 80.6%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.5

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

   5. Applied rewrites 98.5%

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

Alternative 6: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -2.5e-18)
     t_0
     (if (<= m 1.65e-14) (/ a (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -2.5e-18) {
		tmp = t_0;
	} else if (m <= 1.65e-14) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -2.5e-18)
		tmp = t_0;
	elseif (m <= 1.65e-14)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -2.5e-18], t$95$0, If[LessEqual[m, 1.65e-14], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.50000000000000018e-18 or 1.6499999999999999e-14 < m

   1. Initial program 89.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

   if -2.50000000000000018e-18 < m < 1.6499999999999999e-14

   1. Initial program 94.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

Alternative 7: 62.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.38:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 18:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.38)
   (/ (fma (/ (fma -99.0 (/ a k) (* 10.0 a)) k) -1.0 a) (* k k))
   (if (<= m 18.0) (/ a (fma (+ 10.0 k) k 1.0)) (fma (* (* k a) 99.0) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.38) {
		tmp = fma((fma(-99.0, (a / k), (10.0 * a)) / k), -1.0, a) / (k * k);
	} else if (m <= 18.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(((k * a) * 99.0), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.38)
		tmp = Float64(fma(Float64(fma(-99.0, Float64(a / k), Float64(10.0 * a)) / k), -1.0, a) / Float64(k * k));
	elseif (m <= 18.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(Float64(k * a) * 99.0), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.38], N[(N[(N[(N[(-99.0 * N[(a / k), $MachinePrecision] + N[(10.0 * a), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] * -1.0 + a), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 18.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.38

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   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 66.6%

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

   if -0.38 < m < 18

   1. Initial program 94.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.9

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

   5. Applied rewrites 91.9%

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

   if 18 < m

   1. Initial program 80.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 16.0%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 25.4

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

   11. Applied rewrites 25.4%

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

   12. Taylor expanded in k around inf

       \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 25.4

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

   14. Applied rewrites 25.4%

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

Alternative 8: 60.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.38:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 18500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.38)
   (/ a (* k k))
   (if (<= m 18500000.0)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (* (* k a) 99.0) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.38) {
		tmp = a / (k * k);
	} else if (m <= 18500000.0) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma(((k * a) * 99.0), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.38)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 18500000.0)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(Float64(k * a) * 99.0), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.38], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 18500000.0], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.38

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   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. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if -0.38 < m < 1.85e7

   1. Initial program 94.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   if 1.85e7 < m

   1. Initial program 80.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 25.5

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

   11. Applied rewrites 25.5%

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

   12. Taylor expanded in k around inf

       \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 25.5

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

   14. Applied rewrites 25.5%

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

Alternative 9: 59.4% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.38:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 18500000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.38)
   (/ a (* k k))
   (if (<= m 18500000.0) (/ a (fma k k 1.0)) (fma (* (* k a) 99.0) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.38) {
		tmp = a / (k * k);
	} else if (m <= 18500000.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = fma(((k * a) * 99.0), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.38)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 18500000.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = fma(Float64(Float64(k * a) * 99.0), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.38], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 18500000.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.38

   1. Initial program 99.9%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   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. lower-*.f64 61.4

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

   8. Applied rewrites 61.4%

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

   if -0.38 < m < 1.85e7

   1. Initial program 94.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 91.1

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

   5. Applied rewrites 91.1%

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

   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 88.5%

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

   if 1.85e7 < m

   1. Initial program 80.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 16.1%

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

   9. Taylor expanded in k around 0

      \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-*.f64 25.5

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

   11. Applied rewrites 25.5%

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

   12. Taylor expanded in k around inf

       \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 25.5

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

   14. Applied rewrites 25.5%

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

Alternative 10: 26.7% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right) \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (fma (* (* k a) 99.0) k a))

C

double code(double a, double k, double m) {
	return fma(((k * a) * 99.0), k, a);
}

Julia

function code(a, k, m)
	return fma(Float64(Float64(k * a) * 99.0), k, a)
end

Wolfram

code[a_, k_, m_] := N[(N[(N[(k * a), $MachinePrecision] * 99.0), $MachinePrecision] * k + a), $MachinePrecision]

Derivation

1. Initial program 91.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 45.2

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

5. Applied rewrites 45.2%

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

6. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in k around 0

   \[\leadsto \mathsf{fma}\left(-10 \cdot a + 99 \cdot \left(a \cdot k\right), k, a\right) \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lift-*.f64 27.0

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

11. Applied rewrites 27.0%

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

12. Taylor expanded in k around inf

    \[\leadsto \mathsf{fma}\left(99 \cdot \left(a \cdot k\right), k, a\right) \]
13. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lift-*.f64 26.7

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

14. Applied rewrites 26.7%

    \[\leadsto \mathsf{fma}\left(\left(k \cdot a\right) \cdot 99, k, a\right) \]
15. Add Preprocessing

Alternative 11: 21.2% accurate, 11.2× speedup

Math

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

FPCore

(FPCore (a k m) :precision binary64 (fma (* -10.0 a) k a))

C

double code(double a, double k, double m) {
	return fma((-10.0 * a), k, a);
}

Julia

function code(a, k, m)
	return fma(Float64(-10.0 * a), k, a)
end

Wolfram

code[a_, k_, m_] := N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision]

Derivation

1. Initial program 91.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 45.2

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

5. Applied rewrites 45.2%

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

6. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

8. Applied rewrites 23.4%

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

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 21.2

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

11. Applied rewrites 21.2%

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

Alternative 12: 20.2% accurate, 134.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 91.3%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 45.2

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

5. Applied rewrites 45.2%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 20.2%

   \[\leadsto a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))