Falkner and Boettcher, Appendix A

Percentage Accurate: 90.3% → 97.7%

Time: 6.0s

Alternatives: 12

Speedup: 1.1×

• 22.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.85)
   (/ (* 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 <= 2.85) {
		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 <= 2.85d0) 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 <= 2.85) {
		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 <= 2.85:
		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 <= 2.85)
		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 <= 2.85)
		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, 2.85], 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 < 2.85000000000000009

   1. Initial program 98.2%

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

   if 2.85000000000000009 < 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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.85:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.85) (* 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 <= 2.85) {
		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 <= 2.85)
		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, 2.85], 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 < 2.85000000000000009

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 98.1

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

   4. Applied rewrites 98.1%

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

   if 2.85000000000000009 < 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 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.95 \cdot 10^{-51}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 1.15 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -3.95e-51)
   (/ (* a (pow k m)) (fma 10.0 k 1.0))
   (if (<= m 1.15e-12) (/ 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 <= -3.95e-51) {
		tmp = (a * pow(k, m)) / fma(10.0, k, 1.0);
	} else if (m <= 1.15e-12) {
		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 <= -3.95e-51)
		tmp = Float64(Float64(a * (k ^ m)) / fma(10.0, k, 1.0));
	elseif (m <= 1.15e-12)
		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, -3.95e-51], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.15e-12], 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 < -3.9499999999999999e-51

   1. Initial program 98.8%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 -3.9499999999999999e-51 < m < 1.14999999999999995e-12

   1. Initial program 97.3%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

   if 1.14999999999999995e-12 < m

   1. Initial program 81.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 4: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5} \lor \neg \left(m \leq 1.15 \cdot 10^{-12}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (or (<= m -1.5e-5) (not (<= m 1.15e-12)))
   (* (pow k m) a)
   (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if ((m <= -1.5e-5) || !(m <= 1.15e-12)) {
		tmp = pow(k, m) * a;
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if ((m <= -1.5e-5) || !(m <= 1.15e-12))
		tmp = Float64((k ^ m) * a);
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[Or[LessEqual[m, -1.5e-5], N[Not[LessEqual[m, 1.15e-12]], $MachinePrecision]], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1.50000000000000004e-5 or 1.14999999999999995e-12 < m

   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 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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.50000000000000004e-5 < m < 1.14999999999999995e-12

   1. Initial program 96.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{-5} \lor \neg \left(m \leq 1.15 \cdot 10^{-12}\right):\\ \;\;\;\;{k}^{m} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.15:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(-99, \frac{a}{k}, 10 \cdot a\right)}{k}, -1, a\right)}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.94:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.15)
   (/ (fma (/ (fma -99.0 (/ a k) (* 10.0 a)) k) -1.0 a) (* k k))
   (if (<= m 0.94) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.15) {
		tmp = fma((fma(-99.0, (a / k), (10.0 * a)) / k), -1.0, a) / (k * k);
	} else if (m <= 0.94) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.15)
		tmp = Float64(fma(Float64(fma(-99.0, Float64(a / k), Float64(10.0 * a)) / k), -1.0, a) / Float64(k * k));
	elseif (m <= 0.94)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.15], 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, 0.94], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.149999999999999994

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 41.0

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

   5. Applied rewrites 41.0%

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

   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 74.6%

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

   if -0.149999999999999994 < m < 0.93999999999999995

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

   if 0.93999999999999995 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 6: 73.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.15:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.94:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.15)
   (/ a (* k k))
   (if (<= m 0.94) (/ a (fma (+ 10.0 k) k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.15) {
		tmp = a / (k * k);
	} else if (m <= 0.94) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.15)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.94)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.15], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.94], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.149999999999999994

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 41.0

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

   5. Applied rewrites 41.0%

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

   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. flip-+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -0.149999999999999994 < m < 0.93999999999999995

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

   if 0.93999999999999995 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 7: 72.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.15:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.94:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.15)
   (/ a (* k k))
   (if (<= m 0.94) (/ a (fma k k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.15) {
		tmp = a / (k * k);
	} else if (m <= 0.94) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.15)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.94)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.15], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.94], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.149999999999999994

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 41.0

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

   5. Applied rewrites 41.0%

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

   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. flip-+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -0.149999999999999994 < m < 0.93999999999999995

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

   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 91.9%

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

   if 0.93999999999999995 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 8: 62.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.94:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.15e-26)
   (/ a (* k k))
   (if (<= m 0.94) (/ a (fma 10.0 k 1.0)) (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.15e-26) {
		tmp = a / (k * k);
	} else if (m <= 0.94) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.15e-26)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.94)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.15e-26], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.94], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.14999999999999994e-26

   1. Initial program 100.0%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 41.0

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

   5. Applied rewrites 41.0%

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

   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. flip-+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -2.14999999999999994e-26 < m < 0.93999999999999995

   1. Initial program 96.4%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

   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 61.6%

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

   if 0.93999999999999995 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 9: 56.5% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.58:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 8.5e-153) (/ a (* k k)) (if (<= m 0.58) a (* (* (* k k) a) 99.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e-153) {
		tmp = a / (k * k);
	} else if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8.5d-153) then
        tmp = a / (k * k)
    else if (m <= 0.58d0) then
        tmp = a
    else
        tmp = ((k * k) * a) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8.5e-153) {
		tmp = a / (k * k);
	} else if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8.5e-153:
		tmp = a / (k * k)
	elif m <= 0.58:
		tmp = a
	else:
		tmp = ((k * k) * a) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8.5e-153)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.58)
		tmp = a;
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8.5e-153)
		tmp = a / (k * k);
	elseif (m <= 0.58)
		tmp = a;
	else
		tmp = ((k * k) * a) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8.5e-153], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.58], a, N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < 8.4999999999999996e-153

   1. Initial program 98.5%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 64.3

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

   5. Applied rewrites 64.3%

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

   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. flip-+ N/A

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

      7. unpow2 N/A

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

      8. +-commutative N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 8.4999999999999996e-153 < m < 0.57999999999999996

   1. Initial program 96.1%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 89.7

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

   5. Applied rewrites 89.7%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 60.5%

      \[\leadsto a \]

   if 0.57999999999999996 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 10: 39.7% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 0.58:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(k \cdot k\right) \cdot a\right) \cdot 99\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 0.58) a (* (* (* k k) a) 99.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 0.58d0) then
        tmp = a
    else
        tmp = ((k * k) * a) * 99.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 0.58) {
		tmp = a;
	} else {
		tmp = ((k * k) * a) * 99.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 0.58:
		tmp = a
	else:
		tmp = ((k * k) * a) * 99.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 0.58)
		tmp = a;
	else
		tmp = Float64(Float64(Float64(k * k) * a) * 99.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 0.58)
		tmp = a;
	else
		tmp = ((k * k) * a) * 99.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 0.58], a, N[(N[(N[(k * k), $MachinePrecision] * a), $MachinePrecision] * 99.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 0.57999999999999996

   1. Initial program 98.2%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 68.2

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

   5. Applied rewrites 68.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 26.4%

      \[\leadsto a \]

   if 0.57999999999999996 < 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 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 2.9

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

   5. Applied rewrites 2.9%

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

   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. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. lower-*.f64 26.7

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

   8. Applied rewrites 26.7%

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

   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 55.3

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

   11. Applied rewrites 55.3%

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

Alternative 11: 25.4% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 8 \cdot 10^{+16}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(-10 \cdot a\right) \cdot k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 8e+16) a (* (* -10.0 a) k)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 8e+16) {
		tmp = a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (m <= 8d+16) then
        tmp = a
    else
        tmp = ((-10.0d0) * a) * k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 8e+16) {
		tmp = a;
	} else {
		tmp = (-10.0 * a) * k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 8e+16:
		tmp = a
	else:
		tmp = (-10.0 * a) * k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 8e+16)
		tmp = a;
	else
		tmp = Float64(Float64(-10.0 * a) * k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 8e+16)
		tmp = a;
	else
		tmp = (-10.0 * a) * k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 8e+16], a, N[(N[(-10.0 * a), $MachinePrecision] * k), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 8e16

   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 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 25.7%

      \[\leadsto a \]

   if 8e16 < m

   1. Initial program 80.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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 2.9

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

   5. Applied rewrites 2.9%

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

   6. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 10.6

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

   8. Applied rewrites 10.6%

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

   9. Taylor expanded in k around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 16.0

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

   11. Applied rewrites 16.0%

       \[\leadsto \left(-10 \cdot a\right) \cdot k \]
3. Recombined 2 regimes into one program.
4. 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 91.8%

   \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
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 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in k around 0

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

8. Applied rewrites 18.2%

   \[\leadsto a \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025075 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))