Falkner and Boettcher, Appendix A

Percentage Accurate: 90.4% → 96.9%

Time: 4.9s

Alternatives: 14

Speedup: 1.3×

• 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: 90.4% 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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -2.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{k \cdot k}\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -2.75e-8)
   (/ (* a (pow k m)) (* k k))
   (if (<= m 4.6e-26)
     (/ (fma (* (log k) m) a a) (fma (+ 10.0 k) k 1.0))
     (* (pow k m) a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -2.75e-8) {
		tmp = (a * pow(k, m)) / (k * k);
	} else if (m <= 4.6e-26) {
		tmp = fma((log(k) * m), a, a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = pow(k, m) * a;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -2.75e-8)
		tmp = Float64(Float64(a * (k ^ m)) / Float64(k * k));
	elseif (m <= 4.6e-26)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64((k ^ m) * a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -2.75e-8], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 4.6e-26], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -2.7500000000000001e-8

   1. Initial program 99.8%

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

   2. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 99.3

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

   4. Applied rewrites 99.3%

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

   if -2.7500000000000001e-8 < m < 4.60000000000000018e-26

   1. Initial program 93.3%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if 4.60000000000000018e-26 < m

   1. Initial program 78.7%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 2: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.6e-26)
   (/ (* 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 <= 4.6e-26) {
		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 <= 4.6d-26) 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 <= 4.6e-26) {
		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 <= 4.6e-26:
		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 <= 4.6e-26)
		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 <= 4.6e-26)
		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, 4.6e-26], 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 < 4.60000000000000018e-26

   1. Initial program 96.5%

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

   if 4.60000000000000018e-26 < m

   1. Initial program 78.7%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;a \cdot \frac{{k}^{m}}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;{k}^{m} \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 4.6e-26)
   (* 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 <= 4.6e-26) {
		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 <= 4.6e-26)
		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, 4.6e-26], 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 < 4.60000000000000018e-26

   1. Initial program 96.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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

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

      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

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

      8. pow2 N/A

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

      9. associate-+l+ N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. distribute-rgt-in N/A

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

      16. +-commutative N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. lower-+.f64 96.5

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

   3. Applied rewrites 96.5%

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

   if 4.60000000000000018e-26 < m

   1. Initial program 78.7%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 97.7

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

   4. Applied rewrites 97.7%

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

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -3.8 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -3.8e-6)
     t_0
     (if (<= m 4.6e-26)
       (/ (fma (* (log k) m) a a) (fma (+ 10.0 k) k 1.0))
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -3.8e-6) {
		tmp = t_0;
	} else if (m <= 4.6e-26) {
		tmp = fma((log(k) * m), a, a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -3.8e-6)
		tmp = t_0;
	elseif (m <= 4.6e-26)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -3.8e-6], t$95$0, If[LessEqual[m, 4.6e-26], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -3.8e-6 or 4.60000000000000018e-26 < m

   1. Initial program 88.9%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if -3.8e-6 < m < 4.60000000000000018e-26

   1. Initial program 93.2%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 93.2

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

   4. Applied rewrites 93.2%

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

Alternative 5: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -2.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 4.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -2.2e-13)
     t_0
     (if (<= m 4.6e-26) (/ a (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -2.2e-13) {
		tmp = t_0;
	} else if (m <= 4.6e-26) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -2.2e-13)
		tmp = t_0;
	elseif (m <= 4.6e-26)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[m, -2.2e-13], t$95$0, If[LessEqual[m, 4.6e-26], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.19999999999999997e-13 or 4.60000000000000018e-26 < m

   1. Initial program 88.9%

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

   2. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-pow.f64 98.4

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

   4. Applied rewrites 98.4%

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

   if -2.19999999999999997e-13 < m < 4.60000000000000018e-26

   1. Initial program 93.2%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 93.1

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

   4. Applied rewrites 93.1%

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

Alternative 6: 72.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(99 \cdot a\right) \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.4)
   (/ a (* k k))
   (if (<= m 0.75) (/ a (fma (+ 10.0 k) k 1.0)) (* (* 99.0 a) (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.4) {
		tmp = a / (k * k);
	} else if (m <= 0.75) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (99.0 * a) * (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.4)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.75)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(99.0 * a) * Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.4], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.75], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(99.0 * a), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.40000000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

   if -0.40000000000000002 < m < 0.75

   1. Initial program 93.2%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 0.75 < m

   1. Initial program 78.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 25.4

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

         \[\leadsto \left(-10 \cdot \frac{a}{k} + 99 \cdot a\right) \cdot {k}^{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{a}{k} \cdot -10 + 99 \cdot a\right) \cdot {k}^{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 47.5

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

   10. Applied rewrites 47.5%

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

   11. Taylor expanded in k around inf

       \[\leadsto \left(99 \cdot a\right) \cdot \left(k \cdot k\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 60.9

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

   13. Applied rewrites 60.9%

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

Alternative 7: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 0.75:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(99 \cdot a\right) \cdot \left(k \cdot k\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.4)
   (/ a (* k k))
   (if (<= m 0.75) (/ a (fma k k 1.0)) (* (* 99.0 a) (* k k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.4) {
		tmp = a / (k * k);
	} else if (m <= 0.75) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = (99.0 * a) * (k * k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.4)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 0.75)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(99.0 * a) * Float64(k * k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.4], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.75], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(99.0 * a), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.40000000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

   if -0.40000000000000002 < m < 0.75

   1. Initial program 93.2%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 91.7

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

   4. Applied rewrites 91.7%

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

   5. Taylor expanded in k around inf

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

   7. Applied rewrites 89.3%

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

   if 0.75 < m

   1. Initial program 78.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 25.4

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

         \[\leadsto \left(-10 \cdot \frac{a}{k} + 99 \cdot a\right) \cdot {k}^{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{a}{k} \cdot -10 + 99 \cdot a\right) \cdot {k}^{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 47.5

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

   10. Applied rewrites 47.5%

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

   11. Taylor expanded in k around inf

       \[\leadsto \left(99 \cdot a\right) \cdot \left(k \cdot k\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 60.9

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

   13. Applied rewrites 60.9%

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

Alternative 8: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.4:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 400:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.4)
   (/ a (* k k))
   (if (<= m 400.0) (/ a (fma k k 1.0)) (* (* k a) -10.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.4) {
		tmp = a / (k * k);
	} else if (m <= 400.0) {
		tmp = a / fma(k, k, 1.0);
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.4)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 400.0)
		tmp = Float64(a / fma(k, k, 1.0));
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.4], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 400.0], N[(a / N[(k * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.40000000000000002

   1. Initial program 99.9%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 62.5

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

   7. Applied rewrites 62.5%

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

   if -0.40000000000000002 < m < 400

   1. Initial program 93.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in k around inf

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

   7. Applied rewrites 89.1%

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

   if 400 < m

   1. Initial program 78.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

         \[\leadsto \left(-10 \cdot \frac{a}{k} + 99 \cdot a\right) \cdot {k}^{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{a}{k} \cdot -10 + 99 \cdot a\right) \cdot {k}^{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 47.6

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

   10. Applied rewrites 47.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto -10 \cdot \left(a \cdot k\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 19.9

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

   13. Applied rewrites 19.9%

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

Alternative 9: 48.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 90000000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.7e-304)
   (/ a (* k k))
   (if (<= k 90000000.0) (/ a (fma 10.0 k 1.0)) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.7e-304) {
		tmp = a / (k * k);
	} else if (k <= 90000000.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.7e-304)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 90000000.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.7e-304], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 90000000.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-304

   1. Initial program 89.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 21.5

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

   if 2.7000000000000001e-304 < k < 9e7

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 50.3%

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

   if 9e7 < k

   1. Initial program 81.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 63.0

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

   9. Applied rewrites 63.0%

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

Alternative 10: 48.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot a, k, a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.7e-304)
   (/ a (* k k))
   (if (<= k 0.1) (fma (* -10.0 a) k a) (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.7e-304) {
		tmp = a / (k * k);
	} else if (k <= 0.1) {
		tmp = fma((-10.0 * a), k, a);
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.7e-304)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 0.1)
		tmp = fma(Float64(-10.0 * a), k, a);
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.7e-304], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 0.1], N[(N[(-10.0 * a), $MachinePrecision] * k + a), $MachinePrecision], N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-304

   1. Initial program 89.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 21.5

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

   if 2.7000000000000001e-304 < k < 0.10000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   8. Taylor expanded in k around 0

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

      1. lower-*.f64 51.2

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

   10. Applied rewrites 51.2%

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

   if 0.10000000000000001 < k

   1. Initial program 81.8%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 61.8

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

   9. Applied rewrites 61.8%

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

Alternative 11: 48.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;k \leq 90000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{k}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= k 2.7e-304) (/ a (* k k)) (if (<= k 90000000.0) a (/ (/ a k) k))))

C

double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.7e-304) {
		tmp = a / (k * k);
	} else if (k <= 90000000.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 2.7d-304) then
        tmp = a / (k * k)
    else if (k <= 90000000.0d0) then
        tmp = a
    else
        tmp = (a / k) / k
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (k <= 2.7e-304) {
		tmp = a / (k * k);
	} else if (k <= 90000000.0) {
		tmp = a;
	} else {
		tmp = (a / k) / k;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if k <= 2.7e-304:
		tmp = a / (k * k)
	elif k <= 90000000.0:
		tmp = a
	else:
		tmp = (a / k) / k
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (k <= 2.7e-304)
		tmp = Float64(a / Float64(k * k));
	elseif (k <= 90000000.0)
		tmp = a;
	else
		tmp = Float64(Float64(a / k) / k);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (k <= 2.7e-304)
		tmp = a / (k * k);
	elseif (k <= 90000000.0)
		tmp = a;
	else
		tmp = (a / k) / k;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[k, 2.7e-304], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 90000000.0], a, N[(N[(a / k), $MachinePrecision] / k), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 2.7000000000000001e-304

   1. Initial program 89.4%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 21.5

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

   4. Applied rewrites 21.5%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 29.7

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

   7. Applied rewrites 29.7%

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

   if 2.7000000000000001e-304 < k < 9e7

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 49.4%

      \[\leadsto a \]

   if 9e7 < k

   1. Initial program 81.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 59.4

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

   7. Applied rewrites 59.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 63.0

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

   9. Applied rewrites 63.0%

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

Alternative 12: 47.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 2.7 \cdot 10^{-304}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 90000000:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 2.7e-304) t_0 (if (<= k 90000000.0) a t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2.7e-304) {
		tmp = t_0;
	} else if (k <= 90000000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 2.7d-304) then
        tmp = t_0
    else if (k <= 90000000.0d0) then
        tmp = a
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 2.7e-304) {
		tmp = t_0;
	} else if (k <= 90000000.0) {
		tmp = a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 2.7e-304:
		tmp = t_0
	elif k <= 90000000.0:
		tmp = a
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 2.7e-304)
		tmp = t_0;
	elseif (k <= 90000000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 2.7e-304)
		tmp = t_0;
	elseif (k <= 90000000.0)
		tmp = a;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.7e-304], t$95$0, If[LessEqual[k, 90000000.0], a, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7000000000000001e-304 or 9e7 < k

   1. Initial program 85.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in k around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 45.6

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

   7. Applied rewrites 45.6%

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

   if 2.7000000000000001e-304 < k < 9e7

   1. Initial program 100.0%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 51.8

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

   4. Applied rewrites 51.8%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 49.4%

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

Alternative 13: 25.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 400:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(k \cdot a\right) \cdot -10\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m) :precision binary64 (if (<= m 400.0) a (* (* k a) -10.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 400.0) {
		tmp = a;
	} else {
		tmp = (k * a) * -10.0;
	}
	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 <= 400.0d0) then
        tmp = a
    else
        tmp = (k * a) * (-10.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double tmp;
	if (m <= 400.0) {
		tmp = a;
	} else {
		tmp = (k * a) * -10.0;
	}
	return tmp;
}

Python

def code(a, k, m):
	tmp = 0
	if m <= 400.0:
		tmp = a
	else:
		tmp = (k * a) * -10.0
	return tmp

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 400.0)
		tmp = a;
	else
		tmp = Float64(Float64(k * a) * -10.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	tmp = 0.0;
	if (m <= 400.0)
		tmp = a;
	else
		tmp = (k * a) * -10.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 400.0], a, N[(N[(k * a), $MachinePrecision] * -10.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 400

   1. Initial program 96.3%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 66.2

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

   4. Applied rewrites 66.2%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 28.4%

      \[\leadsto a \]

   if 400 < m

   1. Initial program 78.1%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 3.0

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

   4. Applied rewrites 3.0%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 25.5

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

   7. Applied rewrites 25.5%

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

   8. Taylor expanded in k around inf

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

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

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

      6. metadata-eval N/A

         \[\leadsto \left(-10 \cdot \frac{a}{k} + 99 \cdot a\right) \cdot {k}^{2} \]

      7. *-commutative N/A

         \[\leadsto \left(\frac{a}{k} \cdot -10 + 99 \cdot a\right) \cdot {k}^{2} \]

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 47.6

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

   10. Applied rewrites 47.6%

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

   11. Taylor expanded in k around 0

       \[\leadsto -10 \cdot \left(a \cdot k\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lift-*.f64 19.9

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

   13. Applied rewrites 19.9%

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

Alternative 14: 20.3% accurate, 34.5× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, k_, m_] := a

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 45.4

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

4. Applied rewrites 45.4%

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

5. Taylor expanded in k around 0

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

7. Applied rewrites 20.3%

   \[\leadsto a \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025122 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))