Falkner and Boettcher, Appendix A

Percentage Accurate: 89.7% → 99.7%

Time: 4.7s

Alternatives: 10

Speedup: 1.3×

• 77.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;k \leq 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{k - -10}{t\_0}, k, \frac{{k}^{\left(-m\right)}}{a}\right)}\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= k 1e-104)
     t_0
     (/ 1.0 (fma (/ (- k -10.0) t_0) k (/ (pow k (- m)) a))))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (k <= 1e-104) {
		tmp = t_0;
	} else {
		tmp = 1.0 / fma(((k - -10.0) / t_0), k, (pow(k, -m) / a));
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (k <= 1e-104)
		tmp = t_0;
	else
		tmp = Float64(1.0 / fma(Float64(Float64(k - -10.0) / t_0), k, Float64((k ^ Float64(-m)) / a)));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[k, 1e-104], t$95$0, N[(1.0 / N[(N[(N[(k - -10.0), $MachinePrecision] / t$95$0), $MachinePrecision] * k + N[(N[Power[k, (-m)], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 9.99999999999999927e-105

   1. Initial program 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.7%

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

   6. Applied rewrites 82.7%

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

   if 9.99999999999999927e-105 < k

   1. Initial program 89.7%

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \]
   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. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 89.5%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{\left(10 \cdot k + k \cdot k\right) + \color{blue}{\left(0 + 1\right)}}{a \cdot {k}^{m}}} \]

      12. associate-+r+ N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(\left(10 \cdot k + k \cdot k\right) + 0\right) + 1}}{a \cdot {k}^{m}}} \]

      13. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\left(0 + \left(10 \cdot k + k \cdot k\right)\right)} + 1}{a \cdot {k}^{m}}} \]

      14. +-lft-identity N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-out N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. add-flip N/A

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

      22. lower--.f64 N/A

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

      23. metadata-eval 89.5%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

      26. lower-*.f64 89.5%

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

   3. Applied rewrites 89.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-/r* N/A

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

      17. lower-/.f64 N/A

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

      18. lift-pow.f64 N/A

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

      19. pow-flip N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-neg.f64 88.8%

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

   5. Applied rewrites 88.8%

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

Alternative 2: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {k}^{m} \cdot a\\ \mathbf{if}\;m \leq -1.3 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 1.55 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(k - -10, \frac{k}{a}, \frac{1}{a}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* (pow k m) a)))
   (if (<= m -1.3e+15)
     t_0
     (if (<= m 1.55e-18) (/ 1.0 (fma (- k -10.0) (/ k a) (/ 1.0 a))) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = pow(k, m) * a;
	double tmp;
	if (m <= -1.3e+15) {
		tmp = t_0;
	} else if (m <= 1.55e-18) {
		tmp = 1.0 / fma((k - -10.0), (k / a), (1.0 / a));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64((k ^ m) * a)
	tmp = 0.0
	if (m <= -1.3e+15)
		tmp = t_0;
	elseif (m <= 1.55e-18)
		tmp = Float64(1.0 / fma(Float64(k - -10.0), Float64(k / a), Float64(1.0 / a)));
	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, -1.3e+15], t$95$0, If[LessEqual[m, 1.55e-18], N[(1.0 / N[(N[(k - -10.0), $MachinePrecision] * N[(k / a), $MachinePrecision] + N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.3e15 or 1.55000000000000003e-18 < m

   1. Initial program 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in k around 0

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

      1. lower-pow.f64 82.7%

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

   6. Applied rewrites 82.7%

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

   if -1.3e15 < m < 1.55000000000000003e-18

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. distribute-rgt-out N/A

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

      7. +-commutative N/A

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

      8. metadata-eval N/A

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

      9. sub-flip N/A

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

      10. lift--.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

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

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

      16. lower-/.f32 N/A

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

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

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

      18. lower-/.f64 44.8%

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

   6. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 44.7%

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

   8. Applied rewrites 44.7%

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

Alternative 3: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.7%

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

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

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2%

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

   9. Applied rewrites 29.2%

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   3. Applied rewrites 89.6%

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

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

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2%

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

   9. Applied rewrites 29.2%

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 54.5% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 5 \cdot 10^{+256}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot \left|a\right|\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 5e+256)
    (/ (fabs a) (fma (- k -10.0) k 1.0))
    (* (+ 1.0 (* k (- (* 99.0 k) 10.0))) (fabs a)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 5e+256) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = (1.0 + (k * ((99.0 * k) - 10.0))) * fabs(a);
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 5e+256)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(Float64(1.0 + Float64(k * Float64(Float64(99.0 * k) - 10.0))) * abs(a));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+256], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(k * N[(N[(99.0 * k), $MachinePrecision] - 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 5.00000000000000015e256

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   if 5.00000000000000015e256 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. distribute-rgt-out N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. sub-flip N/A

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

      9. lift--.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. mult-flip-rev N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   7. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 29.2%

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

   9. Applied rewrites 29.2%

      \[\leadsto \left(1 + k \cdot \left(99 \cdot k - 10\right)\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 49.8% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, a\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{\left|a\right| \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 4 \cdot 10^{+294}:\\ \;\;\;\;\frac{\left|a\right|}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;k \cdot \mathsf{fma}\left(-10, \left|a\right|, \frac{\left|a\right|}{k}\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (*
  (copysign 1.0 a)
  (if (<= (/ (* (fabs a) (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))) 4e+294)
    (/ (fabs a) (fma (- k -10.0) k 1.0))
    (* k (fma -10.0 (fabs a) (/ (fabs a) k))))))

C

double code(double a, double k, double m) {
	double tmp;
	if (((fabs(a) * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k))) <= 4e+294) {
		tmp = fabs(a) / fma((k - -10.0), k, 1.0);
	} else {
		tmp = k * fma(-10.0, fabs(a), (fabs(a) / k));
	}
	return copysign(1.0, a) * tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (Float64(Float64(abs(a) * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k))) <= 4e+294)
		tmp = Float64(abs(a) / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = Float64(k * fma(-10.0, abs(a), Float64(abs(a) / k)));
	end
	return Float64(copysign(1.0, a) * tmp)
end

Wolfram

code[a_, k_, m_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(N[(N[Abs[a], $MachinePrecision] * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+294], N[(N[Abs[a], $MachinePrecision] / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(k * N[(-10.0 * N[Abs[a], $MachinePrecision] + N[(N[Abs[a], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 4.00000000000000027e294

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   if 4.00000000000000027e294 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k)))

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.0%

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

   7. Applied rewrites 21.0%

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

   8. Taylor expanded in k around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 20.4%

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

   10. Applied rewrites 20.4%

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

Alternative 7: 46.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 24000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(k - -10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a, a\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 24000.0) (/ a (fma (- k -10.0) k 1.0)) (fma (* -10.0 k) a a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 24000.0) {
		tmp = a / fma((k - -10.0), k, 1.0);
	} else {
		tmp = fma((-10.0 * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 24000.0)
		tmp = Float64(a / fma(Float64(k - -10.0), k, 1.0));
	else
		tmp = fma(Float64(-10.0 * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 24000.0], N[(a / N[(N[(k - -10.0), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 24000

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   if 24000 < m

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.0%

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

   7. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 21.0%

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

   9. Applied rewrites 21.0%

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

Alternative 8: 29.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;m \leq 12500:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-10 \cdot k, a, a\right)\\ \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 12500.0) (/ a (fma 10.0 k 1.0)) (fma (* -10.0 k) a a)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 12500.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = fma((-10.0 * k), a, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 12500.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = fma(Float64(-10.0 * k), a, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 12500.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(-10.0 * k), $MachinePrecision] * a + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 12500

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. distribute-rgt-out N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. sub-flip N/A

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

      8. lift--.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   7. Taylor expanded in k around 0

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

   9. Applied rewrites 27.8%

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

   if 12500 < m

   1. Initial program 89.7%

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

   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. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-pow.f64 44.8%

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

   4. Applied rewrites 44.8%

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

   5. Taylor expanded in k around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 21.0%

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

   7. Applied rewrites 21.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. metadata-eval N/A

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

      8. distribute-lft-neg-out N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 21.0%

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

   9. Applied rewrites 21.0%

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

Alternative 9: 21.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

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. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 44.8%

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

4. Applied rewrites 44.8%

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

5. Taylor expanded in k around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 21.0%

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

7. Applied rewrites 21.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. metadata-eval N/A

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

   8. distribute-lft-neg-out N/A

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

   9. lower-fma.f64 N/A

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

   10. distribute-lft-neg-out N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 21.0%

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

9. Applied rewrites 21.0%

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

Alternative 10: 20.0% accurate, 7.9× speedup

Math

\[\frac{a}{1} \]

FPCore

(FPCore (a k m) :precision binary64 (/ a 1.0))

C

double code(double a, double k, double m) {
	return a / 1.0;
}

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 / 1.0d0
end function

Java

public static double code(double a, double k, double m) {
	return a / 1.0;
}

Python

def code(a, k, m):
	return a / 1.0

Julia

function code(a, k, m)
	return Float64(a / 1.0)
end

MATLAB

function tmp = code(a, k, m)
	tmp = a / 1.0;
end

Wolfram

code[a_, k_, m_] := N[(a / 1.0), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

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. lower-+.f64 N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-pow.f64 44.8%

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

4. Applied rewrites 44.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow2 N/A

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

   4. distribute-rgt-out N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. sub-flip N/A

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

   8. lift--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 44.8%

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

6. Applied rewrites 44.8%

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

7. Taylor expanded in k around 0

   \[\leadsto \frac{a}{1} \]
8. Step-by-step derivation

9. Applied rewrites 20.0%

   \[\leadsto \frac{a}{1} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))