Falkner and Boettcher, Appendix A

Percentage Accurate: 89.9% → 99.2%

Time: 5.4s

Alternatives: 4

Speedup: N/A×

• 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: 89.9% 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: 99.2% accurate, N/A× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ \begin{array}{l} t_0 := -1 \cdot \frac{\left(-1 \cdot a\_m\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k}\\ t_1 := \frac{a\_m \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\\ t_2 := \mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)\\ a\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left({\log k}^{2} \cdot m\right) \cdot a\_m, 0.5, \log k \cdot a\_m\right)}{t\_2}, m, \frac{a\_m}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ (* (* -1.0 a_m) (/ (pow k m) (* -1.0 k))) (* -1.0 k))))
        (t_1 (/ (* a_m (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))
        (t_2 (fma k k (fma 10.0 k 1.0))))
   (*
    a_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 1e+302)
        (fma
         (/ (fma (* (* (pow (log k) 2.0) m) a_m) 0.5 (* (log k) a_m)) t_2)
         m
         (/ a_m t_2))
        t_0)))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double t_0 = -1.0 * (((-1.0 * a_m) * (pow(k, m) / (-1.0 * k))) / (-1.0 * k));
	double t_1 = (a_m * pow(k, m)) / ((1.0 + (10.0 * k)) + (k * k));
	double t_2 = fma(k, k, fma(10.0, k, 1.0));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+302) {
		tmp = fma((fma(((pow(log(k), 2.0) * m) * a_m), 0.5, (log(k) * a_m)) / t_2), m, (a_m / t_2));
	} else {
		tmp = t_0;
	}
	return a_s * tmp;
}

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	t_0 = Float64(-1.0 * Float64(Float64(Float64(-1.0 * a_m) * Float64((k ^ m) / Float64(-1.0 * k))) / Float64(-1.0 * k)))
	t_1 = Float64(Float64(a_m * (k ^ m)) / Float64(Float64(1.0 + Float64(10.0 * k)) + Float64(k * k)))
	t_2 = fma(k, k, fma(10.0, k, 1.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+302)
		tmp = fma(Float64(fma(Float64(Float64((log(k) ^ 2.0) * m) * a_m), 0.5, Float64(log(k) * a_m)) / t_2), m, Float64(a_m / t_2));
	else
		tmp = t_0;
	end
	return Float64(a_s * tmp)
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := Block[{t$95$0 = N[(-1.0 * N[(N[(N[(-1.0 * a$95$m), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / N[(-1.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + N[(10.0 * k), $MachinePrecision]), $MachinePrecision] + N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(k * k + N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(a$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+302], N[(N[(N[(N[(N[(N[Power[N[Log[k], $MachinePrecision], 2.0], $MachinePrecision] * m), $MachinePrecision] * a$95$m), $MachinePrecision] * 0.5 + N[(N[Log[k], $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * m + N[(a$95$m / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$0]]), $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))) < 0.0 or 1.0000000000000001e302 < (/.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 88.1%

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

   3. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. sqr-neg-rev N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

         \[\leadsto \frac{a}{-k} \cdot \frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{\mathsf{neg}\left(k\right)} \]

      7. lower-/.f64 N/A

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

      8. exp-prod N/A

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

      9. pow-unpow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

      13. inv-pow N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left({k}^{-1}\right)}}{\mathsf{neg}\left(k\right)} \]

      14. log-pow N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      17. lower-neg.f64 48.8

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{-k} \]

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in k around 0

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

      1. lift-pow.f64 77.4

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

   8. Applied rewrites 77.4%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-neg.f64 N/A

         \[\leadsto \frac{a \cdot \frac{{k}^{m}}{\mathsf{neg}\left(k\right)}}{\mathsf{neg}\left(k\right)} \]

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if 0.0 < (/.f64 (*.f64 a (pow.f64 k m)) (+.f64 (+.f64 #s(literal 1 binary64) (*.f64 #s(literal 10 binary64) k)) (*.f64 k k))) < 1.0000000000000001e302

   1. Initial program 99.9%

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

   3. Taylor expanded in m around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left({\log k}^{2} \cdot m\right) \cdot a, 0.5, \log k \cdot a\right)}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}, m, \frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 0:\\ \;\;\;\;-1 \cdot \frac{\left(-1 \cdot a\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k}\\ \mathbf{elif}\;\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\left({\log k}^{2} \cdot m\right) \cdot a, 0.5, \log k \cdot a\right)}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}, m, \frac{a}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\left(-1 \cdot a\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, N/A× speedup

Math

\[\begin{array}{l} a\_m = \left|a\right| \\ a\_s = \mathsf{copysign}\left(1, a\right) \\ a\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 0.0082:\\ \;\;\;\;\frac{a\_m \cdot {k}^{m}}{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\left(-1 \cdot a\_m\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k}\\ \end{array} \end{array} \]

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (*
  a_s
  (if (<= k 0.0082)
    (/ (* a_m (pow k m)) 1.0)
    (* -1.0 (/ (* (* -1.0 a_m) (/ (pow k m) (* -1.0 k))) (* -1.0 k))))))

C

a\_m = fabs(a);
a\_s = copysign(1.0, a);
double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 0.0082) {
		tmp = (a_m * pow(k, m)) / 1.0;
	} else {
		tmp = -1.0 * (((-1.0 * a_m) * (pow(k, m) / (-1.0 * k))) / (-1.0 * k));
	}
	return a_s * tmp;
}

Fortran

a\_m =     private
a\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: tmp
    if (k <= 0.0082d0) then
        tmp = (a_m * (k ** m)) / 1.0d0
    else
        tmp = (-1.0d0) * ((((-1.0d0) * a_m) * ((k ** m) / ((-1.0d0) * k))) / ((-1.0d0) * k))
    end if
    code = a_s * tmp
end function

Java

a\_m = Math.abs(a);
a\_s = Math.copySign(1.0, a);
public static double code(double a_s, double a_m, double k, double m) {
	double tmp;
	if (k <= 0.0082) {
		tmp = (a_m * Math.pow(k, m)) / 1.0;
	} else {
		tmp = -1.0 * (((-1.0 * a_m) * (Math.pow(k, m) / (-1.0 * k))) / (-1.0 * k));
	}
	return a_s * tmp;
}

Python

a\_m = math.fabs(a)
a\_s = math.copysign(1.0, a)
def code(a_s, a_m, k, m):
	tmp = 0
	if k <= 0.0082:
		tmp = (a_m * math.pow(k, m)) / 1.0
	else:
		tmp = -1.0 * (((-1.0 * a_m) * (math.pow(k, m) / (-1.0 * k))) / (-1.0 * k))
	return a_s * tmp

Julia

a\_m = abs(a)
a\_s = copysign(1.0, a)
function code(a_s, a_m, k, m)
	tmp = 0.0
	if (k <= 0.0082)
		tmp = Float64(Float64(a_m * (k ^ m)) / 1.0);
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(-1.0 * a_m) * Float64((k ^ m) / Float64(-1.0 * k))) / Float64(-1.0 * k)));
	end
	return Float64(a_s * tmp)
end

MATLAB

a\_m = abs(a);
a\_s = sign(a) * abs(1.0);
function tmp_2 = code(a_s, a_m, k, m)
	tmp = 0.0;
	if (k <= 0.0082)
		tmp = (a_m * (k ^ m)) / 1.0;
	else
		tmp = -1.0 * (((-1.0 * a_m) * ((k ^ m) / (-1.0 * k))) / (-1.0 * k));
	end
	tmp_2 = a_s * tmp;
end

Wolfram

a\_m = N[Abs[a], $MachinePrecision]
a\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[a]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[a$95$s_, a$95$m_, k_, m_] := N[(a$95$s * If[LessEqual[k, 0.0082], N[(N[(a$95$m * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision], N[(-1.0 * N[(N[(N[(-1.0 * a$95$m), $MachinePrecision] * N[(N[Power[k, m], $MachinePrecision] / N[(-1.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 0.00820000000000000069

   1. Initial program 94.2%

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

   3. Taylor expanded in k around 0

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

   5. Applied rewrites 98.4%

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

   if 0.00820000000000000069 < k

   1. Initial program 78.2%

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

   3. Taylor expanded in k around inf

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

      1. pow2 N/A

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

      2. sqr-neg-rev N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-neg.f64 N/A

         \[\leadsto \frac{a}{-k} \cdot \frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{\mathsf{neg}\left(k\right)} \]

      7. lower-/.f64 N/A

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

      8. exp-prod N/A

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

      9. pow-unpow N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

      12. lower-exp.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

      13. inv-pow N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left({k}^{-1}\right)}}{\mathsf{neg}\left(k\right)} \]

      14. log-pow N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      16. lower-log.f64 N/A

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

      17. lower-neg.f64 90.1

          \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{-k} \]

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in k around 0

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

      1. lift-pow.f64 90.1

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

   8. Applied rewrites 90.1%

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

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-neg.f64 N/A

         \[\leadsto \frac{a \cdot \frac{{k}^{m}}{\mathsf{neg}\left(k\right)}}{\mathsf{neg}\left(k\right)} \]

      8. mul-1-neg N/A

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

      9. lower-*.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 98.7

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

   10. Applied rewrites 98.7%

       \[\leadsto \frac{a \cdot \frac{{k}^{m}}{-1 \cdot k}}{\color{blue}{-1 \cdot k}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 0.0082:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\left(-1 \cdot a\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, N/A× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (* -1.0 (/ (* (* -1.0 a_m) (/ (pow k m) (* -1.0 k))) (* -1.0 k)))))

C

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

Fortran

a\_m =     private
a\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * ((-1.0d0) * ((((-1.0d0) * a_m) * ((k ** m) / ((-1.0d0) * k))) / ((-1.0d0) * k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in k around inf

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

   1. pow2 N/A

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

   2. sqr-neg-rev N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

      \[\leadsto \frac{a}{-k} \cdot \frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{\mathsf{neg}\left(k\right)} \]

   7. lower-/.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-pow.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

   12. lower-exp.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

   13. inv-pow N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left({k}^{-1}\right)}}{\mathsf{neg}\left(k\right)} \]

   14. log-pow N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   16. lower-log.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   17. lower-neg.f64 46.1

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{-k} \]

5. Applied rewrites 46.1%

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

6. Taylor expanded in k around 0

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

   1. lift-pow.f64 72.3

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

8. Applied rewrites 72.3%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-neg.f64 N/A

      \[\leadsto \frac{a \cdot \frac{{k}^{m}}{\mathsf{neg}\left(k\right)}}{\mathsf{neg}\left(k\right)} \]

   8. mul-1-neg N/A

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

   9. lower-*.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 83.2

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

10. Applied rewrites 83.2%

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

11. Final simplification 83.2%

    \[\leadsto -1 \cdot \frac{\left(-1 \cdot a\right) \cdot \frac{{k}^{m}}{-1 \cdot k}}{-1 \cdot k} \]
12. Add Preprocessing

Alternative 4: 74.9% accurate, N/A× speedup

Math

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

FPCore

a\_m = (fabs.f64 a)
a\_s = (copysign.f64 #s(literal 1 binary64) a)
(FPCore (a_s a_m k m)
 :precision binary64
 (* a_s (* (/ a_m (* -1.0 k)) (/ (pow k m) (* -1.0 k)))))

C

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

Fortran

a\_m =     private
a\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a_s, a_m, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a_s
    real(8), intent (in) :: a_m
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    code = a_s * ((a_m / ((-1.0d0) * k)) * ((k ** m) / ((-1.0d0) * k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.1%

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

3. Taylor expanded in k around inf

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

   1. pow2 N/A

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

   2. sqr-neg-rev N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-neg.f64 N/A

      \[\leadsto \frac{a}{-k} \cdot \frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{\mathsf{neg}\left(k\right)} \]

   7. lower-/.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-pow.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

   12. lower-exp.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left(\frac{1}{k}\right)}}{\mathsf{neg}\left(k\right)} \]

   13. inv-pow N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\log \left({k}^{-1}\right)}}{\mathsf{neg}\left(k\right)} \]

   14. log-pow N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   16. lower-log.f64 N/A

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{\mathsf{neg}\left(k\right)} \]

   17. lower-neg.f64 46.1

       \[\leadsto \frac{a}{-k} \cdot \frac{{\left({\left(e^{-1}\right)}^{m}\right)}^{\left(-1 \cdot \log k\right)}}{-k} \]

5. Applied rewrites 46.1%

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

6. Taylor expanded in k around 0

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

   1. lift-pow.f64 72.3

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

8. Applied rewrites 72.3%

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

9. Final simplification 72.3%

   \[\leadsto \frac{a}{-1 \cdot k} \cdot \frac{{k}^{m}}{-1 \cdot k} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))