Falkner and Boettcher, Appendix A

Percentage Accurate: 90.5% → 97.7%

Time: 4.6s

Alternatives: 12

Speedup: 1.1×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq 2.8:\\ \;\;\;\;\frac{{k}^{m} \cdot a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot {k}^{m}}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m 2.8)
   (/ (* (pow k m) a) (fma (+ 10.0 k) k 1.0))
   (/ (* a (pow k m)) 1.0)))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= 2.8) {
		tmp = (pow(k, m) * a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (a * pow(k, m)) / 1.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= 2.8)
		tmp = Float64(Float64((k ^ m) * a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(a * (k ^ m)) / 1.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, 2.8], N[(N[(N[Power[k, m], $MachinePrecision] * a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < 2.7999999999999998

   1. Initial program 90.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-pow.f64 90.5

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. pow2 N/A

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

      11. associate-+l+ N/A

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

      12. pow2 N/A

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

      13. distribute-rgt-in N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 90.5

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

   3. Applied rewrites 90.5%

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

   if 2.7999999999999998 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.0%

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

Alternative 2: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot {k}^{m}\\ \mathbf{if}\;m \leq -4.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (* a (pow k m))))
   (if (<= m -4.2e-11)
     (/ t_0 (fma 10.0 k 1.0))
     (if (<= m 5e-6)
       (/ (fma (* (log k) m) a a) (fma (+ 10.0 k) k 1.0))
       (/ t_0 1.0)))))

C

double code(double a, double k, double m) {
	double t_0 = a * pow(k, m);
	double tmp;
	if (m <= -4.2e-11) {
		tmp = t_0 / fma(10.0, k, 1.0);
	} else if (m <= 5e-6) {
		tmp = fma((log(k) * m), a, a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0 / 1.0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a * (k ^ m))
	tmp = 0.0
	if (m <= -4.2e-11)
		tmp = Float64(t_0 / fma(10.0, k, 1.0));
	elseif (m <= 5e-6)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(t_0 / 1.0);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -4.2e-11], N[(t$95$0 / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 5e-6], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if m < -4.1999999999999997e-11

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 79.3

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

   4. Applied rewrites 79.3%

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

   if -4.1999999999999997e-11 < m < 5.00000000000000041e-6

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 41.5

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

   4. Applied rewrites 41.5%

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

   if 5.00000000000000041e-6 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.0%

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

Alternative 3: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k \cdot m, a, a\right)}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -2.25e-8)
     t_0
     (if (<= m 5e-6) (/ (fma (* (log k) m) a a) (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -2.25e-8) {
		tmp = t_0;
	} else if (m <= 5e-6) {
		tmp = fma((log(k) * m), a, a) / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -2.25e-8)
		tmp = t_0;
	elseif (m <= 5e-6)
		tmp = Float64(fma(Float64(log(k) * m), a, a) / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -2.25e-8], t$95$0, If[LessEqual[m, 5e-6], N[(N[(N[(N[Log[k], $MachinePrecision] * m), $MachinePrecision] * a + a), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.24999999999999996e-8 or 5.00000000000000041e-6 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.0%

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

   if -2.24999999999999996e-8 < m < 5.00000000000000041e-6

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 41.5

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

   4. Applied rewrites 41.5%

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

Alternative 4: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -2.25 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\log k, m, 1\right)}{\mathsf{fma}\left(10 + k, k, 1\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -2.25e-8)
     t_0
     (if (<= m 5e-6)
       (* (/ (fma (log k) m 1.0) (fma (+ 10.0 k) k 1.0)) a)
       t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -2.25e-8) {
		tmp = t_0;
	} else if (m <= 5e-6) {
		tmp = (fma(log(k), m, 1.0) / fma((10.0 + k), k, 1.0)) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -2.25e-8)
		tmp = t_0;
	elseif (m <= 5e-6)
		tmp = Float64(Float64(fma(log(k), m, 1.0) / fma(Float64(10.0 + k), k, 1.0)) * a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -2.25e-8], t$95$0, If[LessEqual[m, 5e-6], N[(N[(N[(N[Log[k], $MachinePrecision] * m + 1.0), $MachinePrecision] / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -2.24999999999999996e-8 or 5.00000000000000041e-6 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.0%

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

   if -2.24999999999999996e-8 < m < 5.00000000000000041e-6

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 41.5

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

   4. Applied rewrites 41.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-+l+ N/A

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

      11. lower-/.f64 N/A

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

      12. +-commutative N/A

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

      13. pow2 N/A

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

      14. distribute-rgt-in N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-+.f64 42.1

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

   6. Applied rewrites 42.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 42.1

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

      4. pow-to-exp 42.1

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

      5. *-commutative 42.1

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

      6. pow-exp 42.1

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

   8. Applied rewrites 42.1%

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

Alternative 5: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot {k}^{m}}{1}\\ \mathbf{if}\;m \leq -6.4 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ (* a (pow k m)) 1.0)))
   (if (<= m -6.4e-10) t_0 (if (<= m 5e-6) (/ a (fma (+ 10.0 k) k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = (a * pow(k, m)) / 1.0;
	double tmp;
	if (m <= -6.4e-10) {
		tmp = t_0;
	} else if (m <= 5e-6) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(Float64(a * (k ^ m)) / 1.0)
	tmp = 0.0
	if (m <= -6.4e-10)
		tmp = t_0;
	elseif (m <= 5e-6)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(N[(a * N[Power[k, m], $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]}, If[LessEqual[m, -6.4e-10], t$95$0, If[LessEqual[m, 5e-6], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -6.39999999999999961e-10 or 5.00000000000000041e-6 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in k around 0

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

   4. Applied rewrites 82.0%

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

   if -6.39999999999999961e-10 < m < 5.00000000000000041e-6

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

Alternative 6: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.04:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.15 \cdot 10^{+16}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, k, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.04)
   (/ a (* k k))
   (if (<= m 2.15e+16)
     (/ a (fma (+ 10.0 k) k 1.0))
     (fma (- (- (* (* -99.0 a) k)) (* 10.0 a)) k a))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.04) {
		tmp = a / (k * k);
	} else if (m <= 2.15e+16) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = fma((-((-99.0 * a) * k) - (10.0 * a)), k, a);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.04)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.15e+16)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = fma(Float64(Float64(-Float64(Float64(-99.0 * a) * k)) - Float64(10.0 * a)), k, a);
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.04], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.15e+16], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-N[(N[(-99.0 * a), $MachinePrecision] * k), $MachinePrecision]) - N[(10.0 * a), $MachinePrecision]), $MachinePrecision] * k + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0400000000000000008

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if -0.0400000000000000008 < m < 2.15e16

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   if 2.15e16 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.3

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

   7. Applied rewrites 27.3%

      \[\leadsto \mathsf{fma}\left(\left(-\left(-99 \cdot a\right) \cdot k\right) - 10 \cdot a, \color{blue}{k}, a\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.04:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{elif}\;m \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot m\right) \cdot \log k\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.04)
   (/ a (* k k))
   (if (<= m 2.6e+55) (/ a (fma (+ 10.0 k) k 1.0)) (* (* a m) (log k)))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.04) {
		tmp = a / (k * k);
	} else if (m <= 2.6e+55) {
		tmp = a / fma((10.0 + k), k, 1.0);
	} else {
		tmp = (a * m) * log(k);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.04)
		tmp = Float64(a / Float64(k * k));
	elseif (m <= 2.6e+55)
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	else
		tmp = Float64(Float64(a * m) * log(k));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.04], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.6e+55], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * m), $MachinePrecision] * N[Log[k], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.0400000000000000008

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if -0.0400000000000000008 < m < 2.6e55

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   if 2.6e55 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. pow2 N/A

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

      10. distribute-rgt-in N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-+.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in k around 0

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

      1. *-commutative N/A

         \[\leadsto a + \left(m \cdot \log k\right) \cdot a \]

      2. *-commutative N/A

         \[\leadsto a + \left(\log k \cdot m\right) \cdot a \]

      3. +-commutative N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a + a \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\log k \cdot m + 1\right) \cdot a \]

      5. *-commutative N/A

         \[\leadsto \left(m \cdot \log k + 1\right) \cdot a \]

      6. +-commutative N/A

         \[\leadsto \left(1 + m \cdot \log k\right) \cdot a \]

      7. lower-*.f64 N/A

         \[\leadsto \left(1 + m \cdot \log k\right) \cdot a \]

      8. +-commutative N/A

         \[\leadsto \left(m \cdot \log k + 1\right) \cdot a \]

      9. *-commutative N/A

         \[\leadsto \left(\log k \cdot m + 1\right) \cdot a \]

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 23.4

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

   7. Applied rewrites 23.4%

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

   8. Taylor expanded in m around inf

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

      1. *-commutative N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a \]

      2. lower-*.f64 N/A

         \[\leadsto \left(m \cdot \log k\right) \cdot a \]

      3. *-commutative N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

      4. lift-log.f64 N/A

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

      5. lift-*.f64 10.6

         \[\leadsto \left(\log k \cdot m\right) \cdot a \]

   10. Applied rewrites 10.6%

       \[\leadsto \left(\log k \cdot m\right) \cdot a \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(\log k \cdot m\right) \cdot a \]

       2. lift-*.f64 N/A

          \[\leadsto \left(\log k \cdot m\right) \cdot a \]

       3. lift-log.f64 N/A

          \[\leadsto \left(\log k \cdot m\right) \cdot a \]

       4. *-commutative N/A

          \[\leadsto \left(m \cdot \log k\right) \cdot a \]

       5. *-commutative N/A

          \[\leadsto a \cdot \left(m \cdot \log k\right) \]

       6. associate-*r* N/A

          \[\leadsto \left(a \cdot m\right) \cdot \log k \]

       7. lower-*.f64 N/A

          \[\leadsto \left(a \cdot m\right) \cdot \log k \]

       8. lower-*.f64 N/A

          \[\leadsto \left(a \cdot m\right) \cdot \log k \]

       9. lift-log.f64 10.6

          \[\leadsto \left(a \cdot m\right) \cdot \log k \]

   12. Applied rewrites 10.6%

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

Alternative 8: 53.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.04:\\ \;\;\;\;\frac{a}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10 + k, k, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (if (<= m -0.04) (/ a (* k k)) (/ a (fma (+ 10.0 k) k 1.0))))

C

double code(double a, double k, double m) {
	double tmp;
	if (m <= -0.04) {
		tmp = a / (k * k);
	} else {
		tmp = a / fma((10.0 + k), k, 1.0);
	}
	return tmp;
}

Julia

function code(a, k, m)
	tmp = 0.0
	if (m <= -0.04)
		tmp = Float64(a / Float64(k * k));
	else
		tmp = Float64(a / fma(Float64(10.0 + k), k, 1.0));
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := If[LessEqual[m, -0.04], N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(a / N[(N[(10.0 + k), $MachinePrecision] * k + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.0400000000000000008

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if -0.0400000000000000008 < m

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

Alternative 9: 47.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 195000:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(10, k, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 4.2e-307) t_0 (if (<= k 195000.0) (/ a (fma 10.0 k 1.0)) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 4.2e-307) {
		tmp = t_0;
	} else if (k <= 195000.0) {
		tmp = a / fma(10.0, k, 1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 4.2e-307)
		tmp = t_0;
	elseif (k <= 195000.0)
		tmp = Float64(a / fma(10.0, k, 1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.2e-307], t$95$0, If[LessEqual[k, 195000.0], N[(a / N[(10.0 * k + 1.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000002e-307 or 195000 < k

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 4.2000000000000002e-307 < k < 195000

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around 0

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

   7. Applied rewrites 29.0%

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

Alternative 10: 47.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.052:\\ \;\;\;\;\mathsf{fma}\left(a \cdot k, -10, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 4.2e-307) t_0 (if (<= k 0.052) (fma (* a k) -10.0 a) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 4.2e-307) {
		tmp = t_0;
	} else if (k <= 0.052) {
		tmp = fma((a * k), -10.0, a);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 4.2e-307)
		tmp = t_0;
	elseif (k <= 0.052)
		tmp = fma(Float64(a * k), -10.0, a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.2e-307], t$95$0, If[LessEqual[k, 0.052], N[(N[(a * k), $MachinePrecision] * -10.0 + a), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000002e-307 or 0.0519999999999999976 < k

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 4.2000000000000002e-307 < k < 0.0519999999999999976

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 21.2

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

   7. Applied rewrites 21.2%

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

Alternative 11: 47.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{k \cdot k}\\ \mathbf{if}\;k \leq 4.2 \cdot 10^{-307}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;k \leq 0.052:\\ \;\;\;\;\frac{a}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a k m)
 :precision binary64
 (let* ((t_0 (/ a (* k k))))
   (if (<= k 4.2e-307) t_0 (if (<= k 0.052) (/ a 1.0) t_0))))

C

double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 4.2e-307) {
		tmp = t_0;
	} else if (k <= 0.052) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, k, m)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / (k * k)
    if (k <= 4.2d-307) then
        tmp = t_0
    else if (k <= 0.052d0) then
        tmp = a / 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double k, double m) {
	double t_0 = a / (k * k);
	double tmp;
	if (k <= 4.2e-307) {
		tmp = t_0;
	} else if (k <= 0.052) {
		tmp = a / 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, k, m):
	t_0 = a / (k * k)
	tmp = 0
	if k <= 4.2e-307:
		tmp = t_0
	elif k <= 0.052:
		tmp = a / 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(a, k, m)
	t_0 = Float64(a / Float64(k * k))
	tmp = 0.0
	if (k <= 4.2e-307)
		tmp = t_0;
	elseif (k <= 0.052)
		tmp = Float64(a / 1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, k, m)
	t_0 = a / (k * k);
	tmp = 0.0;
	if (k <= 4.2e-307)
		tmp = t_0;
	elseif (k <= 0.052)
		tmp = a / 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, k_, m_] := Block[{t$95$0 = N[(a / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 4.2e-307], t$95$0, If[LessEqual[k, 0.052], N[(a / 1.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if k < 4.2000000000000002e-307 or 0.0519999999999999976 < k

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 4.2000000000000002e-307 < k < 0.0519999999999999976

   1. Initial program 90.5%

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

   2. Taylor expanded in m around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-+.f64 46.1

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. pow2 N/A

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

      5. associate-+l+ N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 36.4

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

   7. Applied rewrites 36.4%

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

   8. Taylor expanded in k around 0

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

   10. Applied rewrites 20.2%

       \[\leadsto \frac{a}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 20.2% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \frac{a}{1} \end{array} \]

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 90.5%

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

2. Taylor expanded in m around 0

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

   1. lower-/.f64 N/A

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

   2. pow2 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-+.f64 46.1

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

4. Applied rewrites 46.1%

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

5. Taylor expanded in k around inf

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. pow2 N/A

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

   5. associate-+l+ N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lower-*.f64 36.4

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

7. Applied rewrites 36.4%

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

8. Taylor expanded in k around 0

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

10. Applied rewrites 20.2%

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

Reproduce

herbie shell --seed 2025135 
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1.0 (* 10.0 k)) (* k k))))