ENA, Section 1.4, Exercise 4b, n=5

Percentage Accurate: 88.7% → 98.2%

Time: 6.0s

Alternatives: 10

Speedup: 5.5×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{5} - {x}^{5} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 5.0) (pow x 5.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 5.0) - pow(x, 5.0);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 5.0) - math.pow(x, 5.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5.2e-43) (not (<= x 2.5e-39)))
   (* (* (* (fma (* eps (+ eps x)) 10.0 (* (* x x) 5.0)) x) x) eps)
   (- (pow (+ x eps) 5.0) (pow x 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5.2e-43) || !(x <= 2.5e-39)) {
		tmp = ((fma((eps * (eps + x)), 10.0, ((x * x) * 5.0)) * x) * x) * eps;
	} else {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5.2e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(fma(Float64(eps * Float64(eps + x)), 10.0, Float64(Float64(x * x) * 5.0)) * x) * x) * eps);
	else
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5.2e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0 + N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 20.3%

      \[\leadsto \left(\left(\left(\left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
   9. Step-by-step derivation

   10. Applied rewrites 20.3%

       \[\leadsto \left(x \cdot \left(\left(10 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]

   11. Taylor expanded in x around 0

       \[\leadsto \left({x}^{2} \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]
   12. Step-by-step derivation

   13. Applied rewrites 95.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \]

   if -5.2e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot {\varepsilon}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (fma (* eps (+ eps x)) 10.0 (* (* x x) 5.0)) x) x) eps)
   (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (pow eps 3.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = ((fma((eps * (eps + x)), 10.0, ((x * x) * 5.0)) * x) * x) * eps;
	} else {
		tmp = fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * pow(eps, 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(fma(Float64(eps * Float64(eps + x)), 10.0, Float64(Float64(x * x) * 5.0)) * x) * x) * eps);
	else
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * (eps ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0 + N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 20.3%

      \[\leadsto \left(\left(\left(\left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
   9. Step-by-step derivation

   10. Applied rewrites 20.3%

       \[\leadsto \left(x \cdot \left(\left(10 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]

   11. Taylor expanded in x around 0

       \[\leadsto \left({x}^{2} \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]
   12. Step-by-step derivation

   13. Applied rewrites 95.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot {\varepsilon}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (fma (* eps (+ eps x)) 10.0 (* (* x x) 5.0)) x) x) eps)
   (* (fma (/ x eps) 5.0 1.0) (pow eps 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = ((fma((eps * (eps + x)), 10.0, ((x * x) * 5.0)) * x) * x) * eps;
	} else {
		tmp = fma((x / eps), 5.0, 1.0) * pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(fma(Float64(eps * Float64(eps + x)), 10.0, Float64(Float64(x * x) * 5.0)) * x) * x) * eps);
	else
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * (eps ^ 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0 + N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 20.3%

      \[\leadsto \left(\left(\left(\left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
   9. Step-by-step derivation

   10. Applied rewrites 20.3%

       \[\leadsto \left(x \cdot \left(\left(10 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]

   11. Taylor expanded in x around 0

       \[\leadsto \left({x}^{2} \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]
   12. Step-by-step derivation

   13. Applied rewrites 95.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \cdot {\varepsilon}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\varepsilon} \cdot 5} + 1\right) \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.8

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (fma (* eps (+ eps x)) 10.0 (* (* x x) 5.0)) x) x) eps)
   (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (* (* eps eps) eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = ((fma((eps * (eps + x)), 10.0, ((x * x) * 5.0)) * x) * x) * eps;
	} else {
		tmp = fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * ((eps * eps) * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(fma(Float64(eps * Float64(eps + x)), 10.0, Float64(Float64(x * x) * 5.0)) * x) * x) * eps);
	else
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * Float64(Float64(eps * eps) * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(N[(eps * N[(eps + x), $MachinePrecision]), $MachinePrecision] * 10.0 + N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto \left(10 \cdot \left({\varepsilon}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
   7. Step-by-step derivation

   8. Applied rewrites 20.3%

      \[\leadsto \left(\left(\left(\left(10 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
   9. Step-by-step derivation

   10. Applied rewrites 20.3%

       \[\leadsto \left(x \cdot \left(\left(10 \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]

   11. Taylor expanded in x around 0

       \[\leadsto \left({x}^{2} \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot \varepsilon \]
   12. Step-by-step derivation

   13. Applied rewrites 95.9%

       \[\leadsto \left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\mathsf{fma}\left(\varepsilon \cdot \left(\varepsilon + x\right), 10, \left(x \cdot x\right) \cdot 5\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-43)
   (* (* (* (fma 10.0 eps (* 5.0 x)) x) eps) (* x x))
   (if (<= x 2.5e-39)
     (* (fma (fma 5.0 x eps) eps (* (* 10.0 x) x)) (* (* eps eps) eps))
     (* (* (* (* x x) 5.0) eps) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-43) {
		tmp = ((fma(10.0, eps, (5.0 * x)) * x) * eps) * (x * x);
	} else if (x <= 2.5e-39) {
		tmp = fma(fma(5.0, x, eps), eps, ((10.0 * x) * x)) * ((eps * eps) * eps);
	} else {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-43)
		tmp = Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps) * Float64(x * x));
	elseif (x <= 2.5e-39)
		tmp = Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(10.0 * x) * x)) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-43], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-39], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(10.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000019e-43

   1. Initial program 25.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if 2.4999999999999999e-39 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.9%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (* x x) 5.0) eps) (* x x))
   (* (* (* (* (fma 5.0 x eps) eps) eps) eps) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (((fma(5.0, x, eps) * eps) * eps) * eps) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * eps) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \cdot {\varepsilon}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\varepsilon} \cdot 5} + 1\right) \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.8

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) + {\varepsilon}^{5}} \]
   7. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {\varepsilon}^{4}\right)} + {\varepsilon}^{5} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{5} \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot 5\right) \cdot {\varepsilon}^{4}} + {\varepsilon}^{5} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right)} \cdot {\varepsilon}^{4} + {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

      6. pow-plus N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x + \varepsilon\right)} \]

      9. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{\left(5 \cdot x + \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      14. lower-pow.f64 99.8

          \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{4}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.7%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(-\varepsilon\right)\right) \cdot \color{blue}{\left(-\varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (* x x) 5.0) eps) (* x x))
   (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \cdot {\varepsilon}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\varepsilon} \cdot 5} + 1\right) \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.8

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) + {\varepsilon}^{5}} \]
   7. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {\varepsilon}^{4}\right)} + {\varepsilon}^{5} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{5} \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot 5\right) \cdot {\varepsilon}^{4}} + {\varepsilon}^{5} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right)} \cdot {\varepsilon}^{4} + {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

      6. pow-plus N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x + \varepsilon\right)} \]

      9. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{\left(5 \cdot x + \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      14. lower-pow.f64 99.8

          \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{4}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5e-43)
   (* (* (* (fma 10.0 eps (* 5.0 x)) x) eps) (* x x))
   (if (<= x 2.5e-39)
     (* (* (* (* (fma 5.0 x eps) eps) eps) eps) eps)
     (* (* (* (* x x) 5.0) eps) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5e-43) {
		tmp = ((fma(10.0, eps, (5.0 * x)) * x) * eps) * (x * x);
	} else if (x <= 2.5e-39) {
		tmp = (((fma(5.0, x, eps) * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5e-43)
		tmp = Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * x) * eps) * Float64(x * x));
	elseif (x <= 2.5e-39)
		tmp = Float64(Float64(Float64(Float64(fma(5.0, x, eps) * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5e-43], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-39], N[(N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.00000000000000019e-43

   1. Initial program 25.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in eps around 0

      \[\leadsto \left(\varepsilon \cdot \left(5 \cdot {x}^{2} + 10 \cdot \left(\varepsilon \cdot x\right)\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.4%

       \[\leadsto \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right) \cdot {\varepsilon}^{5}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right) + 1\right)} \cdot {\varepsilon}^{5} \]

      4. distribute-lft1-in N/A

         \[\leadsto \left(\color{blue}{\left(4 + 1\right) \cdot \frac{x}{\varepsilon}} + 1\right) \cdot {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(\color{blue}{5} \cdot \frac{x}{\varepsilon} + 1\right) \cdot {\varepsilon}^{5} \]

      6. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{x}{\varepsilon} \cdot 5} + 1\right) \cdot {\varepsilon}^{5} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right)} \cdot {\varepsilon}^{5} \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{\varepsilon}}, 5, 1\right) \cdot {\varepsilon}^{5} \]

      9. lower-pow.f64 99.8

         \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \color{blue}{{\varepsilon}^{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + {\varepsilon}^{4}\right) + {\varepsilon}^{5}} \]
   7. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(4 + 1\right) \cdot {\varepsilon}^{4}\right)} + {\varepsilon}^{5} \]

      2. metadata-eval N/A

         \[\leadsto x \cdot \left(\color{blue}{5} \cdot {\varepsilon}^{4}\right) + {\varepsilon}^{5} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot 5\right) \cdot {\varepsilon}^{4}} + {\varepsilon}^{5} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 \cdot x\right)} \cdot {\varepsilon}^{4} + {\varepsilon}^{5} \]

      5. metadata-eval N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

      6. pow-plus N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{{\varepsilon}^{4} \cdot \varepsilon} \]

      7. *-commutative N/A

         \[\leadsto \left(5 \cdot x\right) \cdot {\varepsilon}^{4} + \color{blue}{\varepsilon \cdot {\varepsilon}^{4}} \]

      8. distribute-rgt-in N/A

         \[\leadsto \color{blue}{{\varepsilon}^{4} \cdot \left(5 \cdot x + \varepsilon\right)} \]

      9. +-commutative N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)} \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      11. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\varepsilon + 5 \cdot x\right) \cdot {\varepsilon}^{4}} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{\left(5 \cdot x + \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right)} \cdot {\varepsilon}^{4} \]

      14. lower-pow.f64 99.8

          \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{4}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.7%

       \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 99.7%

       \[\leadsto \left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(-\varepsilon\right)\right) \cdot \color{blue}{\left(-\varepsilon\right)} \]

   if 2.4999999999999999e-39 < x

   1. Initial program 29.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.9%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;\left(\left(\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.5e-39)))
   (* (* (* (* x x) 5.0) eps) (* x x))
   (* (* eps eps) (* (* eps eps) eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (eps * eps) * ((eps * eps) * eps);
	}
	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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-5d-43)) .or. (.not. (x <= 2.5d-39))) then
        tmp = (((x * x) * 5.0d0) * eps) * (x * x)
    else
        tmp = (eps * eps) * ((eps * eps) * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.5e-39)) {
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	} else {
		tmp = (eps * eps) * ((eps * eps) * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -5e-43) or not (x <= 2.5e-39):
		tmp = (((x * x) * 5.0) * eps) * (x * x)
	else:
		tmp = (eps * eps) * ((eps * eps) * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.5e-39))
		tmp = Float64(Float64(Float64(Float64(x * x) * 5.0) * eps) * Float64(x * x));
	else
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -5e-43) || ~((x <= 2.5e-39)))
		tmp = (((x * x) * 5.0) * eps) * (x * x);
	else
		tmp = (eps * eps) * ((eps * eps) * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.5e-39]], $MachinePrecision]], N[(N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.4999999999999999e-39 < x

   1. Initial program 27.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(4 \cdot {x}^{4} + \left(\varepsilon \cdot \left(4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right)\right) + {x}^{4}\right)\right) \cdot \varepsilon} \]

   5. Applied rewrites 95.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(10 \cdot \left(x \cdot x\right), \varepsilon, {x}^{3} \cdot 10\right), \varepsilon, {x}^{4} \cdot 5\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(10 \cdot {\varepsilon}^{3} + x \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.6%

      \[\leadsto \mathsf{fma}\left(5 \cdot \varepsilon, x \cdot x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right) \cdot 10\right) \cdot \color{blue}{\left(x \cdot x\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{2}\right)\right) \cdot \left(x \cdot x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.6%

       \[\leadsto \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right) \]

   if -5.00000000000000019e-43 < x < 2.4999999999999999e-39

   1. Initial program 100.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   2. Add Preprocessing

   3. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5}}\right) \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   5. Applied rewrites 91.8%

      \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

   6. Taylor expanded in x around 0

      \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   9. Step-by-step derivation

   10. Applied rewrites 99.8%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   11. Taylor expanded in x around 0

       \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 99.4%

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.5 \cdot 10^{-39}\right):\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.6% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* (* eps eps) eps)))

C

double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * ((eps * eps) * eps)
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Python

def code(x, eps):
	return (eps * eps) * ((eps * eps) * eps)

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * ((eps * eps) * eps);
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
2. Add Preprocessing

3. Taylor expanded in eps around -inf

   \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)\right) \cdot {\varepsilon}^{5}} \]

5. Applied rewrites 76.5%

   \[\leadsto \color{blue}{\left(-\left(\frac{\mathsf{fma}\left(5, x, \frac{-10 \cdot \left(x \cdot x\right)}{-\varepsilon}\right)}{-\varepsilon} - 1\right)\right) \cdot {\varepsilon}^{5}} \]

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \left(5 \cdot {\varepsilon}^{4} + 10 \cdot \left({\varepsilon}^{3} \cdot x\right)\right) + \color{blue}{{\varepsilon}^{5}} \]
7. Step-by-step derivation

8. Applied rewrites 85.9%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
9. Step-by-step derivation

10. Applied rewrites 85.9%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(10 \cdot x\right) \cdot x\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

11. Taylor expanded in x around 0

    \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
12. Step-by-step derivation

13. Applied rewrites 85.3%

    \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=5"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 5.0) (pow x 5.0)))