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

Percentage Accurate: 88.6% → 97.7%

Time: 4.3s

Alternatives: 11

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: 88.6% 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: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-57}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x, {\varepsilon}^{3} \cdot 5\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-57)
   (*
    (*
     (fma
      (fma (fma 10.0 eps (* 5.0 x)) x (* (* eps eps) 10.0))
      x
      (* (pow eps 3.0) 5.0))
     x)
    eps)
   (if (<= x 1.22e-59)
     (pow eps 5.0)
     (* (* (fma (/ eps x) 10.0 5.0) eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-57) {
		tmp = (fma(fma(fma(10.0, eps, (5.0 * x)), x, ((eps * eps) * 10.0)), x, (pow(eps, 3.0) * 5.0)) * x) * eps;
	} else if (x <= 1.22e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-57)
		tmp = Float64(Float64(fma(fma(fma(10.0, eps, Float64(5.0 * x)), x, Float64(Float64(eps * eps) * 10.0)), x, Float64((eps ^ 3.0) * 5.0)) * x) * eps);
	elseif (x <= 1.22e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-57], N[(N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[eps, 3.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 1.22e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.19999999999999971e-57

   1. Initial program 46.8%

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

   3. Taylor expanded in eps around 0

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 4, \mathsf{fma}\left(\mathsf{fma}\left({x}^{3}, 4, \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 10, \left(5 \cdot x\right) \cdot \varepsilon\right), \varepsilon, \left(\left(x \cdot x\right) \cdot 6\right) \cdot x\right)\right), \varepsilon, {x}^{4}\right)\right) \cdot \varepsilon} \]

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot x\right) \cdot \varepsilon \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot {\varepsilon}^{3} + x \cdot \left(10 \cdot {\varepsilon}^{2} + x \cdot \left(5 \cdot x + 10 \cdot \varepsilon\right)\right)\right) \cdot x\right) \cdot \varepsilon \]

   8. Applied rewrites 90.2%

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

   if -5.19999999999999971e-57 < x < 1.22e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 99.8

         \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.22e-59 < x

   1. Initial program 47.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 5\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 5\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.3

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

   8. Applied rewrites 90.3%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.2e-57)
   (*
    (/ (- (fma (* 5.0 x) eps (* (* eps eps) 6.0)) (* (* eps eps) -4.0)) x)
    (* (* x x) (* x x)))
   (if (<= x 1.22e-59)
     (pow eps 5.0)
     (* (* (fma (/ eps x) 10.0 5.0) eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.2e-57) {
		tmp = ((fma((5.0 * x), eps, ((eps * eps) * 6.0)) - ((eps * eps) * -4.0)) / x) * ((x * x) * (x * x));
	} else if (x <= 1.22e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.2e-57)
		tmp = Float64(Float64(Float64(fma(Float64(5.0 * x), eps, Float64(Float64(eps * eps) * 6.0)) - Float64(Float64(eps * eps) * -4.0)) / x) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 1.22e-59)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.2e-57], N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * eps + N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e-59], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.19999999999999971e-57

   1. Initial program 46.8%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.2

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.2%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 89.2

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 89.2%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

   13. Applied rewrites 89.1%

       \[\leadsto \frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

   if -5.19999999999999971e-57 < x < 1.22e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 99.8

         \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]

   if 1.22e-59 < x

   1. Initial program 47.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 5\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 5\right) \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.3

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

   8. Applied rewrites 90.3%

      \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (- (fma (* 5.0 x) eps (* (* eps eps) 6.0)) (* (* eps eps) -4.0))
           x)
          (* (* x x) (* x x)))))
   (if (<= x -5.2e-57) t_0 (if (<= x 1.22e-59) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma((5.0 * x), eps, ((eps * eps) * 6.0)) - ((eps * eps) * -4.0)) / x) * ((x * x) * (x * x));
	double tmp;
	if (x <= -5.2e-57) {
		tmp = t_0;
	} else if (x <= 1.22e-59) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(Float64(5.0 * x), eps, Float64(Float64(eps * eps) * 6.0)) - Float64(Float64(eps * eps) * -4.0)) / x) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (x <= -5.2e-57)
		tmp = t_0;
	elseif (x <= 1.22e-59)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * eps + N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e-57], t$95$0, If[LessEqual[x, 1.22e-59], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999971e-57 or 1.22e-59 < x

   1. Initial program 46.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 89.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

   13. Applied rewrites 89.6%

       \[\leadsto \frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

   if -5.19999999999999971e-57 < x < 1.22e-59

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   4. Step-by-step derivation

      1. lower-pow.f64 99.8

         \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (- (fma (* 5.0 x) eps (* (* eps eps) 6.0)) (* (* eps eps) -4.0))
           x)
          (* (* x x) (* x x)))))
   (if (<= x -8.5e-57)
     t_0
     (if (<= x 1.22e-59)
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma((5.0 * x), eps, ((eps * eps) * 6.0)) - ((eps * eps) * -4.0)) / x) * ((x * x) * (x * x));
	double tmp;
	if (x <= -8.5e-57) {
		tmp = t_0;
	} else if (x <= 1.22e-59) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(fma(Float64(5.0 * x), eps, Float64(Float64(eps * eps) * 6.0)) - Float64(Float64(eps * eps) * -4.0)) / x) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (x <= -8.5e-57)
		tmp = t_0;
	elseif (x <= 1.22e-59)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(5.0 * x), $MachinePrecision] * eps + N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(eps * eps), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-57], t$95$0, If[LessEqual[x, 1.22e-59], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999955e-57 or 1.22e-59 < x

   1. Initial program 46.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 89.7%

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

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}}{x} \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

   13. Applied rewrites 89.6%

       \[\leadsto \frac{\mathsf{fma}\left(5 \cdot x, \varepsilon, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4}{x} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

   if -8.49999999999999955e-57 < x < 1.22e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

       7. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       12. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       14. pow2 N/A

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

       15. lift-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

Alternative 5: 97.5% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)}{x} \cdot t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -8.5e-57)
     (* (/ (fma (* eps x) 5.0 (* (* eps eps) 10.0)) x) t_0)
     (if (<= x 1.22e-59)
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
       (* (* (fma (/ eps x) 10.0 5.0) eps) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -8.5e-57) {
		tmp = (fma((eps * x), 5.0, ((eps * eps) * 10.0)) / x) * t_0;
	} else if (x <= 1.22e-59) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -8.5e-57)
		tmp = Float64(Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) / x) * t_0);
	elseif (x <= 1.22e-59)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-57], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.22e-59], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999955e-57

   1. Initial program 46.7%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.2

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.2%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(10 \cdot \frac{1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{10 \cdot 1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\frac{10}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{10}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{10}{x} + \frac{5 \cdot 1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{10}{x} + \frac{5}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{10}{x} + \frac{5}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 70.7

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \frac{5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {\varepsilon}^{2}}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \frac{\left(\varepsilon \cdot x\right) \cdot 5 + 10 \cdot {\varepsilon}^{2}}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot x, 5, 10 \cdot {\varepsilon}^{2}\right)}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot x, 5, 10 \cdot {\varepsilon}^{2}\right)}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2} \cdot 10\right)}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\varepsilon \cdot x, 5, {\varepsilon}^{2} \cdot 10\right)}{x} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       7. pow2 N/A

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

       8. lift-*.f64 89.1

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

   13. Applied rewrites 89.1%

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

   if -8.49999999999999955e-57 < x < 1.22e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

       7. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       12. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       14. pow2 N/A

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

       15. lift-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

   if 1.22e-59 < x

   1. Initial program 47.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 90.1

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 90.1%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 90.1

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 90.1%

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

   11. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 90.1

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

   13. Applied rewrites 90.1%

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

Alternative 6: 97.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\varepsilon \cdot \varepsilon}{x}, 10, 5 \cdot \varepsilon\right) \cdot t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -8.5e-57)
     (* (fma (/ (* eps eps) x) 10.0 (* 5.0 eps)) t_0)
     (if (<= x 1.22e-59)
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
       (* (* (fma (/ eps x) 10.0 5.0) eps) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -8.5e-57) {
		tmp = fma(((eps * eps) / x), 10.0, (5.0 * eps)) * t_0;
	} else if (x <= 1.22e-59) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -8.5e-57)
		tmp = Float64(fma(Float64(Float64(eps * eps) / x), 10.0, Float64(5.0 * eps)) * t_0);
	elseif (x <= 1.22e-59)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-57], N[(N[(N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision] * 10.0 + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x, 1.22e-59], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.49999999999999955e-57

   1. Initial program 46.7%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.2

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.2%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around inf

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \frac{1}{\varepsilon} + 10 \cdot \frac{1}{x}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot \color{blue}{x}\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \left(\left(10 \cdot \frac{1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{10 \cdot 1}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(\frac{10}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\left(\frac{10}{x} + 5 \cdot \frac{1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \left(\left(\frac{10}{x} + \frac{5 \cdot 1}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \left(\left(\frac{10}{x} + \frac{5}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(\frac{10}{x} + \frac{5}{\varepsilon}\right) \cdot {\varepsilon}^{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      11. pow2 N/A

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

      12. lift-*.f64 70.7

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

   10. Applied rewrites 70.7%

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

   11. Taylor expanded in x around inf

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

       1. +-commutative N/A

          \[\leadsto \left(10 \cdot \frac{{\varepsilon}^{2}}{x} + 5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{{\varepsilon}^{2}}{x} \cdot 10 + 5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\varepsilon}^{2}}{x}, 10, 5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{{\varepsilon}^{2}}{x}, 10, 5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-*.f64 89.2

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

   13. Applied rewrites 89.2%

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

   if -8.49999999999999955e-57 < x < 1.22e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

       7. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       12. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       14. pow2 N/A

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

       15. lift-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

   if 1.22e-59 < x

   1. Initial program 47.0%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 90.1

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 90.1%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 90.1

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 90.1%

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

   11. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 90.1

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

   13. Applied rewrites 90.1%

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

Alternative 7: 97.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma (/ eps x) 10.0 5.0) eps) (* (* x x) (* x x)))))
   (if (<= x -8.5e-57)
     t_0
     (if (<= x 1.22e-59)
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma((eps / x), 10.0, 5.0) * eps) * ((x * x) * (x * x));
	double tmp;
	if (x <= -8.5e-57) {
		tmp = t_0;
	} else if (x <= 1.22e-59) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (x <= -8.5e-57)
		tmp = t_0;
	elseif (x <= 1.22e-59)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8.5e-57], t$95$0, If[LessEqual[x, 1.22e-59], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999955e-57 or 1.22e-59 < x

   1. Initial program 46.9%

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

   3. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\color{blue}{4}} \]

      2. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      8. lower-*.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   7. Applied rewrites 89.7%

      \[\leadsto \left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(4 + 10 \cdot \frac{\varepsilon}{x}\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \frac{\varepsilon}{x} + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \left(\left(\frac{\varepsilon}{x} \cdot 10 + 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

      6. lift-/.f64 89.7

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 4\right) \cdot \varepsilon + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   10. Applied rewrites 89.7%

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

   11. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lift-/.f64 89.7

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

   13. Applied rewrites 89.7%

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

   if -8.49999999999999955e-57 < x < 1.22e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

       7. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       12. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       14. pow2 N/A

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

       15. lift-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

Alternative 8: 97.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 (* (* x x) (* x x))) eps)))
   (if (<= x -8.5e-57)
     t_0
     (if (<= x 1.35e-59)
       (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (5.0 * ((x * x) * (x * x))) * eps;
	double tmp;
	if (x <= -8.5e-57) {
		tmp = t_0;
	} else if (x <= 1.35e-59) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))) * eps)
	tmp = 0.0
	if (x <= -8.5e-57)
		tmp = t_0;
	elseif (x <= 1.35e-59)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -8.5e-57], t$95$0, If[LessEqual[x, 1.35e-59], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999955e-57 or 1.3499999999999999e-59 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      3. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      4. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      5. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if -8.49999999999999955e-57 < x < 1.3499999999999999e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

       7. pow2 N/A

          \[\leadsto \left(5 \cdot x + \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \]

       8. associate-*r* N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       9. lower-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{\color{blue}{2}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(\left(5 \cdot x + \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       11. lift-fma.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} \]

       12. pow2 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       13. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot {\varepsilon}^{2} \]

       14. pow2 N/A

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

       15. lift-*.f64 99.7

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

   12. Applied rewrites 99.7%

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

Alternative 9: 97.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 (* (* x x) (* x x))) eps)))
   (if (<= x -5.2e-57)
     t_0
     (if (<= x 1.35e-59) (* eps (* (* eps eps) (* eps eps))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (5.0 * ((x * x) * (x * x))) * eps;
	double tmp;
	if (x <= -5.2e-57) {
		tmp = t_0;
	} else if (x <= 1.35e-59) {
		tmp = eps * ((eps * eps) * (eps * eps));
	} 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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (5.0d0 * ((x * x) * (x * x))) * eps
    if (x <= (-5.2d-57)) then
        tmp = t_0
    else if (x <= 1.35d-59) then
        tmp = eps * ((eps * eps) * (eps * eps))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (5.0 * ((x * x) * (x * x))) * eps;
	double tmp;
	if (x <= -5.2e-57) {
		tmp = t_0;
	} else if (x <= 1.35e-59) {
		tmp = eps * ((eps * eps) * (eps * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (5.0 * ((x * x) * (x * x))) * eps
	tmp = 0
	if x <= -5.2e-57:
		tmp = t_0
	elif x <= 1.35e-59:
		tmp = eps * ((eps * eps) * (eps * eps))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))) * eps)
	tmp = 0.0
	if (x <= -5.2e-57)
		tmp = t_0;
	elseif (x <= 1.35e-59)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(eps * eps)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (5.0 * ((x * x) * (x * x))) * eps;
	tmp = 0.0;
	if (x <= -5.2e-57)
		tmp = t_0;
	elseif (x <= 1.35e-59)
		tmp = eps * ((eps * eps) * (eps * eps));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -5.2e-57], t$95$0, If[LessEqual[x, 1.35e-59], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999971e-57 or 1.3499999999999999e-59 < x

   1. Initial program 46.9%

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

   3. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 88.8

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

   5. Applied rewrites 88.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      3. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      4. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      5. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if -5.19999999999999971e-57 < x < 1.3499999999999999e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot \varepsilon\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 97.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{-57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-59}:\\ \;\;\;\;\varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 eps) (* (* x x) (* x x)))))
   (if (<= x -5.2e-57)
     t_0
     (if (<= x 1.35e-59) (* eps (* (* eps eps) (* eps eps))) t_0))))

C

double code(double x, double eps) {
	double t_0 = (5.0 * eps) * ((x * x) * (x * x));
	double tmp;
	if (x <= -5.2e-57) {
		tmp = t_0;
	} else if (x <= 1.35e-59) {
		tmp = eps * ((eps * eps) * (eps * eps));
	} 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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (5.0d0 * eps) * ((x * x) * (x * x))
    if (x <= (-5.2d-57)) then
        tmp = t_0
    else if (x <= 1.35d-59) then
        tmp = eps * ((eps * eps) * (eps * eps))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (5.0 * eps) * ((x * x) * (x * x));
	double tmp;
	if (x <= -5.2e-57) {
		tmp = t_0;
	} else if (x <= 1.35e-59) {
		tmp = eps * ((eps * eps) * (eps * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (5.0 * eps) * ((x * x) * (x * x))
	tmp = 0
	if x <= -5.2e-57:
		tmp = t_0
	elif x <= 1.35e-59:
		tmp = eps * ((eps * eps) * (eps * eps))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(5.0 * eps) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (x <= -5.2e-57)
		tmp = t_0;
	elseif (x <= 1.35e-59)
		tmp = Float64(eps * Float64(Float64(eps * eps) * Float64(eps * eps)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (5.0 * eps) * ((x * x) * (x * x));
	tmp = 0.0;
	if (x <= -5.2e-57)
		tmp = t_0;
	elseif (x <= 1.35e-59)
		tmp = eps * ((eps * eps) * (eps * eps));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.2e-57], t$95$0, If[LessEqual[x, 1.35e-59], N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999971e-57 or 1.3499999999999999e-59 < x

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 88.9

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

   5. Applied rewrites 88.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{\left(2 + \color{blue}{2}\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      7. unpow2 N/A

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

      8. lower-*.f64 88.7

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

   7. Applied rewrites 88.7%

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

   if -5.19999999999999971e-57 < x < 1.3499999999999999e-59

   1. Initial program 99.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
   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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-pow.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

      3. pow-prod-up N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

      7. pow2 N/A

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

      8. lift-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 99.6%

       \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot \varepsilon\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.2% accurate, 10.0× speedup

Math

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

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(eps * Float64(Float64(eps * eps) * Float64(eps * eps)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]
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

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-pow.f64 87.5

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

8. Applied rewrites 87.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{\left(2 + 2\right)} \]

   3. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

   5. pow2 N/A

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \]

   7. pow2 N/A

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

   8. lift-*.f64 87.4

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

10. Applied rewrites 87.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 87.2%

    \[\leadsto \varepsilon \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\color{blue}{\varepsilon} \cdot \varepsilon\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(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)))