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

Percentage Accurate: 88.4% → 97.6%

Time: 5.0s

Alternatives: 12

Speedup: 1.3×

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: 88.4% 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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -10, -\frac{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 6, \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 4\right)}{x}\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (*
    (+
     (fma
      4.0
      eps
      (-
       (/
        (fma
         (* eps eps)
         -10.0
         (- (/ (fma (* (* eps eps) 6.0) eps (* (* (* eps eps) eps) 4.0)) x)))
        x)))
     eps)
    (pow x 4.0))
   (if (<= x 9.5e-34)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = (fma(4.0, eps, -(fma((eps * eps), -10.0, -(fma(((eps * eps) * 6.0), eps, (((eps * eps) * eps) * 4.0)) / x)) / x)) + eps) * pow(x, 4.0);
	} else if (x <= 9.5e-34) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(Float64(fma(4.0, eps, Float64(-Float64(fma(Float64(eps * eps), -10.0, Float64(-Float64(fma(Float64(Float64(eps * eps) * 6.0), eps, Float64(Float64(Float64(eps * eps) * eps) * 4.0)) / x))) / x))) + eps) * (x ^ 4.0));
	elseif (x <= 9.5e-34)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(N[(N[(4.0 * eps + (-N[(N[(N[(eps * eps), $MachinePrecision] * -10.0 + (-N[(N[(N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision] * eps + N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision] + eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower--.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f64 87.5

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + \left(-1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + -1 \cdot \frac{4 \cdot {\varepsilon}^{3} + \varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]

   10. Applied rewrites 80.9%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 2: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left({x}^{4}, 5, \left(10 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.4e-38)
   (* eps (fma (pow x 4.0) 5.0 (* (* 10.0 eps) (* (* x x) x))))
   (if (<= x 9.5e-34)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.4e-38) {
		tmp = eps * fma(pow(x, 4.0), 5.0, ((10.0 * eps) * ((x * x) * x)));
	} else if (x <= 9.5e-34) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.4e-38)
		tmp = Float64(eps * fma((x ^ 4.0), 5.0, Float64(Float64(10.0 * eps) * Float64(Float64(x * x) * x))));
	elseif (x <= 9.5e-34)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.4e-38], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0 + N[(N[(10.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-pow.f64 82.3

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

   if -1.4e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 3: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left({x}^{4}, 5, \left(10 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (* eps (fma (pow x 4.0) 5.0 (* (* 10.0 eps) (* (* x x) x))))
   (if (<= x 9.5e-34)
     (* (fma 5.0 (/ x eps) 1.0) (pow eps 5.0))
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = eps * fma(pow(x, 4.0), 5.0, ((10.0 * eps) * ((x * x) * x)));
	} else if (x <= 9.5e-34) {
		tmp = fma(5.0, (x / eps), 1.0) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(eps * fma((x ^ 4.0), 5.0, Float64(Float64(10.0 * eps) * Float64(Float64(x * x) * x))));
	elseif (x <= 9.5e-34)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0 + N[(N[(10.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(5, {x}^{4}, 10 \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-pow.f64 82.3

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. pow3 N/A

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

      11. associate-*r* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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

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

   4. Applied rewrites 87.6%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 4: 97.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (* (* x x) (* x (fma (* eps x) 5.0 (* (* eps eps) 10.0))))
   (if (<= x 9.5e-34)
     (* (fma 5.0 (/ x eps) 1.0) (pow eps 5.0))
     (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = (x * x) * (x * fma((eps * x), 5.0, ((eps * eps) * 10.0)));
	} else if (x <= 9.5e-34) {
		tmp = fma(5.0, (x / eps), 1.0) * pow(eps, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0))));
	elseif (x <= 9.5e-34)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * (eps ^ 5.0));
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - \color{blue}{-4 \cdot {\varepsilon}^{2}}\right) \]

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt1-in N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   9. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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

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

   4. Applied rewrites 87.6%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 5: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right)\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (* (* x x) (* x (fma (* eps x) 5.0 (* (* eps eps) 10.0))))
   (if (<= x 9.5e-34) (pow eps 5.0) (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = (x * x) * (x * fma((eps * x), 5.0, ((eps * eps) * 10.0)));
	} else if (x <= 9.5e-34) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(Float64(x * x) * Float64(x * fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0))));
	elseif (x <= 9.5e-34)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(N[(x * x), $MachinePrecision] * N[(x * N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[Power[eps, 5.0], $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - \color{blue}{-4 \cdot {\varepsilon}^{2}}\right) \]

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt1-in N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   9. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 6: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (* (* (* x x) x) (* (fma 10.0 eps (* 5.0 x)) eps))
   (if (<= x 9.5e-34) (pow eps 5.0) (* (* 5.0 eps) (pow x 4.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = ((x * x) * x) * (fma(10.0, eps, (5.0 * x)) * eps);
	} else if (x <= 9.5e-34) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (5.0 * eps) * pow(x, 4.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(fma(10.0, eps, Float64(5.0 * x)) * eps));
	elseif (x <= 9.5e-34)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[Power[eps, 5.0], $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - \color{blue}{-4 \cdot {\varepsilon}^{2}}\right) \]

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt1-in N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. 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 82.1

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

   4. Applied rewrites 82.1%

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

Alternative 7: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-38)
   (* (* (* x x) x) (* (fma 10.0 eps (* 5.0 x)) eps))
   (if (<= x 9.5e-34) (pow eps 5.0) (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-38) {
		tmp = ((x * x) * x) * (fma(10.0, eps, (5.0 * x)) * eps);
	} else if (x <= 9.5e-34) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(fma(10.0, eps, Float64(5.0 * x)) * eps));
	elseif (x <= 9.5e-34)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-38], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - \color{blue}{-4 \cdot {\varepsilon}^{2}}\right) \]

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt1-in N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   3. 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

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

   4. Applied rewrites 87.6%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 82.1

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

   7. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-pow.f64 82.1

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

   9. Applied rewrites 82.1%

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

Alternative 8: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ t_1 := \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \varepsilon\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(t\_1 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)) (t_1 (fma 10.0 eps (* 5.0 x))))
   (if (<= x -1.35e-38)
     (* t_0 (* t_1 eps))
     (if (<= x 9.5e-34) (pow eps 5.0) (* eps (* t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double t_1 = fma(10.0, eps, (5.0 * x));
	double tmp;
	if (x <= -1.35e-38) {
		tmp = t_0 * (t_1 * eps);
	} else if (x <= 9.5e-34) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (t_1 * t_0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	t_1 = fma(10.0, eps, Float64(5.0 * x))
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = Float64(t_0 * Float64(t_1 * eps));
	elseif (x <= 9.5e-34)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(t_1 * t_0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$1 = N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-38], N[(t$95$0 * N[(t$95$1 * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-34], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.35000000000000003e-38

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - \color{blue}{-4 \cdot {\varepsilon}^{2}}\right) \]

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt1-in N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\mathsf{fma}\left(6, \varepsilon \cdot \varepsilon, x \cdot \left(5 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{\color{blue}{2}}\right) \]

      14. pow2 N/A

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

      15. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.4

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

   4. Applied rewrites 87.4%

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

   if 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 9: 97.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* (fma 10.0 eps (* 5.0 x)) (* (* x x) x)))))
   (if (<= x -1.35e-38) t_0 (if (<= x 9.5e-34) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (fma(10.0, eps, (5.0 * x)) * ((x * x) * x));
	double tmp;
	if (x <= -1.35e-38) {
		tmp = t_0;
	} else if (x <= 9.5e-34) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64(fma(10.0, eps, Float64(5.0 * x)) * Float64(Float64(x * x) * x)))
	tmp = 0.0
	if (x <= -1.35e-38)
		tmp = t_0;
	elseif (x <= 9.5e-34)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e-38], t$95$0, If[LessEqual[x, 9.5e-34], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e-38 or 9.49999999999999985e-34 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. 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

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. pow3 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 82.3

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

   10. Applied rewrites 82.3%

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

   if -1.35000000000000003e-38 < x < 9.49999999999999985e-34

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.4

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

   4. Applied rewrites 87.4%

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

Alternative 10: 87.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ {\varepsilon}^{5} \end{array} \]

FPCore

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

C

double code(double x, double eps) {
	return pow(eps, 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 = eps ** 5.0d0
end function

Java

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

Python

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

Julia

function code(x, eps)
	return eps ^ 5.0
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Power[eps, 5.0], $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0

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

   1. lower-pow.f64 87.4

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

4. Applied rewrites 87.4%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
5. Add Preprocessing

Alternative 11: 70.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 10 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 10.0 (* (* eps eps) (* (* x x) x))))

C

double code(double x, double eps) {
	return 10.0 * ((eps * eps) * ((x * x) * x));
}

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 = 10.0d0 * ((eps * eps) * ((x * x) * x))
end function

Java

public static double code(double x, double eps) {
	return 10.0 * ((eps * eps) * ((x * x) * x));
}

Python

def code(x, eps):
	return 10.0 * ((eps * eps) * ((x * x) * x))

Julia

function code(x, eps)
	return Float64(10.0 * Float64(Float64(eps * eps) * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 10.0 * ((eps * eps) * ((x * x) * x));
end

Wolfram

code[x_, eps_] := N[(10.0 * N[(N[(eps * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
3. 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

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow3 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 70.4

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

7. Applied rewrites 70.4%

   \[\leadsto 10 \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]
8. Add Preprocessing

Alternative 12: 70.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ 10 \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \left(x \cdot x\right)\right) \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 10.0 (* eps (* (* eps (* x x)) x))))

C

double code(double x, double eps) {
	return 10.0 * (eps * ((eps * (x * x)) * x));
}

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 = 10.0d0 * (eps * ((eps * (x * x)) * x))
end function

Java

public static double code(double x, double eps) {
	return 10.0 * (eps * ((eps * (x * x)) * x));
}

Python

def code(x, eps):
	return 10.0 * (eps * ((eps * (x * x)) * x))

Julia

function code(x, eps)
	return Float64(10.0 * Float64(eps * Float64(Float64(eps * Float64(x * x)) * x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 10.0 * (eps * ((eps * (x * x)) * x));
end

Wolfram

code[x_, eps_] := N[(10.0 * N[(eps * N[(N[(eps * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
3. 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

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

4. Applied rewrites 82.3%

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

5. Taylor expanded in eps around inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. pow2 N/A

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

   4. lift-*.f64 N/A

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

   5. pow3 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 70.4

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

7. Applied rewrites 70.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. pow3 N/A

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

   6. associate-*l* N/A

      \[\leadsto 10 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \color{blue}{{x}^{3}}\right)\right) \]

   7. pow3 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 70.4

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

   12. lift-*.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow3 N/A

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

   16. unpow3 N/A

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

   17. pow2 N/A

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

   18. associate-*r* N/A

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

   19. lower-*.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. pow2 N/A

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

   22. lift-*.f64 70.4

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

9. Applied rewrites 70.4%

   \[\leadsto 10 \cdot \left(\varepsilon \cdot \left(\left(\varepsilon \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{x}\right)\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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)))