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

Percentage Accurate: 89.0% → 99.0%

Time: 3.0s

Alternatives: 7

Speedup: 0.4×

Specification

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

Math

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

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: 89.0% accurate, 1.0× speedup

Math

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

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
  (if (<= t_0 -1e-323)
    (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
    (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.8813129168249309e-324

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   if -9.8813129168249309e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.0%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 82.1%

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.1%

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.1%

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

   6. Applied rewrites 82.1%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 89.0%

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

Alternative 2: 98.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + 5 \cdot \frac{x}{\varepsilon}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
  (if (<= t_0 -1e-323)
    (* (pow eps 5.0) (+ 1.0 (* 5.0 (/ x eps))))
    (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = (eps ** 5.0d0) * (1.0d0 + (5.0d0 * (x / eps)))
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = Math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)));
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = math.pow(eps, 5.0) * (1.0 + (5.0 * (x / eps)))
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + Float64(5.0 * Float64(x / eps))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = (eps ^ 5.0) * (1.0 + (5.0 * (x / eps)));
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(5.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.8813129168249309e-324

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   if -9.8813129168249309e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.0%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 82.1%

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.1%

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.1%

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

   6. Applied rewrites 82.1%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.9%

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

   9. Applied rewrites 87.9%

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

Alternative 3: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\varepsilon}^{4} \cdot \left(\varepsilon + 5 \cdot x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
  (if (<= t_0 -1e-323)
    (* (pow eps 4.0) (+ eps (* 5.0 x)))
    (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = pow(eps, 4.0) * (eps + (5.0 * x));
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = (eps ** 4.0d0) * (eps + (5.0d0 * x))
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = Math.pow(eps, 4.0) * (eps + (5.0 * x));
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = math.pow(eps, 4.0) * (eps + (5.0 * x))
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = Float64((eps ^ 4.0) * Float64(eps + Float64(5.0 * x)));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = (eps ^ 4.0) * (eps + (5.0 * x));
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], N[(N[Power[eps, 4.0], $MachinePrecision] * N[(eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.8813129168249309e-324

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.9%

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

   9. Applied rewrites 87.9%

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

   10. Taylor expanded in eps around 0

       \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\left(\varepsilon + 5 \cdot x\right)} \]
   11. 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. lower-+.f64 N/A

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

       4. lower-*.f64 88.0%

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

   12. Applied rewrites 88.0%

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

   if -9.8813129168249309e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.0%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 82.1%

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.1%

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.1%

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

   6. Applied rewrites 82.1%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.9%

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

   9. Applied rewrites 87.9%

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

Alternative 4: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
  (if (<= t_0 -1e-323)
    (pow eps 5.0)
    (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) eps) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * eps
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.8813129168249309e-324 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.9%

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

   9. Applied rewrites 87.9%

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

   if -9.8813129168249309e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.0%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 82.1%

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.1%

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.1%

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

   6. Applied rewrites 82.1%

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

Alternative 5: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-323}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \]

FPCore

(FPCore (x eps)
  :precision binary64
  (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
  (if (<= t_0 -1e-323)
    (pow eps 5.0)
    (if (<= t_0 0.0)
      (* (* (* (* x x) (* x x)) 5.0) eps)
      (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-323)) then
        tmp = eps ** 5.0d0
    else if (t_0 <= 0.0d0) then
        tmp = (((x * x) * (x * x)) * 5.0d0) * eps
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-323) {
		tmp = Math.pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-323:
		tmp = math.pow(eps, 5.0)
	elif t_0 <= 0.0:
		tmp = (((x * x) * (x * x)) * 5.0) * eps
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-323)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps);
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-323)
		tmp = eps ^ 5.0;
	elseif (t_0 <= 0.0)
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-323], N[Power[eps, 5.0], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -9.8813129168249309e-324 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. lower-unsound-*.f64 N/A

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

      6. lower-unsound-+.f64 N/A

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

      7. lower-unsound-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 11.8%

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

   3. Applied rewrites 11.8%

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

   4. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 88.1%

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.9%

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

   9. Applied rewrites 87.9%

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

   if -9.8813129168249309e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 89.0%

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

   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. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 82.1%

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

   4. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-rgt1-in N/A

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

      5. metadata-eval N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. distribute-lft1-in N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lower-*.f64 82.1%

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

      13. lift-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 82.1%

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

   6. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 82.1%

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

   8. Applied rewrites 82.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. exp-to-pow N/A

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

      5. lift-log.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-exp.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 40.7%

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

   10. Applied rewrites 82.1%

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

Alternative 6: 82.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (((x * x) * (x * x)) * 5.0) * eps;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double eps) {
	return (((x * x) * (x * x)) * 5.0) * eps;
}

Python

def code(x, eps):
	return (((x * x) * (x * x)) * 5.0) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

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. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 82.1%

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

4. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft1-in N/A

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

   10. lift-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 82.1%

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-lft1-in N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 82.1%

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

6. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 82.1%

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

8. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. exp-to-pow N/A

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

   5. lift-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 40.7%

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

10. Applied rewrites 82.1%

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

Alternative 7: 82.1% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return ((x * x) * (x * x)) * (5.0 * eps);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x, double eps) {
	return ((x * x) * (x * x)) * (5.0 * eps);
}

Python

def code(x, eps):
	return ((x * x) * (x * x)) * (5.0 * eps)

Julia

function code(x, eps)
	return Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.0%

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

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. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 82.1%

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

4. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt1-in N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft1-in N/A

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

   10. lift-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-*.f64 82.1%

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

   13. lift-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. distribute-lft1-in N/A

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

   16. metadata-eval N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 82.1%

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

6. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 82.1%

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

8. Applied rewrites 82.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. exp-to-pow N/A

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

   5. lift-log.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 40.7%

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

10. Applied rewrites 82.1%

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

Reproduce

herbie shell --seed 2025258 
(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)))