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

Percentage Accurate: 88.4% → 99.3%

Time: 5.4s

Alternatives: 16

Speedup: 1.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{4} \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right), {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(4, {x}^{3}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(2, {x}^{2}, 8 \cdot {x}^{2}\right), x \cdot \mathsf{fma}\left(2, {x}^{2}, 4 \cdot {x}^{2}\right)\right)\right), {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + x\right) \cdot \left(\left(\varepsilon + x\right) \cdot \left(\left(\mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot \varepsilon\right)\right) + \varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right)\right) - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-316)
     (fma x (* (pow eps 4.0) (+ 5.0 (* 10.0 (/ x eps)))) (pow eps 5.0))
     (if (<= t_0 0.0)
       (*
        eps
        (fma
         4.0
         (pow x 4.0)
         (fma
          eps
          (fma
           4.0
           (pow x 3.0)
           (fma
            eps
            (fma 2.0 (pow x 2.0) (* 8.0 (pow x 2.0)))
            (* x (fma 2.0 (pow x 2.0) (* 4.0 (pow x 2.0))))))
          (pow x 4.0))))
       (-
        (*
         (+ eps x)
         (*
          (+ eps x)
          (* (+ (fma x x (* 2.0 (* x eps))) (* eps eps)) (+ eps x))))
        (pow x 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 <= -2e-316) {
		tmp = fma(x, (pow(eps, 4.0) * (5.0 + (10.0 * (x / eps)))), pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = eps * fma(4.0, pow(x, 4.0), fma(eps, fma(4.0, pow(x, 3.0), fma(eps, fma(2.0, pow(x, 2.0), (8.0 * pow(x, 2.0))), (x * fma(2.0, pow(x, 2.0), (4.0 * pow(x, 2.0)))))), pow(x, 4.0)));
	} else {
		tmp = ((eps + x) * ((eps + x) * ((fma(x, x, (2.0 * (x * eps))) + (eps * eps)) * (eps + x)))) - pow(x, 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 <= -2e-316)
		tmp = fma(x, Float64((eps ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(x / eps)))), (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * fma(4.0, (x ^ 4.0), fma(eps, fma(4.0, (x ^ 3.0), fma(eps, fma(2.0, (x ^ 2.0), Float64(8.0 * (x ^ 2.0))), Float64(x * fma(2.0, (x ^ 2.0), Float64(4.0 * (x ^ 2.0)))))), (x ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(eps + x) * Float64(Float64(eps + x) * Float64(Float64(fma(x, x, Float64(2.0 * Float64(x * eps))) + Float64(eps * eps)) * Float64(eps + x)))) - (x ^ 5.0));
	end
	return 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, -2e-316], N[(x * N[(N[Power[eps, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(eps * N[(4.0 * N[Power[x, 4.0], $MachinePrecision] + N[(eps * N[(4.0 * N[Power[x, 3.0], $MachinePrecision] + N[(eps * N[(2.0 * N[Power[x, 2.0], $MachinePrecision] + N[(8.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(2.0 * N[Power[x, 2.0], $MachinePrecision] + N[(4.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps + x), $MachinePrecision] * N[(N[(eps + x), $MachinePrecision] * N[(N[(N[(x * x + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $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))) < -2.000000017e-316

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)}, {\varepsilon}^{5}\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{4}, \mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(2, {\varepsilon}^{2}, 4 \cdot {\varepsilon}^{2}\right), x \cdot \mathsf{fma}\left(2, {\varepsilon}^{2}, 8 \cdot {\varepsilon}^{2}\right)\right)\right), {\varepsilon}^{4}\right)\right), {\varepsilon}^{5}\right)} \]

   5. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2.000000017e-316 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, \mathsf{fma}\left(\varepsilon, 4 \cdot {x}^{3} + \left(\varepsilon \cdot \left(2 \cdot {x}^{2} + 8 \cdot {x}^{2}\right) + x \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)\right), {x}^{4}\right)\right) \]

   4. Applied rewrites 83.5%

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

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

   1. Initial program 88.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

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

      4. lower-special-*.f64 N/A

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

      5. lower-special-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-special-pow.f64 85.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.8

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

   3. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 86.1

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-*.f64 85.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-square-pow N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-316}:\\ \;\;\;\;\mathsf{fma}\left(x, {\varepsilon}^{4} \cdot \left(5 + 10 \cdot \frac{x}{\varepsilon}\right), {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(4, \varepsilon, \frac{10 \cdot \left(\varepsilon \cdot \varepsilon\right)}{x}\right), {x}^{4}, {x}^{4} \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + x\right) \cdot \left(\left(\varepsilon + x\right) \cdot \left(\left(\mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot \varepsilon\right)\right) + \varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right)\right) - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-316)
     (fma x (* (pow eps 4.0) (+ 5.0 (* 10.0 (/ x eps)))) (pow eps 5.0))
     (if (<= t_0 0.0)
       (fma
        (fma 4.0 eps (/ (* 10.0 (* eps eps)) x))
        (pow x 4.0)
        (* (pow x 4.0) eps))
       (-
        (*
         (+ eps x)
         (*
          (+ eps x)
          (* (+ (fma x x (* 2.0 (* x eps))) (* eps eps)) (+ eps x))))
        (pow x 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 <= -2e-316) {
		tmp = fma(x, (pow(eps, 4.0) * (5.0 + (10.0 * (x / eps)))), pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = fma(fma(4.0, eps, ((10.0 * (eps * eps)) / x)), pow(x, 4.0), (pow(x, 4.0) * eps));
	} else {
		tmp = ((eps + x) * ((eps + x) * ((fma(x, x, (2.0 * (x * eps))) + (eps * eps)) * (eps + x)))) - pow(x, 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 <= -2e-316)
		tmp = fma(x, Float64((eps ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(x / eps)))), (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = fma(fma(4.0, eps, Float64(Float64(10.0 * Float64(eps * eps)) / x)), (x ^ 4.0), Float64((x ^ 4.0) * eps));
	else
		tmp = Float64(Float64(Float64(eps + x) * Float64(Float64(eps + x) * Float64(Float64(fma(x, x, Float64(2.0 * Float64(x * eps))) + Float64(eps * eps)) * Float64(eps + x)))) - (x ^ 5.0));
	end
	return 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, -2e-316], N[(x * N[(N[Power[eps, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(4.0 * eps + N[(N[(10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps + x), $MachinePrecision] * N[(N[(eps + x), $MachinePrecision] * N[(N[(N[(x * x + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $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))) < -2.000000017e-316

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)}, {\varepsilon}^{5}\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{4}, \mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(2, {\varepsilon}^{2}, 4 \cdot {\varepsilon}^{2}\right), x \cdot \mathsf{fma}\left(2, {\varepsilon}^{2}, 8 \cdot {\varepsilon}^{2}\right)\right)\right), {\varepsilon}^{4}\right)\right), {\varepsilon}^{5}\right)} \]

   5. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -2.000000017e-316 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right), \color{blue}{{x}^{4}}, \varepsilon \cdot {x}^{4}\right) \]

   6. Applied rewrites 83.4%

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

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

   1. Initial program 88.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

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

      4. lower-special-*.f64 N/A

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

      5. lower-special-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-special-pow.f64 85.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.8

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

   3. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 86.1

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-*.f64 85.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-square-pow N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

Alternative 3: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + x\right) \cdot \left(\left(\varepsilon + x\right) \cdot \left(\left(\mathsf{fma}\left(x, x, 2 \cdot \left(x \cdot \varepsilon\right)\right) + \varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon + x\right)\right)\right) - {x}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-319)
     t_0
     (if (<= t_0 0.0)
       (* eps (* (pow x 4.0) 5.0))
       (-
        (*
         (+ eps x)
         (*
          (+ eps x)
          (* (+ (fma x x (* 2.0 (* x eps))) (* eps eps)) (+ eps x))))
        (pow x 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 <= -2e-319) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (pow(x, 4.0) * 5.0);
	} else {
		tmp = ((eps + x) * ((eps + x) * ((fma(x, x, (2.0 * (x * eps))) + (eps * eps)) * (eps + x)))) - pow(x, 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 <= -2e-319)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	else
		tmp = Float64(Float64(Float64(eps + x) * Float64(Float64(eps + x) * Float64(Float64(fma(x, x, Float64(2.0 * Float64(x * eps))) + Float64(eps * eps)) * Float64(eps + x)))) - (x ^ 5.0));
	end
	return 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, -2e-319], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps + x), $MachinePrecision] * N[(N[(eps + x), $MachinePrecision] * N[(N[(N[(x * x + N[(2.0 * N[(x * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $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))) < -1.99998e-319

   1. Initial program 88.4%

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

   if -1.99998e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

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

   1. Initial program 88.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto {\left(x + \varepsilon\right)}^{\color{blue}{\left(3 + 2\right)}} - {x}^{5} \]

      3. pow-add N/A

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

      4. lower-special-*.f64 N/A

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

      5. lower-special-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-special-pow.f64 85.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.8

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

   3. Applied rewrites 85.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 86.1

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

      8. lift-pow.f64 N/A

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

      9. unpow3 N/A

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

      10. unpow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. lower-*.f64 85.6

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. sum-square-pow N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

Alternative 4: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-319}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -2e-319)
     t_0
     (if (<= t_0 0.0) (* eps (* (pow x 4.0) 5.0)) 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 <= -2e-319) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (pow(x, 4.0) * 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 <= (-2d-319)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = eps * ((x ** 4.0d0) * 5.0d0)
    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 <= -2e-319) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = eps * (Math.pow(x, 4.0) * 5.0);
	} 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 <= -2e-319:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = eps * (math.pow(x, 4.0) * 5.0)
	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 <= -2e-319)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	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 <= -2e-319)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = eps * ((x ^ 4.0) * 5.0);
	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, -2e-319], t$95$0, If[LessEqual[t$95$0, 0.0], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

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))) < -1.99998e-319 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.4%

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

   if -1.99998e-319 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

Alternative 5: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon, \varepsilon, \left(\left(\left(\mathsf{fma}\left(10, \frac{x}{\varepsilon}, 5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.1e-57)
       (fma
        (* (* (* eps eps) eps) eps)
        eps
        (* (* (* (* (fma 10.0 (/ x eps) 5.0) (* eps eps)) eps) eps) x))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.1e-57) {
		tmp = fma((((eps * eps) * eps) * eps), eps, ((((fma(10.0, (x / eps), 5.0) * (eps * eps)) * eps) * eps) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.1e-57)
		tmp = fma(Float64(Float64(Float64(eps * eps) * eps) * eps), eps, Float64(Float64(Float64(Float64(fma(10.0, Float64(x / eps), 5.0) * Float64(eps * eps)) * eps) * eps) * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.1e-57], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps + N[(N[(N[(N[(N[(10.0 * N[(x / eps), $MachinePrecision] + 5.0), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.09999999999999999e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -6.20000000000000049e-64 < x < 1.09999999999999999e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)\right) + {\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{4 \cdot {\varepsilon}^{4} + \left(x \cdot \left(4 \cdot {\varepsilon}^{3} + \left(\varepsilon \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(2 \cdot {\varepsilon}^{2} + 8 \cdot {\varepsilon}^{2}\right)\right)\right) + {\varepsilon}^{4}\right)}, {\varepsilon}^{5}\right) \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{4}, \mathsf{fma}\left(x, \mathsf{fma}\left(4, {\varepsilon}^{3}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(2, {\varepsilon}^{2}, 4 \cdot {\varepsilon}^{2}\right), x \cdot \mathsf{fma}\left(2, {\varepsilon}^{2}, 8 \cdot {\varepsilon}^{2}\right)\right)\right), {\varepsilon}^{4}\right)\right), {\varepsilon}^{5}\right)} \]

   5. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. exp-to-pow N/A

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

      4. lift-log.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-exp.f64 N/A

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

      7. +-commutative N/A

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

   9. Applied rewrites 87.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. pow2 N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lift-fma.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-fma.f64 N/A

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

       14. pow2 N/A

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

       15. lift-*.f64 87.8

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

   11. Applied rewrites 87.8%

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

Alternative 6: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5} \cdot \left(1 + \mathsf{fma}\left(4, \frac{x}{\varepsilon}, \frac{x}{\varepsilon}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.1e-57)
       (* (pow eps 5.0) (+ 1.0 (fma 4.0 (/ x eps) (/ x eps))))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.1e-57) {
		tmp = pow(eps, 5.0) * (1.0 + fma(4.0, (x / eps), (x / eps)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.1e-57)
		tmp = Float64((eps ^ 5.0) * Float64(1.0 + fma(4.0, Float64(x / eps), Float64(x / eps))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.1e-57], N[(N[Power[eps, 5.0], $MachinePrecision] * N[(1.0 + N[(4.0 * N[(x / eps), $MachinePrecision] + N[(x / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.09999999999999999e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -6.20000000000000049e-64 < x < 1.09999999999999999e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \mathsf{fma}\left(4, \frac{x}{\varepsilon}, \frac{x}{\varepsilon}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x}^{4} \cdot \left(\varepsilon \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (pow x 4.0) (* eps (+ 5.0 (* 10.0 (/ eps x)))))))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.1e-57)
       (* (fma (/ x eps) 5.0 1.0) (* (* eps eps) (* (* eps eps) eps)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow(x, 4.0) * (eps * (5.0 + (10.0 * (eps / x))));
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.1e-57) {
		tmp = fma((x / eps), 5.0, 1.0) * ((eps * eps) * ((eps * eps) * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((x ^ 4.0) * Float64(eps * Float64(5.0 + Float64(10.0 * Float64(eps / x)))))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.1e-57)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[x, 4.0], $MachinePrecision] * N[(eps * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.1e-57], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.09999999999999999e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -6.20000000000000049e-64 < x < 1.09999999999999999e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7

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

      11. lift-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{\left(2 + \color{blue}{3}\right)} \]

      13. pow-add N/A

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

      14. lower-special-pow.f64 N/A

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

      15. lower-special-pow.f64 N/A

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

      16. lower-special-*.f64 87.6

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      19. lower-*.f64 87.6

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{\color{blue}{3}}\right) \]

      21. unpow3 N/A

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

      22. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      23. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      24. lower-*.f64 87.6

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

      25. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      26. unpow2 N/A

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

      27. lower-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

Alternative 8: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\frac{1}{{x}^{-4}} \cdot 5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-64)
   (* eps (* (pow x 4.0) 5.0))
   (if (<= x 1.12e-57)
     (* (fma (/ x eps) 5.0 1.0) (* (* eps eps) (* (* eps eps) eps)))
     (* eps (* (/ 1.0 (pow x -4.0)) 5.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-64) {
		tmp = eps * (pow(x, 4.0) * 5.0);
	} else if (x <= 1.12e-57) {
		tmp = fma((x / eps), 5.0, 1.0) * ((eps * eps) * ((eps * eps) * eps));
	} else {
		tmp = eps * ((1.0 / pow(x, -4.0)) * 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = Float64(eps * Float64((x ^ 4.0) * 5.0));
	elseif (x <= 1.12e-57)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps)));
	else
		tmp = Float64(eps * Float64(Float64(1.0 / (x ^ -4.0)) * 5.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-64], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-57], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[(1.0 / N[Power[x, -4.0], $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000049e-64

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7

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

      11. lift-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{\left(2 + \color{blue}{3}\right)} \]

      13. pow-add N/A

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

      14. lower-special-pow.f64 N/A

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

      15. lower-special-pow.f64 N/A

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

      16. lower-special-*.f64 87.6

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      19. lower-*.f64 87.6

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{\color{blue}{3}}\right) \]

      21. unpow3 N/A

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

      22. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      23. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      24. lower-*.f64 87.6

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

      25. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      26. unpow2 N/A

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

      27. lower-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

   if 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. pow-neg N/A

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

       5. lower-special-/.f64 N/A

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

       6. lower-special-pow.f64 N/A

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

       7. metadata-eval 83.2

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

   11. Applied rewrites 83.2%

       \[\leadsto \varepsilon \cdot \left(\frac{1}{{x}^{-4}} \cdot 5\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 97.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* (pow x 4.0) 5.0))))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.12e-57)
       (* (fma (/ x eps) 5.0 1.0) (* (* eps eps) (* (* eps eps) eps)))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (pow(x, 4.0) * 5.0);
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = fma((x / eps), 5.0, 1.0) * ((eps * eps) * ((eps * eps) * eps));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64((x ^ 4.0) * 5.0))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.12e-57], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in eps around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 87.7

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 87.7

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

      11. lift-pow.f64 N/A

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

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot {\varepsilon}^{\left(2 + \color{blue}{3}\right)} \]

      13. pow-add N/A

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

      14. lower-special-pow.f64 N/A

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

      15. lower-special-pow.f64 N/A

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

      16. lower-special-*.f64 87.6

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

      17. lift-pow.f64 N/A

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

      18. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      19. lower-*.f64 87.6

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{\color{blue}{3}}\right) \]

      21. unpow3 N/A

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

      22. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      23. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      24. lower-*.f64 87.6

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

      25. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right)\right) \]

      26. unpow2 N/A

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

      27. lower-*.f64 87.6

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

   6. Applied rewrites 87.6%

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

Alternative 10: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \varepsilon \cdot \left({x}^{4} \cdot 5\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* eps (* (pow x 4.0) 5.0))))
   (if (<= x -6.2e-64) t_0 (if (<= x 1.12e-57) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = eps * (pow(x, 4.0) * 5.0);
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 = eps * ((x ** 4.0d0) * 5.0d0)
    if (x <= (-6.2d-64)) then
        tmp = t_0
    else if (x <= 1.12d-57) then
        tmp = eps ** 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = eps * (Math.pow(x, 4.0) * 5.0);
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = eps * (math.pow(x, 4.0) * 5.0)
	tmp = 0
	if x <= -6.2e-64:
		tmp = t_0
	elif x <= 1.12e-57:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(eps * Float64((x ^ 4.0) * 5.0))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = eps * ((x ^ 4.0) * 5.0);
	tmp = 0.0;
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.12e-57], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.5

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

   4. Applied rewrites 87.5%

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

Alternative 11: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot \varepsilon\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -6.2e-64)
   (* (* (* (* x x) (* x x)) 5.0) eps)
   (if (<= x 1.12e-57) (pow eps 5.0) (* (* (pow x 4.0) eps) 5.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-64) {
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	} else if (x <= 1.12e-57) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (pow(x, 4.0) * 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) :: tmp
    if (x <= (-6.2d-64)) then
        tmp = (((x * x) * (x * x)) * 5.0d0) * eps
    else if (x <= 1.12d-57) then
        tmp = eps ** 5.0d0
    else
        tmp = ((x ** 4.0d0) * eps) * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -6.2e-64) {
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	} else if (x <= 1.12e-57) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (Math.pow(x, 4.0) * eps) * 5.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -6.2e-64:
		tmp = (((x * x) * (x * x)) * 5.0) * eps
	elif x <= 1.12e-57:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (math.pow(x, 4.0) * eps) * 5.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps);
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64((x ^ 4.0) * eps) * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -6.2e-64)
		tmp = (((x * x) * (x * x)) * 5.0) * eps;
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = ((x ^ 4.0) * eps) * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -6.2e-64], N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 1.12e-57], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.20000000000000049e-64

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 83.2

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

       4. lift-pow.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-add N/A

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

       7. lower-special-pow.f32 N/A

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

       8. lower-pow.f32 N/A

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

       9. pow2 N/A

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

       10. lower-special-pow.f32 N/A

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

       11. lower-pow.f32 N/A

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

       12. pow2 N/A

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

       13. lower-special-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.5

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

   4. Applied rewrites 87.5%

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

   if 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

Alternative 12: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* x x) (* x x)) 5.0) eps)))
   (if (<= x -6.2e-64) t_0 (if (<= x 1.12e-57) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 * x) * (x * x)) * 5.0d0) * eps
    if (x <= (-6.2d-64)) then
        tmp = t_0
    else if (x <= 1.12d-57) then
        tmp = eps ** 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((x * x) * (x * x)) * 5.0) * eps
	tmp = 0
	if x <= -6.2e-64:
		tmp = t_0
	elif x <= 1.12e-57:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps)
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	tmp = 0.0;
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.12e-57], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 83.2

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

       4. lift-pow.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-add N/A

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

       7. lower-special-pow.f32 N/A

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

       8. lower-pow.f32 N/A

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

       9. pow2 N/A

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

       10. lower-special-pow.f32 N/A

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

       11. lower-pow.f32 N/A

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

       12. pow2 N/A

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

       13. lower-special-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.5

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

   4. Applied rewrites 87.5%

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

Alternative 13: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* x x) (* x x)) 5.0) eps)))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.12e-57) (* (* (* (* eps eps) eps) eps) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) * (x * x)) * 5.0d0) * eps
    if (x <= (-6.2d-64)) then
        tmp = t_0
    else if (x <= 1.12d-57) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((x * x) * (x * x)) * 5.0) * eps
	tmp = 0
	if x <= -6.2e-64:
		tmp = t_0
	elif x <= 1.12e-57:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(x * x) * Float64(x * x)) * 5.0) * eps)
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((x * x) * (x * x)) * 5.0) * eps;
	tmp = 0.0;
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.12e-57], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 83.2

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

       4. lift-pow.f64 N/A

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

       5. metadata-eval N/A

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

       6. pow-add N/A

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

       7. lower-special-pow.f32 N/A

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

       8. lower-pow.f32 N/A

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

       9. pow2 N/A

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

       10. lower-special-pow.f32 N/A

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

       11. lower-pow.f32 N/A

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

       12. pow2 N/A

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

       13. lower-special-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. lower-special-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-special-*.f64 87.5

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(3 + 1\right)} \cdot \varepsilon \]

      8. pow-plus-rev N/A

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

      9. lower-special-pow.f64 N/A

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

      10. lower-special-*.f64 87.5

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

      11. lift-pow.f64 N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 87.4

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 87.4

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

   6. Applied rewrites 87.4%

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

Alternative 14: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-57}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* x x) (* x x)) (* 5.0 eps))))
   (if (<= x -6.2e-64)
     t_0
     (if (<= x 1.12e-57) (* (* (* (* eps eps) eps) eps) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((x * x) * (x * x)) * (5.0 * eps);
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * x) * (x * x)) * (5.0d0 * eps)
    if (x <= (-6.2d-64)) then
        tmp = t_0
    else if (x <= 1.12d-57) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((x * x) * (x * x)) * (5.0 * eps);
	double tmp;
	if (x <= -6.2e-64) {
		tmp = t_0;
	} else if (x <= 1.12e-57) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * x) * (x * x)) * (5.0 * eps)
	tmp = 0
	if x <= -6.2e-64:
		tmp = t_0
	elif x <= 1.12e-57:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps))
	tmp = 0.0
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * x) * (x * x)) * (5.0 * eps);
	tmp = 0.0;
	if (x <= -6.2e-64)
		tmp = t_0;
	elseif (x <= 1.12e-57)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e-64], t$95$0, If[LessEqual[x, 1.12e-57], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000049e-64 or 1.12e-57 < x

   1. Initial program 88.4%

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

   2. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, 4 \cdot \varepsilon + 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      9. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

      10. lower-pow.f64 83.4

          \[\leadsto {x}^{4} \cdot \left(\varepsilon + \mathsf{fma}\left(2, \frac{{\varepsilon}^{2}}{x}, \mathsf{fma}\left(4, \varepsilon, 8 \cdot \frac{{\varepsilon}^{2}}{x}\right)\right)\right) \]

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 83.2

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

   7. Applied rewrites 83.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 83.2

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

   9. Applied rewrites 83.2%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lower-*.f64 N/A

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

       6. lift-pow.f64 N/A

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

       7. metadata-eval N/A

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

       8. pow-add N/A

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

       9. lower-special-pow.f32 N/A

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

       10. lower-pow.f32 N/A

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

       11. pow2 N/A

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

       12. lower-special-pow.f32 N/A

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

       13. lower-pow.f32 N/A

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

       14. pow2 N/A

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

       15. lower-special-*.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 83.1

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

   11. Applied rewrites 83.1%

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

   if -6.20000000000000049e-64 < x < 1.12e-57

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. lower-special-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-special-*.f64 87.5

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(3 + 1\right)} \cdot \varepsilon \]

      8. pow-plus-rev N/A

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

      9. lower-special-pow.f64 N/A

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

      10. lower-special-*.f64 87.5

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

      11. lift-pow.f64 N/A

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

      12. unpow3 N/A

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

      13. unpow2 N/A

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

      14. lift-pow.f64 N/A

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

      15. lower-*.f64 87.4

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

      16. lift-pow.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 87.4

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

   6. Applied rewrites 87.4%

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

Alternative 15: 87.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0

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

   1. lower-pow.f64 87.5

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

4. Applied rewrites 87.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. lower-special-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   5. lower-special-*.f64 87.5

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   7. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(3 + 1\right)} \cdot \varepsilon \]

   8. pow-plus-rev N/A

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

   9. lower-special-pow.f64 N/A

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

   10. lower-special-*.f64 87.5

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

   11. lift-pow.f64 N/A

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

   12. unpow3 N/A

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

   13. unpow2 N/A

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

   14. lift-pow.f64 N/A

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

   15. lower-*.f64 87.4

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

   16. lift-pow.f64 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f64 87.4

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

6. Applied rewrites 87.4%

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

Alternative 16: 87.4% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.4%

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

2. Taylor expanded in x around 0

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

   1. lower-pow.f64 87.5

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

4. Applied rewrites 87.5%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-add N/A

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{{\varepsilon}^{2}} \]

   4. lower-special-pow.f64 N/A

      \[\leadsto {\varepsilon}^{3} \cdot {\color{blue}{\varepsilon}}^{2} \]

   5. lower-special-pow.f64 N/A

      \[\leadsto {\varepsilon}^{3} \cdot {\varepsilon}^{\color{blue}{2}} \]

   6. lower-special-*.f64 87.5

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{{\varepsilon}^{2}} \]

   7. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{3} \cdot {\color{blue}{\varepsilon}}^{2} \]

   8. unpow3 N/A

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

   9. unpow2 N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-*.f64 87.4

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 87.4

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

   15. lift-pow.f64 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f64 87.4

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \color{blue}{\varepsilon}\right) \]

6. Applied rewrites 87.4%

   \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025151 
(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)))