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

Percentage Accurate: 88.7% → 98.0%

Time: 4.1s

Alternatives: 17

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: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(4, {x}^{4}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(4, {x}^{3}, \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(2, {x}^{2}, \mathsf{fma}\left(8, {x}^{2}, \varepsilon \cdot \left(x + 4 \cdot x\right)\right)\right), x \cdot \mathsf{fma}\left(2, {x}^{2}, 4 \cdot {x}^{2}\right)\right)\right), {x}^{4}\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (*
    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) (fma 8.0 (pow x 2.0) (* eps (+ x (* 4.0 x)))))
        (* x (fma 2.0 (pow x 2.0) (* 4.0 (pow x 2.0))))))
      (pow x 4.0))))
   (if (<= x 8.8e-65)
     (fma x (* (* (* eps eps) (* eps eps)) 5.0) (pow eps 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		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), fma(8.0, pow(x, 2.0), (eps * (x + (4.0 * x))))), (x * fma(2.0, pow(x, 2.0), (4.0 * pow(x, 2.0)))))), pow(x, 4.0)));
	} else if (x <= 8.8e-65) {
		tmp = fma(x, (((eps * eps) * (eps * eps)) * 5.0), pow(eps, 5.0));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		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), fma(8.0, (x ^ 2.0), Float64(eps * Float64(x + Float64(4.0 * x))))), Float64(x * fma(2.0, (x ^ 2.0), Float64(4.0 * (x ^ 2.0)))))), (x ^ 4.0))));
	elseif (x <= 8.8e-65)
		tmp = fma(x, Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], 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] + N[(eps * N[(x + N[(4.0 * x), $MachinePrecision]), $MachinePrecision]), $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], If[LessEqual[x, 8.8e-65], N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.8

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (*
    eps
    (fma
     5.0
     (pow x 4.0)
     (*
      eps
      (fma
       10.0
       (pow x 3.0)
       (* eps (fma 5.0 (* eps x) (* 10.0 (pow x 2.0))))))))
   (if (<= x 8.8e-65)
     (fma x (* (* (* eps eps) (* eps eps)) 5.0) (pow eps 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = eps * fma(5.0, pow(x, 4.0), (eps * fma(10.0, pow(x, 3.0), (eps * fma(5.0, (eps * x), (10.0 * pow(x, 2.0)))))));
	} else if (x <= 8.8e-65) {
		tmp = fma(x, (((eps * eps) * (eps * eps)) * 5.0), pow(eps, 5.0));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(eps * fma(5.0, (x ^ 4.0), Float64(eps * fma(10.0, (x ^ 3.0), Float64(eps * fma(5.0, Float64(eps * x), Float64(10.0 * (x ^ 2.0))))))));
	elseif (x <= 8.8e-65)
		tmp = fma(x, Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(eps * N[(5.0 * N[Power[x, 4.0], $MachinePrecision] + N[(eps * N[(10.0 * N[Power[x, 3.0], $MachinePrecision] + N[(eps * N[(5.0 * N[(eps * x), $MachinePrecision] + N[(10.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. sum-to-mult N/A

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

      4. unpow-prod-down N/A

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

      5. lift-pow.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 55.5

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

   3. Applied rewrites 55.5%

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

   4. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \color{blue}{\left(5 \cdot {x}^{4} + \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, \color{blue}{{x}^{4}}, \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{\color{blue}{4}}, \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \left(10 \cdot {x}^{3} + \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      6. lower-pow.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \left(5 \cdot \left(\varepsilon \cdot x\right) + 10 \cdot {x}^{2}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right) \]

      11. lower-pow.f64 82.7

          \[\leadsto \varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right) \]

   6. Applied rewrites 82.7%

      \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(5, {x}^{4}, \varepsilon \cdot \mathsf{fma}\left(10, {x}^{3}, \varepsilon \cdot \mathsf{fma}\left(5, \varepsilon \cdot x, 10 \cdot {x}^{2}\right)\right)\right)} \]

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.8

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 3: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left({x}^{2} \cdot \mathsf{fma}\left(10, {\varepsilon}^{2}, x \cdot \mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right)\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (*
    (* (pow x 2.0) (fma 10.0 (pow eps 2.0) (* x (fma 5.0 x (* 10.0 eps)))))
    eps)
   (if (<= x 8.8e-65)
     (fma x (* (* (* eps eps) (* eps eps)) 5.0) (pow eps 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (pow(x, 2.0) * fma(10.0, pow(eps, 2.0), (x * fma(5.0, x, (10.0 * eps))))) * eps;
	} else if (x <= 8.8e-65) {
		tmp = fma(x, (((eps * eps) * (eps * eps)) * 5.0), pow(eps, 5.0));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64((x ^ 2.0) * fma(10.0, (eps ^ 2.0), Float64(x * fma(5.0, x, Float64(10.0 * eps))))) * eps);
	elseif (x <= 8.8e-65)
		tmp = fma(x, Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(10.0 * N[Power[eps, 2.0], $MachinePrecision] + N[(x * N[(5.0 * x + N[(10.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   10. Taylor expanded in x around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-*.f64 82.7

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

   12. Applied rewrites 82.7%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.8

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 4: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;{x}^{4} \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (pow x 4.0) (fma 5.0 eps (* 10.0 (/ (pow eps 2.0) x))))
   (if (<= x 8.8e-65)
     (fma x (* (* (* eps eps) (* eps eps)) 5.0) (pow eps 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = pow(x, 4.0) * fma(5.0, eps, (10.0 * (pow(eps, 2.0) / x)));
	} else if (x <= 8.8e-65) {
		tmp = fma(x, (((eps * eps) * (eps * eps)) * 5.0), pow(eps, 5.0));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64((x ^ 4.0) * fma(5.0, eps, Float64(10.0 * Float64((eps ^ 2.0) / x))));
	elseif (x <= 8.8e-65)
		tmp = fma(x, Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 * eps + N[(10.0 * N[(N[Power[eps, 2.0], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon + 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(5 \cdot \varepsilon + \color{blue}{10 \cdot \frac{{\varepsilon}^{2}}{x}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto {x}^{4} \cdot \left(5 \cdot \varepsilon + \color{blue}{10} \cdot \frac{{\varepsilon}^{2}}{x}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto {x}^{4} \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto {x}^{4} \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto {x}^{4} \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right) \]

      6. lower-pow.f64 82.6

         \[\leadsto {x}^{4} \cdot \mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right) \]

   7. Applied rewrites 82.6%

      \[\leadsto {x}^{4} \cdot \color{blue}{\mathsf{fma}\left(5, \varepsilon, 10 \cdot \frac{{\varepsilon}^{2}}{x}\right)} \]

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.8

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 5: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(10, \frac{\varepsilon}{x}, 5\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot 5, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (* (* (* (* (fma 10.0 (/ eps x) 5.0) x) x) x) x) eps)
   (if (<= x 8.8e-65)
     (fma x (* (* (* eps eps) (* eps eps)) 5.0) (pow eps 5.0))
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = ((((fma(10.0, (eps / x), 5.0) * x) * x) * x) * x) * eps;
	} else if (x <= 8.8e-65) {
		tmp = fma(x, (((eps * eps) * (eps * eps)) * 5.0), pow(eps, 5.0));
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(10.0, Float64(eps / x), 5.0) * x) * x) * x) * x) * eps);
	elseif (x <= 8.8e-65)
		tmp = fma(x, Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * 5.0), (eps ^ 5.0));
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(N[(N[(N[(N[(10.0 * N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(x * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   11. Applied rewrites 82.5%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 87.8

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

   4. Applied rewrites 87.8%

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

      1. lift-fma.f64 N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.8

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 6: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(10, \frac{\varepsilon}{x}, 5\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left({x}^{4} \cdot \left(5 + 10 \cdot \frac{\varepsilon}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (* (* (* (* (fma 10.0 (/ eps x) 5.0) x) x) x) x) eps)
   (if (<= x 8.8e-65)
     (pow eps 5.0)
     (* eps (* (pow x 4.0) (+ 5.0 (* 10.0 (/ eps x))))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = ((((fma(10.0, (eps / x), 5.0) * x) * x) * x) * x) * eps;
	} else if (x <= 8.8e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (pow(x, 4.0) * (5.0 + (10.0 * (eps / x))));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(10.0, Float64(eps / x), 5.0) * x) * x) * x) * x) * eps);
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64((x ^ 4.0) * Float64(5.0 + Float64(10.0 * Float64(eps / x)))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(N[(N[(N[(N[(10.0 * N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[Power[x, 4.0], $MachinePrecision] * N[(5.0 + N[(10.0 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   11. Applied rewrites 82.5%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

Alternative 7: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(\left(\left(\left(\mathsf{fma}\left(10, \frac{\varepsilon}{x}, 5\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (* (* (* (* (fma 10.0 (/ eps x) 5.0) x) x) x) x) eps)
   (if (<= x 8.8e-65)
     (pow eps 5.0)
     (* eps (* (fma (/ eps x) 10.0 5.0) (* (* x x) (* x x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = ((((fma(10.0, (eps / x), 5.0) * x) * x) * x) * x) * eps;
	} else if (x <= 8.8e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = eps * (fma((eps / x), 10.0, 5.0) * ((x * x) * (x * x)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(10.0, Float64(eps / x), 5.0) * x) * x) * x) * x) * eps);
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(eps * Float64(fma(Float64(eps / x), 10.0, 5.0) * Float64(Float64(x * x) * Float64(x * x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(N[(N[(N[(N[(10.0 * N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[Power[eps, 5.0], $MachinePrecision], N[(eps * N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   11. Applied rewrites 82.5%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 82.6

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

      9. lift-pow.f64 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. unpow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. lift-*.f64 82.5

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

   9. Applied rewrites 82.5%

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

Alternative 8: 97.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(\mathsf{fma}\left(10, \frac{\varepsilon}{x}, 5\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* (fma 10.0 (/ eps x) 5.0) x) x) x) x) eps)))
   (if (<= x -2.4e-52) t_0 (if (<= x 8.8e-65) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((fma(10.0, (eps / x), 5.0) * x) * x) * x) * x) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(Float64(fma(10.0, Float64(eps / x), 5.0) * x) * x) * x) * x) * eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(N[(10.0 * N[(eps / x), $MachinePrecision] + 5.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], t$95$0, If[LessEqual[x, 8.8e-65], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e-52 or 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   11. Applied rewrites 82.5%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

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

Alternative 9: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(5, x, 10 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma 5.0 x (* 10.0 eps)) (* (* x x) x)) eps)))
   (if (<= x -2.4e-52) t_0 (if (<= x 8.8e-65) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma(5.0, x, (10.0 * eps)) * ((x * x) * x)) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(fma(5.0, x, Float64(10.0 * eps)) * Float64(Float64(x * x) * x)) * eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(5.0 * x + N[(10.0 * eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], t$95$0, If[LessEqual[x, 8.8e-65], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4000000000000002e-52 or 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.6

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

   9. Applied rewrites 82.5%

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

   10. Taylor expanded in x around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 82.5

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

   12. Applied rewrites 82.5%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

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

Alternative 10: 97.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(5 \cdot x\right) \cdot \left(x \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(x \cdot x\right)\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) eps)))
   (if (<= x -2.4e-52)
     (* (* 5.0 x) (* x t_0))
     (if (<= x 8.8e-65) (pow eps 5.0) (* (* t_0 (* x x)) 5.0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * t_0);
	} else if (x <= 8.8e-65) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (t_0 * (x * x)) * 5.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * eps
    if (x <= (-2.4d-52)) then
        tmp = (5.0d0 * x) * (x * t_0)
    else if (x <= 8.8d-65) then
        tmp = eps ** 5.0d0
    else
        tmp = (t_0 * (x * x)) * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * x) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * t_0);
	} else if (x <= 8.8e-65) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (t_0 * (x * x)) * 5.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * x) * eps
	tmp = 0
	if x <= -2.4e-52:
		tmp = (5.0 * x) * (x * t_0)
	elif x <= 8.8e-65:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (t_0 * (x * x)) * 5.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(5.0 * x) * Float64(x * t_0));
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(t_0 * Float64(x * x)) * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * x) * eps;
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = (5.0 * x) * (x * t_0);
	elseif (x <= 8.8e-65)
		tmp = eps ^ 5.0;
	else
		tmp = (t_0 * (x * x)) * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], N[(N[(5.0 * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[Power[eps, 5.0], $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.3

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. unpow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

Alternative 11: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(5 \cdot x\right) \cdot \left(x \cdot t\_0\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(x \cdot x\right)\right) \cdot 5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) eps)))
   (if (<= x -2.4e-52)
     (* (* 5.0 x) (* x t_0))
     (if (<= x 8.8e-65)
       (* (* (* (* eps eps) eps) eps) eps)
       (* (* t_0 (* x x)) 5.0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * t_0);
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (t_0 * (x * x)) * 5.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * eps
    if (x <= (-2.4d-52)) then
        tmp = (5.0d0 * x) * (x * t_0)
    else if (x <= 8.8d-65) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (t_0 * (x * x)) * 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * x) * eps;
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * t_0);
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (t_0 * (x * x)) * 5.0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * x) * eps
	tmp = 0
	if x <= -2.4e-52:
		tmp = (5.0 * x) * (x * t_0)
	elif x <= 8.8e-65:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (t_0 * (x * x)) * 5.0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(5.0 * x) * Float64(x * t_0));
	elseif (x <= 8.8e-65)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(t_0 * Float64(x * x)) * 5.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * x) * eps;
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = (5.0 * x) * (x * t_0);
	elseif (x <= 8.8e-65)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (t_0 * (x * x)) * 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], N[(N[(5.0 * x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] * 5.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-*.f64 87.6

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.3

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

      4. lift-*.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. unpow-prod-down N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

Alternative 12: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(5 \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(5 \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (* 5.0 x) (* x (* (* x x) eps)))
   (if (<= x 8.8e-65)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* x x) (* x x)) (* 5.0 eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.4d-52)) then
        tmp = (5.0d0 * x) * (x * ((x * x) * eps))
    else if (x <= 8.8d-65) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = ((x * x) * (x * x)) * (5.0d0 * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.4e-52:
		tmp = (5.0 * x) * (x * ((x * x) * eps))
	elif x <= 8.8e-65:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = ((x * x) * (x * x)) * (5.0 * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(5.0 * x) * Float64(x * Float64(Float64(x * x) * eps)));
	elseif (x <= 8.8e-65)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * Float64(5.0 * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	elseif (x <= 8.8e-65)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = ((x * x) * (x * x)) * (5.0 * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(5.0 * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-*.f64 87.6

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      4. *-commutative 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-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

Alternative 13: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;\left(5 \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \left(\left(\left(x \cdot x\right) \cdot 5\right) \cdot x\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.4e-52)
   (* (* 5.0 x) (* x (* (* x x) eps)))
   (if (<= x 8.8e-65)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* eps (* (* (* x x) 5.0) x)) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (eps * (((x * x) * 5.0) * x)) * x;
	}
	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 <= (-2.4d-52)) then
        tmp = (5.0d0 * x) * (x * ((x * x) * eps))
    else if (x <= 8.8d-65) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (eps * (((x * x) * 5.0d0) * x)) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.4e-52) {
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (eps * (((x * x) * 5.0) * x)) * x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.4e-52:
		tmp = (5.0 * x) * (x * ((x * x) * eps))
	elif x <= 8.8e-65:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (eps * (((x * x) * 5.0) * x)) * x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = Float64(Float64(5.0 * x) * Float64(x * Float64(Float64(x * x) * eps)));
	elseif (x <= 8.8e-65)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(eps * Float64(Float64(Float64(x * x) * 5.0) * x)) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = (5.0 * x) * (x * ((x * x) * eps));
	elseif (x <= 8.8e-65)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (eps * (((x * x) * 5.0) * x)) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.4e-52], N[(N[(5.0 * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.8e-65], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(eps * N[(N[(N[(x * x), $MachinePrecision] * 5.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.4000000000000002e-52

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-*.f64 87.6

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

   if 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. associate-*r* N/A

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

       6. associate-*r* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 82.3

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

       10. lift-*.f64 N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

Alternative 14: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot x\right) \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \varepsilon\right)\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 x) (* x (* (* x x) eps)))))
   (if (<= x -2.4e-52)
     t_0
     (if (<= x 8.8e-65) (* (* (* (* eps eps) eps) eps) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (5.0 * x) * (x * ((x * x) * eps));
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (5.0d0 * x) * (x * ((x * x) * eps))
    if (x <= (-2.4d-52)) then
        tmp = t_0
    else if (x <= 8.8d-65) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (5.0 * x) * (x * ((x * x) * eps));
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (5.0 * x) * (x * ((x * x) * eps))
	tmp = 0
	if x <= -2.4e-52:
		tmp = t_0
	elif x <= 8.8e-65:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(5.0 * x) * Float64(x * Float64(Float64(x * x) * eps)))
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		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 = (5.0 * x) * (x * ((x * x) * eps));
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(5.0 * x), $MachinePrecision] * N[(x * N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], t$95$0, If[LessEqual[x, 8.8e-65], 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 < -2.4000000000000002e-52 or 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. unpow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. associate-*r* N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-*.f64 82.3

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

   11. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-*.f64 87.6

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 15: 97.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 5 \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\varepsilon \cdot x\right)\right)\\ \mathbf{if}\;x \leq -2.4 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-65}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* 5.0 (* (* (* x x) x) (* eps x)))))
   (if (<= x -2.4e-52)
     t_0
     (if (<= x 8.8e-65) (* (* (* (* eps eps) eps) eps) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = 5.0 * (((x * x) * x) * (eps * x));
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 5.0d0 * (((x * x) * x) * (eps * x))
    if (x <= (-2.4d-52)) then
        tmp = t_0
    else if (x <= 8.8d-65) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = 5.0 * (((x * x) * x) * (eps * x));
	double tmp;
	if (x <= -2.4e-52) {
		tmp = t_0;
	} else if (x <= 8.8e-65) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = 5.0 * (((x * x) * x) * (eps * x))
	tmp = 0
	if x <= -2.4e-52:
		tmp = t_0
	elif x <= 8.8e-65:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(5.0 * Float64(Float64(Float64(x * x) * x) * Float64(eps * x)))
	tmp = 0.0
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		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 = 5.0 * (((x * x) * x) * (eps * x));
	tmp = 0.0;
	if (x <= -2.4e-52)
		tmp = t_0;
	elseif (x <= 8.8e-65)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(5.0 * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(eps * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.4e-52], t$95$0, If[LessEqual[x, 8.8e-65], 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 < -2.4000000000000002e-52 or 8.80000000000000084e-65 < x

   1. Initial program 88.7%

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

   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 82.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto 5 \cdot \color{blue}{\left(\varepsilon \cdot {x}^{4}\right)} \]
   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 82.3

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

   7. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. metadata-eval N/A

         \[\leadsto 5 \cdot \left({x}^{\left(3 + 1\right)} \cdot \varepsilon\right) \]

      5. pow-plus N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 82.3

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

   9. Applied rewrites 82.3%

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

   if -2.4000000000000002e-52 < x < 8.80000000000000084e-65

   1. Initial program 88.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 87.7

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

   4. Applied rewrites 87.7%

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   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. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      5. lower-*.f64 87.6

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      7. metadata-eval N/A

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

      8. pow-sqr N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 87.5

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

   6. Applied rewrites 87.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. cube-unmult N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 87.6

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

   8. Applied rewrites 87.6%

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

Alternative 16: 87.6% 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 88.7%

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

2. Taylor expanded in x around 0

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

   1. lower-pow.f64 87.7

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

4. Applied rewrites 87.7%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
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. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   5. lower-*.f64 87.6

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 87.5

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

6. Applied rewrites 87.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. cube-unmult N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 87.6

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

8. Applied rewrites 87.6%

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

Alternative 17: 87.5% 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 88.7%

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

2. Taylor expanded in x around 0

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

   1. lower-pow.f64 87.7

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

4. Applied rewrites 87.7%

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
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. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   5. lower-*.f64 87.6

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

   7. metadata-eval N/A

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

   8. pow-sqr N/A

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

   9. pow2 N/A

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

   10. pow2 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 87.5

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

6. Applied rewrites 87.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. cube-unmult N/A

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

   7. lower-*.f64 N/A

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

   8. pow3 N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 87.5

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

8. Applied rewrites 87.5%

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

Reproduce

herbie shell --seed 2025149 
(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)))