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

Percentage Accurate: 88.6% → 99.0%

Time: 3.6s

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: 88.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\varepsilon \cdot \varepsilon\right) \cdot 6\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, \left(-\frac{\mathsf{fma}\left(t\_0, \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 4\right)}{x}\right) + \left(-t\_0\right)\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* eps eps) 6.0)))
   (if (<= x -1.45e-32)
     (*
      (+
       (fma
        4.0
        eps
        (-
         (/
          (fma
           -4.0
           (* eps eps)
           (+ (- (/ (fma t_0 eps (* (* (* eps eps) eps) 4.0)) x)) (- t_0)))
          x)))
       eps)
      (* (* (* x x) x) x))
     (if (<= x 1.9e-56)
       (- (pow (+ x eps) 5.0) (pow x 5.0))
       (* (* (* (* x x) eps) 5.0) (* x x))))))

C

double code(double x, double eps) {
	double t_0 = (eps * eps) * 6.0;
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (fma(4.0, eps, -(fma(-4.0, (eps * eps), (-(fma(t_0, eps, (((eps * eps) * eps) * 4.0)) / x) + -t_0)) / x)) + eps) * (((x * x) * x) * x);
	} else if (x <= 1.9e-56) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(eps * eps) * 6.0)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(fma(4.0, eps, Float64(-Float64(fma(-4.0, Float64(eps * eps), Float64(Float64(-Float64(fma(t_0, eps, Float64(Float64(Float64(eps * eps) * eps) * 4.0)) / x)) + Float64(-t_0))) / x))) + eps) * Float64(Float64(Float64(x * x) * x) * x));
	elseif (x <= 1.9e-56)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision]}, If[LessEqual[x, -1.45e-32], N[(N[(N[(4.0 * eps + (-N[(N[(-4.0 * N[(eps * eps), $MachinePrecision] + N[((-N[(N[(t$95$0 * eps + N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]) + (-t$95$0)), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision] + eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x around -inf

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

   7. Applied rewrites 95.2%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 2: 97.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (*
    (+
     (fma 4.0 eps (- (/ (fma -4.0 (* eps eps) (- (* (* eps eps) 6.0))) x)))
     eps)
    (* (* (* x x) x) x))
   (if (<= x 1.9e-56)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (fma(4.0, eps, -(fma(-4.0, (eps * eps), -((eps * eps) * 6.0)) / x)) + eps) * (((x * x) * x) * x);
	} else if (x <= 1.9e-56) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(fma(4.0, eps, Float64(-Float64(fma(-4.0, Float64(eps * eps), Float64(-Float64(Float64(eps * eps) * 6.0))) / x))) + eps) * Float64(Float64(Float64(x * x) * x) * x));
	elseif (x <= 1.9e-56)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(4.0 * eps + (-N[(N[(-4.0 * N[(eps * eps), $MachinePrecision] + (-N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision])), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision] + eps), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 21.7

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

   4. Applied rewrites 21.7%

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

   5. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 94.5%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 3: 97.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (*
    (+
     (fma 4.0 eps (- (/ (fma -4.0 (* eps eps) (- (* (* eps eps) 6.0))) x)))
     eps)
    (* (* x x) (* x x)))
   (if (<= x 1.9e-56)
     (- (pow (+ x eps) 5.0) (pow x 5.0))
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (fma(4.0, eps, -(fma(-4.0, (eps * eps), -((eps * eps) * 6.0)) / x)) + eps) * ((x * x) * (x * x));
	} else if (x <= 1.9e-56) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(fma(4.0, eps, Float64(-Float64(fma(-4.0, Float64(eps * eps), Float64(-Float64(Float64(eps * eps) * 6.0))) / x))) + eps) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 1.9e-56)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(4.0 * eps + (-N[(N[(-4.0 * N[(eps * eps), $MachinePrecision] + (-N[(N[(eps * eps), $MachinePrecision] * 6.0), $MachinePrecision])), $MachinePrecision] / x), $MachinePrecision])), $MachinePrecision] + eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.4%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 4: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t\_0 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot {x}^{4}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0)) (t_1 (- t_0 (pow x 5.0))))
   (if (<= t_1 -5e-324)
     (- t_0 (* (* (* x x) (* x x)) x))
     (if (<= t_1 0.0) (* (* 5.0 eps) (pow x 4.0)) (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0);
	double t_1 = t_0 - pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_0 - (((x * x) * (x * x)) * x);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * pow(x, 4.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    if (t_1 <= (-5d-324)) then
        tmp = t_0 - (((x * x) * (x * x)) * x)
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * eps) * (x ** 4.0d0)
    else
        tmp = eps ** 5.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0);
	double t_1 = t_0 - Math.pow(x, 5.0);
	double tmp;
	if (t_1 <= -5e-324) {
		tmp = t_0 - (((x * x) * (x * x)) * x);
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * eps) * Math.pow(x, 4.0);
	} else {
		tmp = Math.pow(eps, 5.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	tmp = 0
	if t_1 <= -5e-324:
		tmp = t_0 - (((x * x) * (x * x)) * x)
	elif t_1 <= 0.0:
		tmp = (5.0 * eps) * math.pow(x, 4.0)
	else:
		tmp = math.pow(eps, 5.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	tmp = 0.0
	if (t_1 <= -5e-324)
		tmp = Float64(t_0 - Float64(Float64(Float64(x * x) * Float64(x * x)) * x));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * eps) * (x ^ 4.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	tmp = 0.0;
	if (t_1 <= -5e-324)
		tmp = t_0 - (((x * x) * (x * x)) * x);
	elseif (t_1 <= 0.0)
		tmp = (5.0 * eps) * (x ^ 4.0);
	else
		tmp = eps ^ 5.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-324], N[(t$95$0 - N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * eps), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.94066e-324

   1. Initial program 97.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-plus-rev N/A

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

      4. lower-*.f64 N/A

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

      5. sqr-pow N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 97.8

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

   3. Applied rewrites 97.8%

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

   if -4.94066e-324 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 86.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. pow-pow N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 99.9

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

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

      3. lower-*.f64 N/A

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

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 91.4

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

   4. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow-plus-rev N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 92.1

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

   6. Applied rewrites 92.1%

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

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (pow x 4.0) 5.0) eps)
   (if (<= x 1.9e-56)
     (* (fma 5.0 x eps) (* (* (* eps eps) eps) eps))
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else if (x <= 1.9e-56) {
		tmp = fma(5.0, x, eps) * (((eps * eps) * eps) * eps);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	elseif (x <= 1.9e-56)
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(Float64(eps * eps) * eps) * eps));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.5

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

   4. Applied rewrites 93.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-pow N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

          \[\leadsto \left({x}^{\left(3 + 1\right)} \cdot 5\right) \cdot \varepsilon \]

      11. pow-plus N/A

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

      12. lower-*.f64 N/A

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

      13. unpow3 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 N/A

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

      16. pow2 N/A

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

      17. lift-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow3 N/A

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

      5. pow-plus N/A

         \[\leadsto \left({x}^{\left(3 + 1\right)} \cdot 5\right) \cdot \varepsilon \]

      6. metadata-eval N/A

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

      7. lift-pow.f64 93.8

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

   8. Applied rewrites 93.8%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 6: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* (* (* x x) x) x) eps) 5.0)
   (if (<= x 1.9e-56)
     (* (fma 5.0 x eps) (* (* (* eps eps) eps) eps))
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	} else if (x <= 1.9e-56) {
		tmp = fma(5.0, x, eps) * (((eps * eps) * eps) * eps);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * eps) * 5.0);
	elseif (x <= 1.9e-56)
		tmp = Float64(fma(5.0, x, eps) * Float64(Float64(Float64(eps * eps) * eps) * eps));
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 93.6%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. sqr-pow N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot \left(\left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon\right) \]

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-plus N/A

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

      7. lower-*.f64 N/A

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

      8. unpow3 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 7: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* (* (* x x) x) x) eps) 5.0)
   (if (<= x 1.9e-56) (pow eps 5.0) (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	} else if (x <= 1.9e-56) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (((x * x) * eps) * 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 <= (-1.45d-32)) then
        tmp = ((((x * x) * x) * x) * eps) * 5.0d0
    else if (x <= 1.9d-56) then
        tmp = eps ** 5.0d0
    else
        tmp = (((x * x) * eps) * 5.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	} else if (x <= 1.9e-56) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = ((((x * x) * x) * x) * eps) * 5.0
	elif x <= 1.9e-56:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = (((x * x) * eps) * 5.0) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * eps) * 5.0);
	elseif (x <= 1.9e-56)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	elseif (x <= 1.9e-56)
		tmp = eps ^ 5.0;
	else
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[Power[eps, 5.0], $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 93.6%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

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

      3. lower-*.f64 N/A

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

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-pow N/A

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

      7. metadata-eval N/A

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

      8. pow-plus-rev N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 98.5

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

   6. Applied rewrites 98.5%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 8: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* (* (* x x) x) x) eps) 5.0)
   (if (<= x 1.9e-56)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * eps) * 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 <= (-1.45d-32)) then
        tmp = ((((x * x) * x) * x) * eps) * 5.0d0
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (((x * x) * eps) * 5.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = ((((x * x) * x) * x) * eps) * 5.0
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (((x * x) * eps) * 5.0) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * eps) * 5.0);
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = ((((x * x) * x) * x) * eps) * 5.0;
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-rgt1-in N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. *-commutative N/A

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

      15. associate-*r* N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 93.6%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

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

      3. lower-*.f64 N/A

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

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

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

      4. lift-*.f64 N/A

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

      5. unpow3 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 9: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* x x) x) (* x (* 5.0 eps)))
   (if (<= x 1.9e-56)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* (* x x) eps) 5.0) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * eps) * 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 <= (-1.45d-32)) then
        tmp = ((x * x) * x) * (x * (5.0d0 * eps))
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (((x * x) * eps) * 5.0d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = ((x * x) * x) * (x * (5.0 * eps))
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (((x * x) * eps) * 5.0) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(x * Float64(5.0 * eps)));
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(x * x) * eps) * 5.0) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (((x * x) * eps) * 5.0) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-pow N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      12. pow-plus N/A

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

      13. lower-*.f64 N/A

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

      14. unpow3 N/A

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

      15. pow2 N/A

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

      16. lower-*.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 93.7

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow3 N/A

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

      8. lower-*.f64 N/A

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

      9. pow3 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 93.7

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

   8. Applied rewrites 93.7%

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

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow-plus-rev N/A

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

      3. lower-*.f64 N/A

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

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 98.3

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

   4. Applied rewrites 98.3%

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

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

      4. lift-*.f64 N/A

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

      5. unpow3 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-*.f64 98.3

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

   6. Applied rewrites 98.3%

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

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.4

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

   4. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow-prod-down N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 89.4

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

   6. Applied rewrites 89.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 89.4

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

   8. Applied rewrites 89.4%

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

Alternative 10: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* x x) x) (* x (* 5.0 eps)))
   (if (<= x 1.9e-56)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* (* 5.0 eps) x) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((5.0 * eps) * x) * x) * (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 <= (-1.45d-32)) then
        tmp = ((x * x) * x) * (x * (5.0d0 * eps))
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (((5.0d0 * eps) * x) * x) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (((5.0 * eps) * x) * x) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = ((x * x) * x) * (x * (5.0 * eps))
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (((5.0 * eps) * x) * x) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(x * Float64(5.0 * eps)));
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(Float64(5.0 * eps) * x) * x) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (((5.0 * eps) * x) * x) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-rgt1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. sqr-pow N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto {\left(x \cdot x\right)}^{2} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      6. lift-*.f64 N/A

         \[\leadsto {\left(x \cdot x\right)}^{2} \cdot \left(5 \cdot \varepsilon\right) \]

      7. pow2 N/A

         \[\leadsto {\left({x}^{2}\right)}^{2} \cdot \left(5 \cdot \varepsilon\right) \]

      8. pow-pow N/A

         \[\leadsto {x}^{\left(2 \cdot 2\right)} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(5 \cdot \varepsilon\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      11. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      12. pow-plus N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      14. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      15. pow2 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      19. lift-*.f64 93.7

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

   6. Applied rewrites 93.7%

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right) \]

      7. pow3 N/A

         \[\leadsto {x}^{3} \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

      9. pow3 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]

      13. lift-*.f64 93.7

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \color{blue}{\varepsilon}\right)\right) \]

   8. Applied rewrites 93.7%

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 98.3

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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-*r* N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      5. unpow3 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      7. unpow3 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      9. lower-*.f64 98.3

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   6. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 89.4

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{\color{blue}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{2} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {\color{blue}{x}}^{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

      11. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      13. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      14. lift-*.f64 89.4

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   6. Applied rewrites 89.4%

      \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot x\right) \]

      5. lower-*.f64 89.5

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right) \]

   8. Applied rewrites 89.5%

      \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* (* x x) x) (* x (* 5.0 eps)))
   (if (<= x 1.9e-56)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* 5.0 eps) (* x x)) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((5.0 * eps) * (x * x)) * (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 <= (-1.45d-32)) then
        tmp = ((x * x) * x) * (x * (5.0d0 * eps))
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = ((5.0d0 * eps) * (x * x)) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = ((x * x) * x) * (x * (5.0 * eps))
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = ((5.0 * eps) * (x * x)) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(Float64(x * x) * x) * Float64(x * Float64(5.0 * eps)));
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(5.0 * eps) * Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = ((x * x) * x) * (x * (5.0 * eps));
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 93.6

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 93.6%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto {\left(x \cdot x\right)}^{2} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      6. lift-*.f64 N/A

         \[\leadsto {\left(x \cdot x\right)}^{2} \cdot \left(5 \cdot \varepsilon\right) \]

      7. pow2 N/A

         \[\leadsto {\left({x}^{2}\right)}^{2} \cdot \left(5 \cdot \varepsilon\right) \]

      8. pow-pow N/A

         \[\leadsto {x}^{\left(2 \cdot 2\right)} \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      9. metadata-eval N/A

         \[\leadsto {x}^{4} \cdot \left(5 \cdot \varepsilon\right) \]

      10. lower-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      11. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      12. pow-plus N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      14. unpow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      15. pow2 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{2} \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      17. pow2 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      19. lift-*.f64 93.7

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

   6. Applied rewrites 93.7%

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right) \]

      7. pow3 N/A

         \[\leadsto {x}^{3} \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto {x}^{3} \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

      9. pow3 N/A

         \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \varepsilon\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\color{blue}{x} \cdot \left(5 \cdot \varepsilon\right)\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(5 \cdot \varepsilon\right)}\right) \]

      13. lift-*.f64 93.7

          \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(5 \cdot \color{blue}{\varepsilon}\right)\right) \]

   8. Applied rewrites 93.7%

      \[\leadsto \left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot \left(5 \cdot \varepsilon\right)\right)} \]

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 98.3

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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-*r* N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      5. unpow3 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      7. unpow3 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      9. lower-*.f64 98.3

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   6. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 89.4

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{\color{blue}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{2} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {\color{blue}{x}}^{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

      11. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      13. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      14. lift-*.f64 89.4

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   6. Applied rewrites 89.4%

      \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.45e-32)
   (* (* 5.0 (* (* x x) (* x x))) eps)
   (if (<= x 1.9e-56)
     (* (* (* (* eps eps) eps) eps) eps)
     (* (* (* 5.0 eps) (* x x)) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (5.0 * ((x * x) * (x * x))) * eps;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((5.0 * eps) * (x * x)) * (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 <= (-1.45d-32)) then
        tmp = (5.0d0 * ((x * x) * (x * x))) * eps
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = ((5.0d0 * eps) * (x * x)) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (5.0 * ((x * x) * (x * x))) * eps;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.45e-32:
		tmp = (5.0 * ((x * x) * (x * x))) * eps
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = ((5.0 * eps) * (x * x)) * (x * x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(5.0 * Float64(Float64(x * x) * Float64(x * x))) * eps);
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(Float64(5.0 * eps) * Float64(x * x)) * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = (5.0 * ((x * x) * (x * x))) * eps;
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = ((5.0 * eps) * (x * x)) * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.45e-32], N[(N[(5.0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(5.0 * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) \cdot \varepsilon \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) \cdot \varepsilon \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      13. lower-*.f64 93.5

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 98.3

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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-*r* N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      5. unpow3 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      7. unpow3 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      9. lower-*.f64 98.3

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   6. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 89.4

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot x\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

      4. pow2 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{\color{blue}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\left(x \cdot x\right)}^{2} \]

      6. unpow-prod-down N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{2}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {\color{blue}{x}}^{2} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot {x}^{2}\right) \cdot {x}^{2} \]

      11. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{2} \]

      13. pow2 N/A

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

      14. lift-*.f64 89.4

          \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{x}\right) \]

   6. Applied rewrites 89.4%

      \[\leadsto \left(\left(5 \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{\left(x \cdot x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;\left(5 \cdot t\_0\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (if (<= x -1.45e-32)
     (* (* 5.0 t_0) eps)
     (if (<= x 1.9e-56)
       (* (* (* (* eps eps) eps) eps) eps)
       (* (* 5.0 eps) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (5.0 * t_0) * eps;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (5.0 * eps) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * x) * (x * x)
    if (x <= (-1.45d-32)) then
        tmp = (5.0d0 * t_0) * eps
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = (5.0d0 * eps) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (x * x) * (x * x);
	double tmp;
	if (x <= -1.45e-32) {
		tmp = (5.0 * t_0) * eps;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = (5.0 * eps) * t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (x * x) * (x * x)
	tmp = 0
	if x <= -1.45e-32:
		tmp = (5.0 * t_0) * eps
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = (5.0 * eps) * t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = Float64(Float64(5.0 * t_0) * eps);
	elseif (x <= 1.9e-56)
		tmp = Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps);
	else
		tmp = Float64(Float64(5.0 * eps) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x * x) * (x * x);
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = (5.0 * t_0) * eps;
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = (5.0 * eps) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-32], N[(N[(5.0 * t$95$0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 1.9e-56], N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], N[(N[(5.0 * eps), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999998e-32

   1. Initial program 25.2%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) \cdot \varepsilon \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right)\right) \cdot \varepsilon \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      13. lower-*.f64 93.5

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 93.5%

      \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 98.3

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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-*r* N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      5. unpow3 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      7. unpow3 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      9. lower-*.f64 98.3

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   6. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   if 1.9000000000000001e-56 < x

   1. Initial program 42.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 89.4

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 97.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-56}:\\ \;\;\;\;\left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 eps) (* (* x x) (* x x)))))
   (if (<= x -1.45e-32)
     t_0
     (if (<= x 1.9e-56) (* (* (* (* eps eps) eps) eps) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (5.0 * eps) * ((x * x) * (x * x));
	double tmp;
	if (x <= -1.45e-32) {
		tmp = t_0;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (5.0d0 * eps) * ((x * x) * (x * x))
    if (x <= (-1.45d-32)) then
        tmp = t_0
    else if (x <= 1.9d-56) then
        tmp = (((eps * eps) * eps) * eps) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (5.0 * eps) * ((x * x) * (x * x));
	double tmp;
	if (x <= -1.45e-32) {
		tmp = t_0;
	} else if (x <= 1.9e-56) {
		tmp = (((eps * eps) * eps) * eps) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (5.0 * eps) * ((x * x) * (x * x))
	tmp = 0
	if x <= -1.45e-32:
		tmp = t_0
	elif x <= 1.9e-56:
		tmp = (((eps * eps) * eps) * eps) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(5.0 * eps) * Float64(Float64(x * x) * Float64(x * x)))
	tmp = 0.0
	if (x <= -1.45e-32)
		tmp = t_0;
	elseif (x <= 1.9e-56)
		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 * eps) * ((x * x) * (x * x));
	tmp = 0.0;
	if (x <= -1.45e-32)
		tmp = t_0;
	elseif (x <= 1.9e-56)
		tmp = (((eps * eps) * eps) * eps) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e-32], t$95$0, If[LessEqual[x, 1.9e-56], 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 < -1.44999999999999998e-32 or 1.9000000000000001e-56 < x

   1. Initial program 35.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + 4 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      3. distribute-rgt1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {x}^{4} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot {\color{blue}{x}}^{4} \]

      6. sqr-pow N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{\left(\frac{4}{2}\right)} \cdot \color{blue}{{x}^{\left(\frac{4}{2}\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{\left(\frac{4}{2}\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot {x}^{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      13. lower-*.f64 91.0

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   if -1.44999999999999998e-32 < x < 1.9000000000000001e-56

   1. Initial program 98.8%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. sqr-pow N/A

         \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      5. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      10. unpow2 N/A

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      11. lower-*.f64 98.3

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. 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-*r* N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      5. unpow3 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

      7. unpow3 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

      9. lower-*.f64 98.3

         \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   6. Applied rewrites 98.3%

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 87.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (* (* eps eps) eps) eps) eps))

C

double code(double x, double eps) {
	return (((eps * eps) * eps) * eps) * eps;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((eps * eps) * eps) * eps) * eps
end function

Java

public static double code(double x, double eps) {
	return (((eps * eps) * eps) * eps) * eps;
}

Python

def code(x, eps):
	return (((eps * eps) * eps) * eps) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(eps * eps) * eps) * eps) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((eps * eps) * eps) * eps) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. sqr-pow N/A

      \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   5. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   6. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   11. lower-*.f64 87.4

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 87.4%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. 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-*r* N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   5. unpow3 N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{3} \cdot \varepsilon\right) \cdot \varepsilon \]

   7. unpow3 N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

   9. lower-*.f64 87.4

      \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]

6. Applied rewrites 87.4%

   \[\leadsto \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon \]
7. Add Preprocessing

Alternative 16: 87.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \end{array} \]

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(eps * eps) * Float64(eps * 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] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. sqr-pow N/A

      \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   5. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   6. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   11. lower-*.f64 87.4

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 87.4%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. Add Preprocessing

Alternative 17: 87.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

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(eps * eps) * Float64(Float64(eps * eps) * eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * ((eps * eps) * eps);
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. sqr-pow N/A

      \[\leadsto \left({\varepsilon}^{\left(\frac{4}{2}\right)} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   5. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{\left(\frac{4}{2}\right)}\right) \cdot \varepsilon \]

   6. metadata-eval N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   10. unpow2 N/A

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   11. lower-*.f64 87.4

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 87.4%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. 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. 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

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   5. unpow3 N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{\color{blue}{3}} \]

   6. lower-*.f64 N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{{\varepsilon}^{3}} \]

   7. unpow3 N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   9. lower-*.f64 87.4

      \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

6. Applied rewrites 87.4%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025116 
(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)))