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

Percentage Accurate: 88.9% → 99.4%

Time: 3.9s

Alternatives: 15

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: 88.9% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(x \cdot x, -\left(x \cdot x\right) \cdot x, {\left(\varepsilon + x\right)}^{5}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left({x}^{3}, x, \left({x}^{3} \cdot x\right) \cdot 4\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (fma (* x x) (- (* (* x x) x)) (pow (+ eps x) 5.0))))
   (if (<= t_0 -5e-309)
     t_1
     (if (<= t_0 0.0)
       (* (fma (pow x 3.0) x (* (* (pow x 3.0) x) 4.0)) eps)
       t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma((x * x), -((x * x) * x), pow((eps + x), 5.0));
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = fma(pow(x, 3.0), x, ((pow(x, 3.0) * x) * 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = fma(Float64(x * x), Float64(-Float64(Float64(x * x) * x)), (Float64(eps + x) ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma((x ^ 3.0), x, Float64(Float64((x ^ 3.0) * x) * 4.0)) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * (-N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]) + N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-309], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 3.0], $MachinePrecision] * x + N[(N[(N[Power[x, 3.0], $MachinePrecision] * x), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.9%

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

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.6

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

   3. Applied rewrites 85.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. lift-pow.f64 N/A

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

      9. sub-flip-reverse N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft1-in N/A

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

      11. +-commutative N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. pow2 N/A

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

      15. associate-*r* N/A

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

      16. pow2 N/A

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

      17. unpow3 N/A

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

      18. lower-fma.f64 N/A

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

      19. pow3 N/A

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

      20. lift-*.f64 N/A

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 N/A

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

   6. Applied rewrites 82.3%

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 4\right) \cdot \varepsilon \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 82.3

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

   8. Applied rewrites 82.3%

      \[\leadsto \mathsf{fma}\left({x}^{3}, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 4\right) \cdot \varepsilon \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow3 N/A

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

      4. lower-pow.f64 82.3

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

   10. Applied rewrites 82.3%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \mathsf{fma}\left(x \cdot x, -\left(x \cdot x\right) \cdot x, {\left(\varepsilon + x\right)}^{5}\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1 (fma (* x x) (- (* (* x x) x)) (pow (+ eps x) 5.0))))
   (if (<= t_0 -5e-309)
     t_1
     (if (<= t_0 0.0) (* (* 5.0 (pow x 4.0)) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = fma((x * x), -((x * x) * x), pow((eps + x), 5.0));
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = fma(Float64(x * x), Float64(-Float64(Float64(x * x) * x)), (Float64(eps + x) ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * (-N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]) + N[Power[N[(eps + x), $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-309], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.9%

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

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.6

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

   3. Applied rewrites 85.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-plus N/A

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

      8. lift-pow.f64 N/A

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

      9. sub-flip-reverse N/A

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

      10. lift-neg.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 82.8%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5}\\ t_1 := t\_0 - {x}^{5}\\ t_2 := t\_0 - \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (pow (+ x eps) 5.0))
        (t_1 (- t_0 (pow x 5.0)))
        (t_2 (- t_0 (* (* x x) (* (* x x) x)))))
   (if (<= t_1 -5e-309)
     t_2
     (if (<= t_1 0.0) (* (* 5.0 (pow x 4.0)) eps) t_2))))

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 t_2 = t_0 - ((x * x) * ((x * x) * x));
	double tmp;
	if (t_1 <= -5e-309) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (x + eps) ** 5.0d0
    t_1 = t_0 - (x ** 5.0d0)
    t_2 = t_0 - ((x * x) * ((x * x) * x))
    if (t_1 <= (-5d-309)) then
        tmp = t_2
    else if (t_1 <= 0.0d0) then
        tmp = (5.0d0 * (x ** 4.0d0)) * eps
    else
        tmp = t_2
    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 t_2 = t_0 - ((x * x) * ((x * x) * x));
	double tmp;
	if (t_1 <= -5e-309) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (5.0 * Math.pow(x, 4.0)) * eps;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0)
	t_1 = t_0 - math.pow(x, 5.0)
	t_2 = t_0 - ((x * x) * ((x * x) * x))
	tmp = 0
	if t_1 <= -5e-309:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (5.0 * math.pow(x, 4.0)) * eps
	else:
		tmp = t_2
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x + eps) ^ 5.0
	t_1 = Float64(t_0 - (x ^ 5.0))
	t_2 = Float64(t_0 - Float64(Float64(x * x) * Float64(Float64(x * x) * x)))
	tmp = 0.0
	if (t_1 <= -5e-309)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (x + eps) ^ 5.0;
	t_1 = t_0 - (x ^ 5.0);
	t_2 = t_0 - ((x * x) * ((x * x) * x));
	tmp = 0.0;
	if (t_1 <= -5e-309)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (5.0 * (x ^ 4.0)) * eps;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-309], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.9%

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

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 85.6

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

   3. Applied rewrites 85.6%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 4: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon, \mathsf{fma}\left(\varepsilon \cdot x, 10, \left(\varepsilon \cdot \varepsilon\right) \cdot 5\right) \cdot x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-309)
     (*
      (* (fma (* (fma 10.0 x (* 5.0 eps)) eps) x (* (* eps eps) eps)) eps)
      eps)
     (if (<= t_0 0.0)
       (* (* 5.0 (pow x 4.0)) eps)
       (*
        (fma (* eps eps) eps (* (fma (* eps x) 10.0 (* (* eps eps) 5.0)) x))
        (* eps eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = (fma((fma(10.0, x, (5.0 * eps)) * eps), x, ((eps * eps) * eps)) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = fma((eps * eps), eps, (fma((eps * x), 10.0, ((eps * eps) * 5.0)) * x)) * (eps * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = Float64(Float64(fma(Float64(fma(10.0, x, Float64(5.0 * eps)) * eps), x, Float64(Float64(eps * eps) * eps)) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = Float64(fma(Float64(eps * eps), eps, Float64(fma(Float64(eps * x), 10.0, Float64(Float64(eps * eps) * 5.0)) * x)) * Float64(eps * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-309], N[(N[(N[(N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(eps * eps), $MachinePrecision] * eps + N[(N[(N[(eps * x), $MachinePrecision] * 10.0 + N[(N[(eps * eps), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

   12. Applied rewrites 88.3%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 82.3

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

   6. Applied rewrites 82.3%

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

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

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

Alternative 5: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(10, x, 5 \cdot \varepsilon\right) \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \varepsilon\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -5e-309)
     (*
      (* (fma (* (fma 10.0 x (* 5.0 eps)) eps) x (* (* eps eps) eps)) eps)
      eps)
     (if (<= t_0 0.0)
       (* (* 5.0 (pow x 4.0)) eps)
       (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) (* eps eps))))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = (fma((fma(10.0, x, (5.0 * eps)) * eps), x, ((eps * eps) * eps)) * eps) * eps;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * (eps * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = Float64(Float64(fma(Float64(fma(10.0, x, Float64(5.0 * eps)) * eps), x, Float64(Float64(eps * eps) * eps)) * eps) * eps);
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * Float64(eps * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-309], N[(N[(N[(N[(N[(10.0 * x + N[(5.0 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

   12. Applied rewrites 88.3%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 82.3

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

   6. Applied rewrites 82.3%

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

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

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

   11. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 88.2

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

   13. Applied rewrites 88.2%

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

Alternative 6: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\mathsf{fma}\left(\mathsf{fma}\left(5, x, \varepsilon\right), \varepsilon, \left(x \cdot x\right) \cdot 10\right) \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-309}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(5 \cdot {x}^{4}\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (* (* (fma (fma 5.0 x eps) eps (* (* x x) 10.0)) eps) (* eps eps))))
   (if (<= t_0 -5e-309)
     t_1
     (if (<= t_0 0.0) (* (* 5.0 (pow x 4.0)) eps) t_1))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double t_1 = (fma(fma(5.0, x, eps), eps, ((x * x) * 10.0)) * eps) * (eps * eps);
	double tmp;
	if (t_0 <= -5e-309) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (5.0 * pow(x, 4.0)) * eps;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	t_1 = Float64(Float64(fma(fma(5.0, x, eps), eps, Float64(Float64(x * x) * 10.0)) * eps) * Float64(eps * eps))
	tmp = 0.0
	if (t_0 <= -5e-309)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(5.0 * (x ^ 4.0)) * eps);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(5.0 * x + eps), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-309], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(5.0 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -4.9999999999999995e-309 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 88.2

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

   10. Applied rewrites 88.2%

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

   11. Taylor expanded in eps around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. pow2 N/A

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

       11. lift-*.f64 88.2

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

   13. Applied rewrites 88.2%

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

   if -4.9999999999999995e-309 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 7: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot t\_0\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\varepsilon}, 5, 1\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -2.3e-32)
     (* (* (fma 10.0 eps (* 5.0 x)) t_0) eps)
     (if (<= x 4.4e-41)
       (* (fma (/ x eps) 5.0 1.0) (* (* eps eps) (* (* eps eps) eps)))
       (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (fma(10.0, eps, (5.0 * x)) * t_0) * eps;
	} else if (x <= 4.4e-41) {
		tmp = fma((x / eps), 5.0, 1.0) * ((eps * eps) * ((eps * eps) * eps));
	} else {
		tmp = fma((eps * x), 5.0, ((eps * eps) * 10.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * t_0) * eps);
	elseif (x <= 4.4e-41)
		tmp = Float64(fma(Float64(x / eps), 5.0, 1.0) * Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps)));
	else
		tmp = Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 4.4e-41], N[(N[(N[(x / eps), $MachinePrecision] * 5.0 + 1.0), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow3 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 82.5

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

   13. Applied rewrites 82.5%

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

   if -2.3000000000000001e-32 < x < 4.4e-41

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{{\varepsilon}^{5} \cdot \left(1 + \left(4 \cdot \frac{x}{\varepsilon} + \frac{x}{\varepsilon}\right)\right)} \]
   6. 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. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. pow-prod-up N/A

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

      11. lower-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 88.1

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

   7. Applied rewrites 88.1%

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

   if 4.4e-41 < x

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow3 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 82.5

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

   13. Applied rewrites 82.5%

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

Alternative 8: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot t\_0\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, \varepsilon, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot x\right) \cdot 5\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -2.3e-32)
     (* (* (fma 10.0 eps (* 5.0 x)) t_0) eps)
     (if (<= x 4.4e-41)
       (* (fma (* eps eps) eps (* (* (* eps eps) x) 5.0)) (* eps eps))
       (* (fma (* eps x) 5.0 (* (* eps eps) 10.0)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (fma(10.0, eps, (5.0 * x)) * t_0) * eps;
	} else if (x <= 4.4e-41) {
		tmp = fma((eps * eps), eps, (((eps * eps) * x) * 5.0)) * (eps * eps);
	} else {
		tmp = fma((eps * x), 5.0, ((eps * eps) * 10.0)) * t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * t_0) * eps);
	elseif (x <= 4.4e-41)
		tmp = Float64(fma(Float64(eps * eps), eps, Float64(Float64(Float64(eps * eps) * x) * 5.0)) * Float64(eps * eps));
	else
		tmp = Float64(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)) * t_0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 4.4e-41], N[(N[(N[(eps * eps), $MachinePrecision] * eps + N[(N[(N[(eps * eps), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow3 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 82.5

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

   13. Applied rewrites 82.5%

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

   if -2.3000000000000001e-32 < x < 4.4e-41

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow3 N/A

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

      3. pow2 N/A

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

      4. lower-fma.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 88.1

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

   10. Applied rewrites 88.1%

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

   if 4.4e-41 < x

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 N/A

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

       10. pow3 N/A

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

       11. lift-*.f64 N/A

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

       12. lift-*.f64 82.5

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

   13. Applied rewrites 82.5%

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

Alternative 9: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot t\_0\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -2.3e-32)
     (* (* (fma 10.0 eps (* 5.0 x)) t_0) eps)
     (if (<= x 4e-49) (pow eps 5.0) (* (* t_0 x) (* 5.0 eps))))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (fma(10.0, eps, (5.0 * x)) * t_0) * eps;
	} else if (x <= 4e-49) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = (t_0 * x) * (5.0 * eps);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * t_0) * eps);
	elseif (x <= 4e-49)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(Float64(t_0 * x) * Float64(5.0 * eps));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 4e-49], N[Power[eps, 5.0], $MachinePrecision], N[(N[(t$95$0 * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 88.2%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 88.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. pow3 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-*.f64 82.5

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

   13. Applied rewrites 82.5%

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

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 88.0

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

   6. Applied rewrites 88.0%

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

   if 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lower-*.f64 N/A

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

      18. pow3 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 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 10: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(t\_0 \cdot 5\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(5 \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (if (<= x -2.3e-32)
     (* (* t_0 5.0) eps)
     (if (<= x 4e-49) (pow eps 5.0) (* t_0 (* 5.0 eps))))))

C

double code(double x, double eps) {
	double t_0 = ((x * x) * x) * x;
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (t_0 * 5.0) * eps;
	} else if (x <= 4e-49) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0 * (5.0 * eps);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * x) * x) * x
    if (x <= (-2.3d-32)) then
        tmp = (t_0 * 5.0d0) * eps
    else if (x <= 4d-49) then
        tmp = eps ** 5.0d0
    else
        tmp = t_0 * (5.0d0 * eps)
    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 <= -2.3e-32) {
		tmp = (t_0 * 5.0) * eps;
	} else if (x <= 4e-49) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0 * (5.0 * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * x) * x) * x
	tmp = 0
	if x <= -2.3e-32:
		tmp = (t_0 * 5.0) * eps
	elif x <= 4e-49:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0 * (5.0 * eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(t_0 * 5.0) * eps);
	elseif (x <= 4e-49)
		tmp = eps ^ 5.0;
	else
		tmp = Float64(t_0 * Float64(5.0 * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * x) * x) * x;
	tmp = 0.0;
	if (x <= -2.3e-32)
		tmp = (t_0 * 5.0) * eps;
	elseif (x <= 4e-49)
		tmp = eps ^ 5.0;
	else
		tmp = t_0 * (5.0 * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], N[(N[(t$95$0 * 5.0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 4e-49], N[Power[eps, 5.0], $MachinePrecision], N[(t$95$0 * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. pow-prod-up N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. metadata-eval N/A

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

      11. lower-pow.f64 88.0

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

   6. Applied rewrites 88.0%

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

   if 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lower-*.f64 N/A

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

      18. pow3 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 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 11: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(x \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(t\_0 \cdot 5\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(5 \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* x x) x) x)))
   (if (<= x -2.3e-32)
     (* (* t_0 5.0) eps)
     (if (<= x 4e-49)
       (* (* eps eps) (* (* eps eps) eps))
       (* t_0 (* 5.0 eps))))))

C

double code(double x, double eps) {
	double t_0 = ((x * x) * x) * x;
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (t_0 * 5.0) * eps;
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = t_0 * (5.0 * eps);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x * x) * x) * x
    if (x <= (-2.3d-32)) then
        tmp = (t_0 * 5.0d0) * eps
    else if (x <= 4d-49) then
        tmp = (eps * eps) * ((eps * eps) * eps)
    else
        tmp = t_0 * (5.0d0 * eps)
    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 <= -2.3e-32) {
		tmp = (t_0 * 5.0) * eps;
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = t_0 * (5.0 * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((x * x) * x) * x
	tmp = 0
	if x <= -2.3e-32:
		tmp = (t_0 * 5.0) * eps
	elif x <= 4e-49:
		tmp = (eps * eps) * ((eps * eps) * eps)
	else:
		tmp = t_0 * (5.0 * eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(x * x) * x) * x)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(t_0 * 5.0) * eps);
	elseif (x <= 4e-49)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(t_0 * Float64(5.0 * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x * x) * x) * x;
	tmp = 0.0;
	if (x <= -2.3e-32)
		tmp = (t_0 * 5.0) * eps;
	elseif (x <= 4e-49)
		tmp = (eps * eps) * ((eps * eps) * eps);
	else
		tmp = t_0 * (5.0 * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], N[(N[(t$95$0 * 5.0), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 4e-49], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lower-*.f64 N/A

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

      18. pow3 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 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 12: 97.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* x x) x) x) (* 5.0 eps))))
   (if (<= x -2.3e-32)
     t_0
     (if (<= x 4e-49) (* (* eps eps) (* (* eps eps) eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((x * x) * x) * x) * (5.0 * eps);
	double tmp;
	if (x <= -2.3e-32) {
		tmp = t_0;
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) * x) * x) * (5.0d0 * eps)
    if (x <= (-2.3d-32)) then
        tmp = t_0
    else if (x <= 4d-49) then
        tmp = (eps * eps) * ((eps * eps) * eps)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((x * x) * x) * x) * (5.0 * eps);
	double tmp;
	if (x <= -2.3e-32) {
		tmp = t_0;
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((x * x) * x) * x) * (5.0 * eps)
	tmp = 0
	if x <= -2.3e-32:
		tmp = t_0
	elif x <= 4e-49:
		tmp = (eps * eps) * ((eps * eps) * eps)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(5.0 * eps))
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = t_0;
	elseif (x <= 4e-49)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((x * x) * x) * x) * (5.0 * eps);
	tmp = 0.0;
	if (x <= -2.3e-32)
		tmp = t_0;
	elseif (x <= 4e-49)
		tmp = (eps * eps) * ((eps * eps) * eps);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e-32], t$95$0, If[LessEqual[x, 4e-49], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e-32 or 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-up N/A

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

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lower-*.f64 N/A

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

      18. pow3 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 N/A

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

      20. lift-*.f64 N/A

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

      21. lower-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

Alternative 13: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(5 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.3e-32)
   (* (* 5.0 eps) (* (* x x) (* x x)))
   (if (<= x 4e-49)
     (* (* eps eps) (* (* eps eps) eps))
     (* (* (* 5.0 (* x x)) (* x x)) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (5.0 * eps) * ((x * x) * (x * x));
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = ((5.0 * (x * x)) * (x * x)) * eps;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-2.3d-32)) then
        tmp = (5.0d0 * eps) * ((x * x) * (x * x))
    else if (x <= 4d-49) then
        tmp = (eps * eps) * ((eps * eps) * eps)
    else
        tmp = ((5.0d0 * (x * x)) * (x * x)) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -2.3e-32) {
		tmp = (5.0 * eps) * ((x * x) * (x * x));
	} else if (x <= 4e-49) {
		tmp = (eps * eps) * ((eps * eps) * eps);
	} else {
		tmp = ((5.0 * (x * x)) * (x * x)) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -2.3e-32:
		tmp = (5.0 * eps) * ((x * x) * (x * x))
	elif x <= 4e-49:
		tmp = (eps * eps) * ((eps * eps) * eps)
	else:
		tmp = ((5.0 * (x * x)) * (x * x)) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.3e-32)
		tmp = Float64(Float64(5.0 * eps) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 4e-49)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps));
	else
		tmp = Float64(Float64(Float64(5.0 * Float64(x * x)) * Float64(x * x)) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -2.3e-32)
		tmp = (5.0 * eps) * ((x * x) * (x * x));
	elseif (x <= 4e-49)
		tmp = (eps * eps) * ((eps * eps) * eps);
	else
		tmp = ((5.0 * (x * x)) * (x * x)) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.3e-32], N[(N[(5.0 * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4e-49], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(5.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3000000000000001e-32

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      5. pow-prod-up N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 87.9

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

   4. Applied rewrites 87.9%

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

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*l* N/A

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

      8. pow2 N/A

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

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow3 N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 87.9

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

   6. Applied rewrites 87.9%

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

   if 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 82.3

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

   4. Applied rewrites 82.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 82.3

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

   6. Applied rewrites 82.3%

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

Alternative 14: 97.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-49}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* 5.0 eps) (* (* x x) (* x x)))))
   (if (<= x -2.3e-32)
     t_0
     (if (<= x 4e-49) (* (* 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 <= -2.3e-32) {
		tmp = t_0;
	} else if (x <= 4e-49) {
		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 <= (-2.3d-32)) then
        tmp = t_0
    else if (x <= 4d-49) 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 <= -2.3e-32) {
		tmp = t_0;
	} else if (x <= 4e-49) {
		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 <= -2.3e-32:
		tmp = t_0
	elif x <= 4e-49:
		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 <= -2.3e-32)
		tmp = t_0;
	elseif (x <= 4e-49)
		tmp = Float64(Float64(eps * eps) * Float64(Float64(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 <= -2.3e-32)
		tmp = t_0;
	elseif (x <= 4e-49)
		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, -2.3e-32], t$95$0, If[LessEqual[x, 4e-49], N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3000000000000001e-32 or 3.99999999999999975e-49 < x

   1. Initial program 88.9%

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

   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. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot {\color{blue}{x}}^{2}\right) \]

      11. unpow2 N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

      12. lower-*.f64 82.3

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{x}\right)\right) \]

   4. Applied rewrites 82.3%

      \[\leadsto \color{blue}{\left(5 \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   if -2.3000000000000001e-32 < x < 3.99999999999999975e-49

   1. Initial program 88.9%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   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. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      5. pow-prod-up N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 87.9

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   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. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. associate-*l* N/A

         \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \varepsilon\right)} \]

      8. pow2 N/A

         \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      9. unpow3 N/A

         \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

      10. lower-*.f64 N/A

          \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

      11. pow2 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

      13. pow3 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

      15. lift-*.f64 87.9

          \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   6. Applied rewrites 87.9%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 87.9% 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 88.9%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

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. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

   5. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   10. lower-*.f64 87.9

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 87.9%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
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. lift-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   5. pow2 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   6. pow2 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. associate-*l* N/A

      \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \varepsilon\right)} \]

   8. pow2 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   9. unpow3 N/A

      \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

   10. lower-*.f64 N/A

       \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

   11. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   13. pow3 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

   15. lift-*.f64 87.9

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

6. Applied rewrites 87.9%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(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)))