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

Percentage Accurate: 88.3% → 98.7%

Time: 3.6s

Alternatives: 11

Speedup: 1.4×

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: 88.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left({x}^{4}, 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 6, x, {x}^{3} \cdot 4\right), \varepsilon, {x}^{4}\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     t_0
     (if (<= t_0 0.0)
       (*
        (fma
         (pow x 4.0)
         4.0
         (fma (fma (* (* x x) 6.0) x (* (pow x 3.0) 4.0)) eps (pow x 4.0)))
        eps)
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = fma(pow(x, 4.0), 4.0, fma(fma(((x * x) * 6.0), x, (pow(x, 3.0) * 4.0)), eps, pow(x, 4.0))) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma((x ^ 4.0), 4.0, fma(fma(Float64(Float64(x * x) * 6.0), x, Float64((x ^ 3.0) * 4.0)), eps, (x ^ 4.0))) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 4.0 + N[(N[(N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision] * x + N[(N[Power[x, 3.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] * eps + N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.8%

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

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in eps around 0

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

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{4}, 4, \mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 6, x, {x}^{3} \cdot 4\right), \varepsilon, {x}^{4}\right)\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, \left(5 \cdot \varepsilon\right) \cdot x\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     t_0
     (if (<= t_0 0.0)
       (* (fma (* eps eps) 10.0 (* (* 5.0 eps) x)) (pow x 3.0))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = fma((eps * eps), 10.0, ((5.0 * eps) * x)) * pow(x, 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(eps * eps), 10.0, Float64(Float64(5.0 * eps) * x)) * (x ^ 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[(eps * eps), $MachinePrecision] * 10.0 + N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.8%

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

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) \cdot {x}^{3} \]

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 98.9

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

   7. Applied rewrites 98.9%

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

Alternative 3: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, {\varepsilon}^{5}\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, \left(5 \cdot \varepsilon\right) \cdot x\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     (fma (* 5.0 (* (* eps eps) (* eps eps))) x (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (fma (* eps eps) 10.0 (* (* 5.0 eps) x)) (pow x 3.0))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = fma((5.0 * ((eps * eps) * (eps * eps))), x, pow(eps, 5.0));
	} else if (t_0 <= 0.0) {
		tmp = fma((eps * eps), 10.0, ((5.0 * eps) * x)) * pow(x, 3.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = fma(Float64(5.0 * Float64(Float64(eps * eps) * Float64(eps * eps))), x, (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(eps * eps), 10.0, Float64(Float64(5.0 * eps) * x)) * (x ^ 3.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], N[(N[(5.0 * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(eps * eps), $MachinePrecision] * 10.0 + N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284

   1. Initial program 97.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 93.4

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

   4. Applied rewrites 93.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 93.4

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

   6. Applied rewrites 93.4%

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

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) \cdot {x}^{3} \]

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 98.9

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

   7. Applied rewrites 98.9%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 4: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, \left(5 \cdot \varepsilon\right) \cdot x\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     (* (fma 5.0 (/ x eps) 1.0) (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (fma (* eps eps) 10.0 (* (* 5.0 eps) x)) (pow x 3.0))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = fma(5.0, (x / eps), 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = fma((eps * eps), 10.0, ((5.0 * eps) * x)) * pow(x, 3.0);
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(fma(Float64(eps * eps), 10.0, Float64(Float64(5.0 * eps) * x)) * (x ^ 3.0));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(eps * eps), $MachinePrecision] * 10.0 + N[(N[(5.0 * eps), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt1-in N/A

         \[\leadsto \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) \cdot {x}^{3} \]

      10. metadata-eval N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lower-pow.f64 98.9

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

   7. Applied rewrites 98.9%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 5: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(5, \frac{x}{\varepsilon}, 1\right) \cdot {\varepsilon}^{5}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     (* (fma 5.0 (/ x eps) 1.0) (pow eps 5.0))
     (if (<= t_0 0.0)
       (* (* (fma (/ eps x) 10.0 5.0) eps) (* (* x x) (* x x)))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = fma(5.0, (x / eps), 1.0) * pow(eps, 5.0);
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * ((x * x) * (x * x));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = Float64(fma(5.0, Float64(x / eps), 1.0) * (eps ^ 5.0));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], N[(N[(5.0 * N[(x / eps), $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 93.4

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

   4. Applied rewrites 93.4%

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

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   9. Applied rewrites 98.8%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 6: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;\left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\varepsilon}^{5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-284)
     (*
      (- (fma (- eps (- (* 5.0 x))) eps (* (* x x) 6.0)) (* (* x x) -4.0))
      (* (* eps eps) eps))
     (if (<= t_0 0.0)
       (* (* (fma (/ eps x) 10.0 5.0) eps) (* (* x x) (* x x)))
       (pow eps 5.0)))))

C

double code(double x, double eps) {
	double t_0 = pow((x + eps), 5.0) - pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = (fma((eps - -(5.0 * x)), eps, ((x * x) * 6.0)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * ((x * x) * (x * x));
	} else {
		tmp = pow(eps, 5.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = Float64(Float64(fma(Float64(eps - Float64(-Float64(5.0 * x))), eps, Float64(Float64(x * x) * 6.0)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = eps ^ 5.0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], N[(N[(N[(N[(eps - (-N[(5.0 * x), $MachinePrecision])), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[eps, 5.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -1.00000000000000004e-284

   1. Initial program 97.9%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   4. Applied rewrites 94.0%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   5. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right) \cdot {\varepsilon}^{\color{blue}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right) \cdot {\varepsilon}^{\color{blue}{3}} \]

   7. Applied rewrites 93.5%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lower-*.f64 93.3

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   9. Applied rewrites 93.3%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   9. Applied rewrites 98.8%

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

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

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. lower-pow.f64 92.7

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

   4. Applied rewrites 92.7%

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

Alternative 7: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ t_1 := \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0)))
        (t_1
         (*
          (- (fma (- eps (- (* 5.0 x))) eps (* (* x x) 6.0)) (* (* x x) -4.0))
          (* (* eps eps) eps))))
   (if (<= t_0 -1e-284)
     t_1
     (if (<= t_0 0.0)
       (* (* (fma (/ eps x) 10.0 5.0) eps) (* (* x x) (* x x)))
       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((eps - -(5.0 * x)), eps, ((x * x) * 6.0)) - ((x * x) * -4.0)) * ((eps * eps) * eps);
	double tmp;
	if (t_0 <= -1e-284) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (fma((eps / x), 10.0, 5.0) * eps) * ((x * x) * (x * x));
	} 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(Float64(eps - Float64(-Float64(5.0 * x))), eps, Float64(Float64(x * x) * 6.0)) - Float64(Float64(x * x) * -4.0)) * Float64(Float64(eps * eps) * eps))
	tmp = 0.0
	if (t_0 <= -1e-284)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * Float64(Float64(x * x) * Float64(x * x)));
	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[(eps - (-N[(5.0 * x), $MachinePrecision])), $MachinePrecision] * eps + N[(N[(x * x), $MachinePrecision] * 6.0), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-284], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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))) < -1.00000000000000004e-284 or -0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.8%

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

   2. Taylor expanded in eps around -inf

      \[\leadsto \color{blue}{-1 \cdot \left({\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -{\varepsilon}^{5} \cdot \left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \]

      3. *-commutative N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

      4. lower-*.f64 N/A

         \[\leadsto -\left(-1 \cdot \frac{x + \left(-1 \cdot \frac{-4 \cdot {x}^{2} + -1 \cdot \left(2 \cdot {x}^{2} + 4 \cdot {x}^{2}\right)}{\varepsilon} + 4 \cdot x\right)}{\varepsilon} - 1\right) \cdot {\varepsilon}^{5} \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{-\left(\left(-\frac{\mathsf{fma}\left(4, x, -\frac{\mathsf{fma}\left(-4, x \cdot x, -\left(x \cdot x\right) \cdot 6\right)}{\varepsilon}\right) + x}{\varepsilon}\right) - 1\right) \cdot {\varepsilon}^{5}} \]

   5. Taylor expanded in eps around 0

      \[\leadsto {\varepsilon}^{3} \cdot \color{blue}{\left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right) \cdot {\varepsilon}^{\color{blue}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(6 \cdot {x}^{2} + \varepsilon \cdot \left(\varepsilon - -1 \cdot \left(x + 4 \cdot x\right)\right)\right) - -4 \cdot {x}^{2}\right) \cdot {\varepsilon}^{\color{blue}{3}} \]

   7. Applied rewrites 93.5%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \color{blue}{{\varepsilon}^{3}} \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot {\varepsilon}^{3} \]

      2. unpow3 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      3. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

      5. pow2 N/A

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

      6. lower-*.f64 93.2

         \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   9. Applied rewrites 93.2%

      \[\leadsto \left(\mathsf{fma}\left(\varepsilon - \left(-5 \cdot x\right), \varepsilon, \left(x \cdot x\right) \cdot 6\right) - \left(x \cdot x\right) \cdot -4\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   if -1.00000000000000004e-284 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < -0.0

   1. Initial program 86.2%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 98.8

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

   9. Applied rewrites 98.8%

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

Alternative 8: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.75 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(10, {\varepsilon}^{3}, \mathsf{fma}\left(5 \cdot \varepsilon, x, --10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-92}:\\ \;\;\;\;\mathsf{fma}\left(5 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right), x, {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon \cdot \varepsilon, 10, \left(\varepsilon \cdot x\right) \cdot 5\right) \cdot {x}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2.75e-53)
   (*
    (fma
     10.0
     (pow eps 3.0)
     (* (fma (* 5.0 eps) x (- (* -10.0 (* eps eps)))) x))
    (* x x))
   (if (<= x 1.55e-92)
     (fma (* 5.0 (* (* eps eps) (* eps eps))) x (pow eps 5.0))
     (* (fma (* eps eps) 10.0 (* (* eps x) 5.0)) (pow x 3.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2.75e-53) {
		tmp = fma(10.0, pow(eps, 3.0), (fma((5.0 * eps), x, -(-10.0 * (eps * eps))) * x)) * (x * x);
	} else if (x <= 1.55e-92) {
		tmp = fma((5.0 * ((eps * eps) * (eps * eps))), x, pow(eps, 5.0));
	} else {
		tmp = fma((eps * eps), 10.0, ((eps * x) * 5.0)) * pow(x, 3.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2.75e-53)
		tmp = Float64(fma(10.0, (eps ^ 3.0), Float64(fma(Float64(5.0 * eps), x, Float64(-Float64(-10.0 * Float64(eps * eps)))) * x)) * Float64(x * x));
	elseif (x <= 1.55e-92)
		tmp = fma(Float64(5.0 * Float64(Float64(eps * eps) * Float64(eps * eps))), x, (eps ^ 5.0));
	else
		tmp = Float64(fma(Float64(eps * eps), 10.0, Float64(Float64(eps * x) * 5.0)) * (x ^ 3.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2.75e-53], N[(N[(10.0 * N[Power[eps, 3.0], $MachinePrecision] + N[(N[(N[(5.0 * eps), $MachinePrecision] * x + (-N[(-10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-92], N[(N[(5.0 * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[(eps * eps), $MachinePrecision] * 10.0 + N[(N[(eps * x), $MachinePrecision] * 5.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.75000000000000011e-53

   1. Initial program 41.5%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -1 \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 6, \frac{\mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 6, \varepsilon, {\varepsilon}^{3} \cdot 4\right)}{x}\right)\right)}{x}\right) + \varepsilon\right) \cdot {x}^{4}} \]

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(4 \cdot {\varepsilon}^{3} + \left(6 \cdot {\varepsilon}^{3} + x \cdot \left(-1 \cdot \left(-6 \cdot {\varepsilon}^{2} + -4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {\varepsilon}^{3} + \left(6 \cdot {\varepsilon}^{3} + x \cdot \left(-1 \cdot \left(-6 \cdot {\varepsilon}^{2} + -4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\right)\right) \cdot {x}^{\color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {\varepsilon}^{3} + \left(6 \cdot {\varepsilon}^{3} + x \cdot \left(-1 \cdot \left(-6 \cdot {\varepsilon}^{2} + -4 \cdot {\varepsilon}^{2}\right) + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right)\right)\right) \cdot {x}^{\color{blue}{2}} \]

   7. Applied rewrites 89.9%

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

   if -2.75000000000000011e-53 < x < 1.55e-92

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if 1.55e-92 < x

   1. Initial program 63.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 87.8%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 87.8

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-+r+ N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot 10 + x \cdot \left(\left(4 + 1\right) \cdot \varepsilon\right)\right) \cdot {x}^{3} \]

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-pow.f64 87.8

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

   10. Applied rewrites 87.8%

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

Alternative 9: 83.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (fma (/ eps x) 10.0 5.0) eps) (* (* x x) (* x x))))

C

double code(double x, double eps) {
	return (fma((eps / x), 10.0, 5.0) * eps) * ((x * x) * (x * x));
}

Julia

function code(x, eps)
	return Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * eps) * Float64(Float64(x * x) * Float64(x * x)))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

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

2. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 83.1%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 83.1

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

7. Applied rewrites 83.1%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. pow2 N/A

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

   6. lift-*.f64 N/A

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

   7. pow2 N/A

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

   8. lift-*.f64 83.1

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

9. Applied rewrites 83.1%

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

Alternative 10: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

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 = ((5.0d0 * eps) * (x * x)) * (x * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

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. lower-pow.f64 82.9

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

4. Applied rewrites 82.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. pow2 N/A

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

   15. lift-*.f64 82.9

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

8. Applied rewrites 82.9%

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

Alternative 11: 82.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

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 = (5.0d0 * eps) * ((x * x) * (x * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

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. lower-pow.f64 82.9

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

4. Applied rewrites 82.9%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 82.9

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

6. Applied rewrites 82.9%

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

Reproduce

herbie shell --seed 2025105 
(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)))