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

Percentage Accurate: 87.6% → 98.9%

Time: 5.4s

Alternatives: 10

Speedup: 5.4×

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: 87.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-294}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (pow (+ x eps) 5.0) (pow x 5.0))))
   (if (<= t_0 -1e-294)
     t_0
     (if (<= t_0 0.0) (* (* (pow x 4.0) 5.0) 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-294) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    if (t_0 <= (-1d-294)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((x ** 4.0d0) * 5.0d0) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	double tmp;
	if (t_0 <= -1e-294) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (Math.pow(x, 4.0) * 5.0) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	tmp = 0
	if t_0 <= -1e-294:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (math.pow(x, 4.0) * 5.0) * 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-294)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((x + eps) ^ 5.0) - (x ^ 5.0);
	tmp = 0.0;
	if (t_0 <= -1e-294)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((x ^ 4.0) * 5.0) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-294], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $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))) < -1.00000000000000002e-294 or 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64)))

   1. Initial program 97.5%

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

   if -1.00000000000000002e-294 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 5 binary64)) (pow.f64 x #s(literal 5 binary64))) < 0.0

   1. Initial program 88.8%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

Alternative 2: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -4, \left(\varepsilon \cdot \varepsilon\right) \cdot -6\right)}{x}, -1, \mathsf{fma}\left(4, \varepsilon, \varepsilon\right)\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (*
    (fma
     (/ (fma (* eps eps) -4.0 (* (* eps eps) -6.0)) x)
     -1.0
     (fma 4.0 eps eps))
    (pow x 4.0))
   (if (<= x 7.2e-46)
     (fma x (* (* 5.0 (* eps eps)) (* eps eps)) (pow eps 5.0))
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma((fma((eps * eps), -4.0, ((eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * pow(x, 4.0);
	} else if (x <= 7.2e-46) {
		tmp = fma(x, ((5.0 * (eps * eps)) * (eps * eps)), pow(eps, 5.0));
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(Float64(fma(Float64(eps * eps), -4.0, Float64(Float64(eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * (x ^ 4.0));
	elseif (x <= 7.2e-46)
		tmp = fma(x, Float64(Float64(5.0 * Float64(eps * eps)) * Float64(eps * eps)), (eps ^ 5.0));
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -4.0 + N[(N[(eps * eps), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(4.0 * eps + eps), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(x * N[(N[(5.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   if -3.10000000000000011e-40 < x < 7.2e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 7.2e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 3: 97.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(5, \varepsilon, \frac{\left(\varepsilon \cdot \varepsilon\right) \cdot 10}{x}\right) \cdot {x}^{4}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (* (fma 5.0 eps (/ (* (* eps eps) 10.0) x)) (pow x 4.0))
   (if (<= x 7.2e-46)
     (fma x (* (* 5.0 (* eps eps)) (* eps eps)) (pow eps 5.0))
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma(5.0, eps, (((eps * eps) * 10.0) / x)) * pow(x, 4.0);
	} else if (x <= 7.2e-46) {
		tmp = fma(x, ((5.0 * (eps * eps)) * (eps * eps)), pow(eps, 5.0));
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(5.0, eps, Float64(Float64(Float64(eps * eps) * 10.0) / x)) * (x ^ 4.0));
	elseif (x <= 7.2e-46)
		tmp = fma(x, Float64(Float64(5.0 * Float64(eps * eps)) * Float64(eps * eps)), (eps ^ 5.0));
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(5.0 * eps + N[(N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(x * N[(N[(5.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   8. Applied rewrites 89.9%

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

   if -3.10000000000000011e-40 < x < 7.2e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 7.2e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 4: 97.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -4, \left(\varepsilon \cdot \varepsilon\right) \cdot -6\right)}{x}, -1, \mathsf{fma}\left(4, \varepsilon, \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(x, \left(5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right), {\varepsilon}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (*
    (fma
     (/ (fma (* eps eps) -4.0 (* (* eps eps) -6.0)) x)
     -1.0
     (fma 4.0 eps eps))
    (* (* x x) (* x x)))
   (if (<= x 7.2e-46)
     (fma x (* (* 5.0 (* eps eps)) (* eps eps)) (pow eps 5.0))
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma((fma((eps * eps), -4.0, ((eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * ((x * x) * (x * x));
	} else if (x <= 7.2e-46) {
		tmp = fma(x, ((5.0 * (eps * eps)) * (eps * eps)), pow(eps, 5.0));
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(Float64(fma(Float64(eps * eps), -4.0, Float64(Float64(eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 7.2e-46)
		tmp = fma(x, Float64(Float64(5.0 * Float64(eps * eps)) * Float64(eps * eps)), (eps ^ 5.0));
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -4.0 + N[(N[(eps * eps), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(4.0 * eps + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(x * N[(N[(5.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + N[Power[eps, 5.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   8. Applied rewrites 89.5%

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

   if -3.10000000000000011e-40 < x < 7.2e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   if 7.2e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 5: 97.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -4, \left(\varepsilon \cdot \varepsilon\right) \cdot -6\right)}{x}, -1, \mathsf{fma}\left(4, \varepsilon, \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot {\varepsilon}^{4}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (*
    (fma
     (/ (fma (* eps eps) -4.0 (* (* eps eps) -6.0)) x)
     -1.0
     (fma 4.0 eps eps))
    (* (* x x) (* x x)))
   (if (<= x 7.2e-46)
     (* (fma 5.0 x eps) (pow eps 4.0))
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma((fma((eps * eps), -4.0, ((eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * ((x * x) * (x * x));
	} else if (x <= 7.2e-46) {
		tmp = fma(5.0, x, eps) * pow(eps, 4.0);
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(Float64(fma(Float64(eps * eps), -4.0, Float64(Float64(eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 7.2e-46)
		tmp = Float64(fma(5.0, x, eps) * (eps ^ 4.0));
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -4.0 + N[(N[(eps * eps), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(4.0 * eps + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(N[(5.0 * x + eps), $MachinePrecision] * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   8. Applied rewrites 89.5%

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

   if -3.10000000000000011e-40 < x < 7.2e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 99.9%

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

   if 7.2e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 6: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -4, \left(\varepsilon \cdot \varepsilon\right) \cdot -6\right)}{x}, -1, \mathsf{fma}\left(4, \varepsilon, \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-46}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{4} \cdot 5\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (*
    (fma
     (/ (fma (* eps eps) -4.0 (* (* eps eps) -6.0)) x)
     -1.0
     (fma 4.0 eps eps))
    (* (* x x) (* x x)))
   (if (<= x 7.2e-46)
     (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
     (* (* (pow x 4.0) 5.0) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma((fma((eps * eps), -4.0, ((eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * ((x * x) * (x * x));
	} else if (x <= 7.2e-46) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = (pow(x, 4.0) * 5.0) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(Float64(fma(Float64(eps * eps), -4.0, Float64(Float64(eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 7.2e-46)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64((x ^ 4.0) * 5.0) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -4.0 + N[(N[(eps * eps), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(4.0 * eps + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-46], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, 4.0], $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   8. Applied rewrites 89.5%

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

   if -3.10000000000000011e-40 < x < 7.2e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 99.9%

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

   if 7.2e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

Alternative 7: 97.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\varepsilon \cdot \varepsilon, -4, \left(\varepsilon \cdot \varepsilon\right) \cdot -6\right)}{x}, -1, \mathsf{fma}\left(4, \varepsilon, \varepsilon\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(5 \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -3.1e-40)
   (*
    (fma
     (/ (fma (* eps eps) -4.0 (* (* eps eps) -6.0)) x)
     -1.0
     (fma 4.0 eps eps))
    (* (* x x) (* x x)))
   (if (<= x 5.5e-46)
     (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps))
     (* (* (* (* 5.0 x) eps) x) (* x x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -3.1e-40) {
		tmp = fma((fma((eps * eps), -4.0, ((eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * ((x * x) * (x * x));
	} else if (x <= 5.5e-46) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = (((5.0 * x) * eps) * x) * (x * x);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = Float64(fma(Float64(fma(Float64(eps * eps), -4.0, Float64(Float64(eps * eps) * -6.0)) / x), -1.0, fma(4.0, eps, eps)) * Float64(Float64(x * x) * Float64(x * x)));
	elseif (x <= 5.5e-46)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = Float64(Float64(Float64(Float64(5.0 * x) * eps) * x) * Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -3.1e-40], N[(N[(N[(N[(N[(eps * eps), $MachinePrecision] * -4.0 + N[(N[(eps * eps), $MachinePrecision] * -6.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] * -1.0 + N[(4.0 * eps + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e-46], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(5.0 * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.10000000000000011e-40

   1. Initial program 46.4%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

   4. Applied rewrites 90.0%

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

   6. Applied rewrites 90.0%

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

   8. Applied rewrites 89.5%

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

   if -3.10000000000000011e-40 < x < 5.49999999999999983e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 99.9%

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

   if 5.49999999999999983e-46 < x

   1. Initial program 28.7%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 99.7

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.4%

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

   8. Applied rewrites 99.6%

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

Alternative 8: 97.4% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(5 \cdot x\right) \cdot \varepsilon\right) \cdot x\right) \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-46}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* 5.0 x) eps) x) (* x x))))
   (if (<= x -3.1e-40)
     t_0
     (if (<= x 5.5e-46) (* (* (fma 5.0 x eps) (* eps eps)) (* eps eps)) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((5.0 * x) * eps) * x) * (x * x);
	double tmp;
	if (x <= -3.1e-40) {
		tmp = t_0;
	} else if (x <= 5.5e-46) {
		tmp = (fma(5.0, x, eps) * (eps * eps)) * (eps * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(5.0 * x) * eps) * x) * Float64(x * x))
	tmp = 0.0
	if (x <= -3.1e-40)
		tmp = t_0;
	elseif (x <= 5.5e-46)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(eps * eps)) * Float64(eps * eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(5.0 * x), $MachinePrecision] * eps), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e-40], t$95$0, If[LessEqual[x, 5.5e-46], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.10000000000000011e-40 or 5.49999999999999983e-46 < x

   1. Initial program 36.8%

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

   2. Taylor expanded in eps around 0

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 93.8

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

   4. Applied rewrites 93.8%

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

   6. Applied rewrites 93.6%

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

   8. Applied rewrites 93.8%

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

   if -3.10000000000000011e-40 < x < 5.49999999999999983e-46

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 99.9%

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

Alternative 9: 83.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (((5.0 * x) * eps) * 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 * x) * eps) * x) * (x * x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 84.4

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

4. Applied rewrites 84.4%

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

6. Applied rewrites 84.4%

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

8. Applied rewrites 84.4%

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

Alternative 10: 83.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return eps * ((5.0 * (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 = eps * ((5.0d0 * (x * x)) * (x * x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.4%

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

2. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-pow.f64 84.4

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

4. Applied rewrites 84.4%

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

6. Applied rewrites 84.4%

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

Reproduce

herbie shell --seed 2024363 -o localize:costs -o setup:simplify -o generate:simplify
(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)))