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

Percentage Accurate: 89.2% → 98.5%

Time: 3.6s

Alternatives: 19

Speedup: 1.6×

Specification

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% 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: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(-\frac{-10 \cdot \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{x}\right)}{x}\right) + 5\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0
         (*
          (*
           (+ (- (/ (* -10.0 (+ eps (/ (* eps eps) x))) x)) 5.0)
           (* (* (* x x) x) x))
          eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (- (pow (+ x eps) 5.0) (pow x 5.0)) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((-((-10.0 * (eps + ((eps * eps) / x))) / x) + 5.0) * (((x * x) * x) * x)) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = pow((x + eps), 5.0) - pow(x, 5.0);
	} 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 = ((-(((-10.0d0) * (eps + ((eps * eps) / x))) / x) + 5.0d0) * (((x * x) * x) * x)) * eps
    if (x <= (-1.65d-44)) then
        tmp = t_0
    else if (x <= 9.5d-50) then
        tmp = ((x + eps) ** 5.0d0) - (x ** 5.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((-((-10.0 * (eps + ((eps * eps) / x))) / x) + 5.0) * (((x * x) * x) * x)) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = Math.pow((x + eps), 5.0) - Math.pow(x, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((-((-10.0 * (eps + ((eps * eps) / x))) / x) + 5.0) * (((x * x) * x) * x)) * eps
	tmp = 0
	if x <= -1.65e-44:
		tmp = t_0
	elif x <= 9.5e-50:
		tmp = math.pow((x + eps), 5.0) - math.pow(x, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(-Float64(Float64(-10.0 * Float64(eps + Float64(Float64(eps * eps) / x))) / x)) + 5.0) * Float64(Float64(Float64(x * x) * x) * x)) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64((Float64(x + eps) ^ 5.0) - (x ^ 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((-((-10.0 * (eps + ((eps * eps) / x))) / x) + 5.0) * (((x * x) * x) * x)) * eps;
	tmp = 0.0;
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = ((x + eps) ^ 5.0) - (x ^ 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[((-N[(N[(-10.0 * N[(eps + N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]) + 5.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \left({x}^{4} \cdot \left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right)\right) \cdot \varepsilon \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

   7. Applied rewrites 94.1%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(-\frac{-10 \cdot \left(\varepsilon + \frac{\varepsilon \cdot \varepsilon}{x}\right)}{x}\right) + 5\right) \cdot \left(t\_0 \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -3.2e-61)
     (*
      (fma
       (fma (* eps x) 5.0 (* (* eps eps) 10.0))
       x
       (* (* (* eps eps) eps) 10.0))
      (* x x))
     (if (<= x 9.5e-78)
       (- (pow (+ x eps) 5.0) (* (* x x) t_0))
       (*
        (* (+ (- (/ (* -10.0 (+ eps (/ (* eps eps) x))) x)) 5.0) (* t_0 x))
        eps)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -3.2e-61) {
		tmp = fma(fma((eps * x), 5.0, ((eps * eps) * 10.0)), x, (((eps * eps) * eps) * 10.0)) * (x * x);
	} else if (x <= 9.5e-78) {
		tmp = pow((x + eps), 5.0) - ((x * x) * t_0);
	} else {
		tmp = ((-((-10.0 * (eps + ((eps * eps) / x))) / x) + 5.0) * (t_0 * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -3.2e-61)
		tmp = Float64(fma(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)), x, Float64(Float64(Float64(eps * eps) * eps) * 10.0)) * Float64(x * x));
	elseif (x <= 9.5e-78)
		tmp = Float64((Float64(x + eps) ^ 5.0) - Float64(Float64(x * x) * t_0));
	else
		tmp = Float64(Float64(Float64(Float64(-Float64(Float64(-10.0 * Float64(eps + Float64(Float64(eps * eps) / x))) / x)) + 5.0) * Float64(t_0 * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e-61], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-78], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[(N[(-10.0 * N[(eps + N[(N[(eps * eps), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]) + 5.0), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-61

   1. Initial program 48.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.8%

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

   if -3.2000000000000001e-61 < x < 9.4999999999999997e-78

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   if 9.4999999999999997e-78 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \left({x}^{4} \cdot \left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right)\right) \cdot \varepsilon \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 + -1 \cdot \frac{-10 \cdot \varepsilon + -10 \cdot \frac{{\varepsilon}^{2}}{x}}{x}\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

   7. Applied rewrites 91.0%

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

Alternative 3: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\varepsilon \cdot x, 5, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right), x, \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot 10\right) \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(t\_0 \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -3.2e-61)
     (*
      (fma
       (fma (* eps x) 5.0 (* (* eps eps) 10.0))
       x
       (* (* (* eps eps) eps) 10.0))
      (* x x))
     (if (<= x 9.5e-78)
       (- (pow (+ x eps) 5.0) (* (* x x) t_0))
       (* (* (fma (/ eps x) 10.0 5.0) (* t_0 x)) eps)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -3.2e-61) {
		tmp = fma(fma((eps * x), 5.0, ((eps * eps) * 10.0)), x, (((eps * eps) * eps) * 10.0)) * (x * x);
	} else if (x <= 9.5e-78) {
		tmp = pow((x + eps), 5.0) - ((x * x) * t_0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * (t_0 * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -3.2e-61)
		tmp = Float64(fma(fma(Float64(eps * x), 5.0, Float64(Float64(eps * eps) * 10.0)), x, Float64(Float64(Float64(eps * eps) * eps) * 10.0)) * Float64(x * x));
	elseif (x <= 9.5e-78)
		tmp = Float64((Float64(x + eps) ^ 5.0) - Float64(Float64(x * x) * t_0));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * Float64(t_0 * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e-61], N[(N[(N[(N[(eps * x), $MachinePrecision] * 5.0 + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * x + N[(N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-78], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-61

   1. Initial program 48.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.8%

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

   if -3.2000000000000001e-61 < x < 9.4999999999999997e-78

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   if 9.4999999999999997e-78 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 90.4

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

   7. Applied rewrites 90.4%

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

Alternative 4: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-61}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-78}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{5} - \left(x \cdot x\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(t\_0 \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= x -3.2e-61)
     (* (* (fma (fma 10.0 eps (* 5.0 x)) x (* (* eps eps) 10.0)) (* x x)) eps)
     (if (<= x 9.5e-78)
       (- (pow (+ x eps) 5.0) (* (* x x) t_0))
       (* (* (fma (/ eps x) 10.0 5.0) (* t_0 x)) eps)))))

C

double code(double x, double eps) {
	double t_0 = (x * x) * x;
	double tmp;
	if (x <= -3.2e-61) {
		tmp = (fma(fma(10.0, eps, (5.0 * x)), x, ((eps * eps) * 10.0)) * (x * x)) * eps;
	} else if (x <= 9.5e-78) {
		tmp = pow((x + eps), 5.0) - ((x * x) * t_0);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * (t_0 * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= -3.2e-61)
		tmp = Float64(Float64(fma(fma(10.0, eps, Float64(5.0 * x)), x, Float64(Float64(eps * eps) * 10.0)) * Float64(x * x)) * eps);
	elseif (x <= 9.5e-78)
		tmp = Float64((Float64(x + eps) ^ 5.0) - Float64(Float64(x * x) * t_0));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * Float64(t_0 * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e-61], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 9.5e-78], N[(N[Power[N[(x + eps), $MachinePrecision], 5.0], $MachinePrecision] - N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(t$95$0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-61

   1. Initial program 48.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 92.9

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

   7. Applied rewrites 92.9%

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

   if -3.2000000000000001e-61 < x < 9.4999999999999997e-78

   1. Initial program 100.0%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. pow-prod-up N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow3 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 100.0

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

   3. Applied rewrites 100.0%

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

   if 9.4999999999999997e-78 < x

   1. Initial program 57.9%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 90.4

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

   7. Applied rewrites 90.4%

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

Alternative 5: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right), x, \left(\varepsilon \cdot \varepsilon\right) \cdot 10\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.65e-44)
   (* (* (fma (fma 10.0 eps (* 5.0 x)) x (* (* eps eps) 10.0)) (* x x)) eps)
   (if (<= x 9.5e-50)
     (* (/ (fma 5.0 x eps) eps) (* (* (* eps eps) (* eps eps)) eps))
     (* (* (fma (/ eps x) 10.0 5.0) (* (* (* x x) x) x)) eps))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.65e-44) {
		tmp = (fma(fma(10.0, eps, (5.0 * x)), x, ((eps * eps) * 10.0)) * (x * x)) * eps;
	} else if (x <= 9.5e-50) {
		tmp = (fma(5.0, x, eps) / eps) * (((eps * eps) * (eps * eps)) * eps);
	} else {
		tmp = (fma((eps / x), 10.0, 5.0) * (((x * x) * x) * x)) * eps;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = Float64(Float64(fma(fma(10.0, eps, Float64(5.0 * x)), x, Float64(Float64(eps * eps) * 10.0)) * Float64(x * x)) * eps);
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) / eps) * Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps));
	else
		tmp = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * Float64(Float64(Float64(x * x) * x) * x)) * eps);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.65e-44], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000003e-44

   1. Initial program 37.3%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 95.0

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

   7. Applied rewrites 95.0%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if 9.4999999999999993e-50 < x

   1. Initial program 40.4%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 93.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 92.7

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

   7. Applied rewrites 92.7%

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

Alternative 6: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{\mathsf{fma}\left(5, x, \varepsilon\right)}{\varepsilon} \cdot \left(\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma (/ eps x) 10.0 5.0) (* (* (* x x) x) x)) eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50)
       (* (/ (fma 5.0 x eps) eps) (* (* (* eps eps) (* eps eps)) eps))
       t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma((eps / x), 10.0, 5.0) * (((x * x) * x) * x)) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = (fma(5.0, x, eps) / eps) * (((eps * eps) * (eps * eps)) * eps);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(Float64(fma(Float64(eps / x), 10.0, 5.0) * Float64(Float64(Float64(x * x) * x) * x)) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) / eps) * Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] / eps), $MachinePrecision] * N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 7: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (fma (/ eps x) 10.0 5.0) (* (* (* x x) x) x)) eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (fma((eps / x), 10.0, 5.0) * (((x * x) * x) * x)) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		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(fma(Float64(eps / x), 10.0, 5.0) * Float64(Float64(Float64(x * x) * x) * x)) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(eps / x), $MachinePrecision] * 10.0 + 5.0), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.0%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{fma}\left(\frac{\varepsilon}{x}, 10, 5\right) \cdot {x}^{\left(3 + 1\right)}\right) \cdot \varepsilon \]

      8. pow-plus N/A

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

      9. lower-*.f64 N/A

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

      10. pow3 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 93.4

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

   7. Applied rewrites 93.4%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

Alternative 8: 98.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 5, 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(5 \cdot \varepsilon, x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.65e-44)
   (* (* (fma x 5.0 (* 10.0 eps)) eps) (* (* x x) x))
   (if (<= x 9.5e-50)
     (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps)
     (* (* (fma (* 5.0 eps) x (* 10.0 (* eps eps))) (* x x)) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.65e-44) {
		tmp = (fma(x, 5.0, (10.0 * eps)) * eps) * ((x * x) * x);
	} else if (x <= 9.5e-50) {
		tmp = (fma(5.0, x, eps) * ((eps * eps) * eps)) * eps;
	} else {
		tmp = (fma((5.0 * eps), x, (10.0 * (eps * eps))) * (x * x)) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = Float64(Float64(fma(x, 5.0, Float64(10.0 * eps)) * eps) * Float64(Float64(x * x) * x));
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = Float64(Float64(fma(Float64(5.0 * eps), x, Float64(10.0 * Float64(eps * eps))) * Float64(x * x)) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.65e-44], N[(N[(N[(x * 5.0 + N[(10.0 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(5.0 * eps), $MachinePrecision] * x + N[(10.0 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000003e-44

   1. Initial program 37.3%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.2

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

   10. Applied rewrites 94.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

          \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(x \cdot 5 + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 94.2

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

   12. Applied rewrites 94.2%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

   if 9.4999999999999993e-50 < x

   1. Initial program 40.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   9. Applied rewrites 92.6%

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

Alternative 9: 98.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, 5, 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.65e-44)
   (* (* (fma x 5.0 (* 10.0 eps)) eps) (* (* x x) x))
   (if (<= x 9.5e-50)
     (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps)
     (* (* (* (fma 10.0 eps (* 5.0 x)) eps) (* x x)) x))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.65e-44) {
		tmp = (fma(x, 5.0, (10.0 * eps)) * eps) * ((x * x) * x);
	} else if (x <= 9.5e-50) {
		tmp = (fma(5.0, x, eps) * ((eps * eps) * eps)) * eps;
	} else {
		tmp = ((fma(10.0, eps, (5.0 * x)) * eps) * (x * x)) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = Float64(Float64(fma(x, 5.0, Float64(10.0 * eps)) * eps) * Float64(Float64(x * x) * x));
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = Float64(Float64(Float64(fma(10.0, eps, Float64(5.0 * x)) * eps) * Float64(x * x)) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.65e-44], N[(N[(N[(x * 5.0 + N[(10.0 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000003e-44

   1. Initial program 37.3%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.2

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

   10. Applied rewrites 94.2%

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

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

          \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       3. +-commutative N/A

          \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       4. *-commutative N/A

          \[\leadsto \left(\left(x \cdot 5 + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 94.2

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

   12. Applied rewrites 94.2%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

   if 9.4999999999999993e-50 < x

   1. Initial program 40.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 92.6

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

   10. Applied rewrites 92.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 92.6

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

   12. Applied rewrites 92.6%

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

Alternative 10: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0 \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 \cdot \left(x \cdot x\right)\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (fma 10.0 eps (* 5.0 x)) eps)))
   (if (<= x -1.65e-44)
     (* t_0 (* (* x x) x))
     (if (<= x 9.5e-50)
       (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps)
       (* (* t_0 (* x x)) x)))))

C

double code(double x, double eps) {
	double t_0 = fma(10.0, eps, (5.0 * x)) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0 * ((x * x) * x);
	} else if (x <= 9.5e-50) {
		tmp = (fma(5.0, x, eps) * ((eps * eps) * eps)) * eps;
	} else {
		tmp = (t_0 * (x * x)) * x;
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = Float64(fma(10.0, eps, Float64(5.0 * x)) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = Float64(t_0 * Float64(Float64(x * x) * x));
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = Float64(Float64(t_0 * Float64(x * x)) * x);
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], N[(N[(t$95$0 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65000000000000003e-44

   1. Initial program 37.3%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 94.2

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

   10. Applied rewrites 94.2%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

   if 9.4999999999999993e-50 < x

   1. Initial program 40.4%

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

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 92.6%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 92.6

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

   10. Applied rewrites 92.6%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 92.6

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

   12. Applied rewrites 92.6%

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

Alternative 11: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\mathsf{fma}\left(10, \varepsilon, 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(x \cdot x\right)\right) \cdot x\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (fma 10.0 eps (* 5.0 x)) eps) (* x x)) x)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((fma(10.0, eps, (5.0 * x)) * eps) * (x * x)) * x;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		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(fma(10.0, eps, Float64(5.0 * x)) * eps) * Float64(x * x)) * x)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(10.0 * eps + N[(5.0 * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in x around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in eps around 0

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

      1. *-commutative N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(5 \cdot x + 10 \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\left(10 \cdot \varepsilon + 5 \cdot x\right) \cdot \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 93.4

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

   10. Applied rewrites 93.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 93.4

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

   12. Applied rewrites 93.4%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

Alternative 12: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* x x) x) x) 5.0) eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (fma 5.0 x eps) (* (* eps eps) eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		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(Float64(x * x) * x) * x) * 5.0) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(fma(5.0, x, eps) * Float64(Float64(eps * eps) * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 92.3

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

   6. Applied rewrites 92.3%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow-prod-up N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. pow-plus N/A

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

      14. lower-*.f64 N/A

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

      15. pow3 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 99.1

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in eps around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. pow3 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 99.1

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

   12. Applied rewrites 99.1%

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

Alternative 13: 98.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(5, x, \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* x x) x) x) 5.0) eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (fma 5.0 x eps) (* (* eps eps) (* eps eps))) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		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(Float64(x * x) * x) * x) * 5.0) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(fma(5.0, x, eps) * Float64(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[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(5.0 * x + eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

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

   2. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. pow-prod-up N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-plus N/A

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

      13. lower-*.f64 N/A

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

      14. pow3 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 92.3

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

   6. Applied rewrites 92.3%

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

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

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

   2. Taylor expanded in eps around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. metadata-eval N/A

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

      9. pow-plus-rev N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. pow-prod-up N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in eps around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. metadata-eval N/A

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

      6. pow-pow N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

Alternative 14: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;{\varepsilon}^{5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* x x) x) x) 5.0) eps)))
   (if (<= x -1.65e-44) t_0 (if (<= x 9.5e-50) (pow eps 5.0) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((x * x) * x) * x) * 5.0d0) * eps
    if (x <= (-1.65d-44)) then
        tmp = t_0
    else if (x <= 9.5d-50) then
        tmp = eps ** 5.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = Math.pow(eps, 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((((x * x) * x) * x) * 5.0) * eps
	tmp = 0
	if x <= -1.65e-44:
		tmp = t_0
	elif x <= 9.5e-50:
		tmp = math.pow(eps, 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 5.0) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	tmp = 0.0;
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = eps ^ 5.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[Power[eps, 5.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      7. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      11. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      12. lower-*.f64 92.1

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left(\left({x}^{2} \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      7. pow2 N/A

         \[\leadsto \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot 5\right) \cdot \varepsilon \]

      8. pow-prod-up N/A

         \[\leadsto \left({x}^{\left(2 + 2\right)} \cdot 5\right) \cdot \varepsilon \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{4} \cdot 5\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left({x}^{4} \cdot 5\right) \cdot \varepsilon \]

      11. metadata-eval N/A

          \[\leadsto \left({x}^{\left(3 + 1\right)} \cdot 5\right) \cdot \varepsilon \]

      12. pow-plus N/A

          \[\leadsto \left(\left({x}^{3} \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{3} \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      14. pow3 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      16. lift-*.f64 92.3

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

   6. Applied rewrites 92.3%

      \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      5. pow-prod-up N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 99.0

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\varepsilon} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      5. pow2 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. pow-prod-up N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      8. metadata-eval N/A

         \[\leadsto {\varepsilon}^{4} \cdot \varepsilon \]

      9. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{\color{blue}{\left(4 + 1\right)}} \]

      10. metadata-eval N/A

          \[\leadsto {\varepsilon}^{5} \]

      11. lower-pow.f64 99.2

          \[\leadsto {\varepsilon}^{\color{blue}{5}} \]

   6. Applied rewrites 99.2%

      \[\leadsto {\varepsilon}^{\color{blue}{5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* x x) x) x) 5.0) eps)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (* eps eps) (* eps eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((x * x) * x) * x) * 5.0d0) * eps
    if (x <= (-1.65d-44)) then
        tmp = t_0
    else if (x <= 9.5d-50) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((((x * x) * x) * x) * 5.0) * eps
	tmp = 0
	if x <= -1.65e-44:
		tmp = t_0
	elif x <= 9.5e-50:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * 5.0) * eps)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((((x * x) * x) * x) * 5.0) * eps;
	tmp = 0.0;
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 5.0), $MachinePrecision] * eps), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      7. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      11. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      12. lower-*.f64 92.1

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      2. *-commutative N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      6. pow2 N/A

         \[\leadsto \left(\left({x}^{2} \cdot \left(x \cdot x\right)\right) \cdot 5\right) \cdot \varepsilon \]

      7. pow2 N/A

         \[\leadsto \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot 5\right) \cdot \varepsilon \]

      8. pow-prod-up N/A

         \[\leadsto \left({x}^{\left(2 + 2\right)} \cdot 5\right) \cdot \varepsilon \]

      9. metadata-eval N/A

         \[\leadsto \left({x}^{4} \cdot 5\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left({x}^{4} \cdot 5\right) \cdot \varepsilon \]

      11. metadata-eval N/A

          \[\leadsto \left({x}^{\left(3 + 1\right)} \cdot 5\right) \cdot \varepsilon \]

      12. pow-plus N/A

          \[\leadsto \left(\left({x}^{3} \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{3} \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      14. pow3 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      15. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

      16. lift-*.f64 92.3

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

   6. Applied rewrites 92.3%

      \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot 5\right) \cdot \varepsilon \]

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      5. pow-prod-up N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 99.0

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* x x) x) x) (* 5.0 eps))))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (* eps eps) (* eps eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = (((x * x) * x) * x) * (5.0 * eps);
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) * x) * x) * (5.0d0 * eps)
    if (x <= (-1.65d-44)) then
        tmp = t_0
    else if (x <= 9.5d-50) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = (((x * x) * x) * x) * (5.0 * eps);
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = (((x * x) * x) * x) * (5.0 * eps)
	tmp = 0
	if x <= -1.65e-44:
		tmp = t_0
	elif x <= 9.5e-50:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(x * x) * x) * x) * Float64(5.0 * eps))
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = (((x * x) * x) * x) * (5.0 * eps);
	tmp = 0.0;
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(5.0 * eps), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\varepsilon \cdot \left(4 \cdot {x}^{4} + {x}^{4}\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(4 \cdot {x}^{4} + {x}^{4}\right) \cdot \color{blue}{\varepsilon} \]

      3. distribute-lft1-in N/A

         \[\leadsto \left(\left(4 + 1\right) \cdot {x}^{4}\right) \cdot \varepsilon \]

      4. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      5. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot {x}^{4}\right) \cdot \varepsilon \]

      6. metadata-eval N/A

         \[\leadsto \left(5 \cdot {x}^{\left(2 + 2\right)}\right) \cdot \varepsilon \]

      7. pow-prod-up N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(5 \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right) \cdot \varepsilon \]

      11. unpow2 N/A

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      12. lower-*.f64 92.1

          \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \color{blue}{\varepsilon} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(5 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon \]

      3. associate-*l* N/A

         \[\leadsto 5 \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto 5 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \]

      5. lift-*.f64 N/A

         \[\leadsto 5 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 5 \cdot \left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \]

      7. pow2 N/A

         \[\leadsto 5 \cdot \left(\left({x}^{2} \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) \]

      8. pow2 N/A

         \[\leadsto 5 \cdot \left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \varepsilon\right) \]

      9. pow-prod-up N/A

         \[\leadsto 5 \cdot \left({x}^{\left(2 + 2\right)} \cdot \varepsilon\right) \]

      10. metadata-eval N/A

          \[\leadsto 5 \cdot \left({x}^{4} \cdot \varepsilon\right) \]

      11. *-commutative N/A

          \[\leadsto 5 \cdot \left(\varepsilon \cdot \color{blue}{{x}^{4}}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(5 \cdot \varepsilon\right) \cdot \color{blue}{{x}^{4}} \]

      13. *-commutative N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      14. lower-*.f64 N/A

          \[\leadsto {x}^{4} \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

      15. metadata-eval N/A

          \[\leadsto {x}^{\left(3 + 1\right)} \cdot \left(5 \cdot \varepsilon\right) \]

      16. pow-plus N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      17. lower-*.f64 N/A

          \[\leadsto \left({x}^{3} \cdot x\right) \cdot \left(\color{blue}{5} \cdot \varepsilon\right) \]

      18. pow3 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      19. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \varepsilon\right) \]

      21. lower-*.f64 92.3

          \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(5 \cdot \color{blue}{\varepsilon}\right) \]

   6. Applied rewrites 92.3%

      \[\leadsto \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \color{blue}{\left(5 \cdot \varepsilon\right)} \]

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      5. pow-prod-up N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 99.0

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 97.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot x\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-50}:\\ \;\;\;\;\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (* (* (* (* x x) x) eps) 5.0) x)))
   (if (<= x -1.65e-44)
     t_0
     (if (<= x 9.5e-50) (* (* (* eps eps) (* eps eps)) eps) t_0))))

C

double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * eps) * 5.0) * x;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((((x * x) * x) * eps) * 5.0d0) * x
    if (x <= (-1.65d-44)) then
        tmp = t_0
    else if (x <= 9.5d-50) then
        tmp = ((eps * eps) * (eps * eps)) * eps
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = ((((x * x) * x) * eps) * 5.0) * x;
	double tmp;
	if (x <= -1.65e-44) {
		tmp = t_0;
	} else if (x <= 9.5e-50) {
		tmp = ((eps * eps) * (eps * eps)) * eps;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = ((((x * x) * x) * eps) * 5.0) * x
	tmp = 0
	if x <= -1.65e-44:
		tmp = t_0
	elif x <= 9.5e-50:
		tmp = ((eps * eps) * (eps * eps)) * eps
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(Float64(Float64(Float64(Float64(x * x) * x) * eps) * 5.0) * x)
	tmp = 0.0
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = ((((x * x) * x) * eps) * 5.0) * x;
	tmp = 0.0;
	if (x <= -1.65e-44)
		tmp = t_0;
	elseif (x <= 9.5e-50)
		tmp = ((eps * eps) * (eps * eps)) * eps;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision] * 5.0), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.65e-44], t$95$0, If[LessEqual[x, 9.5e-50], N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65000000000000003e-44 or 9.4999999999999993e-50 < x

   1. Initial program 39.0%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{{x}^{4} \cdot \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \cdot \color{blue}{{x}^{4}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\varepsilon + \left(-1 \cdot \frac{-4 \cdot {\varepsilon}^{2} + -1 \cdot \left(2 \cdot {\varepsilon}^{2} + 4 \cdot {\varepsilon}^{2}\right)}{x} + 4 \cdot \varepsilon\right)\right) \cdot \color{blue}{{x}^{4}} \]

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(4, \varepsilon, -\frac{\mathsf{fma}\left(-4, \varepsilon \cdot \varepsilon, -\left(\varepsilon \cdot \varepsilon\right) \cdot 6\right)}{x}\right) + \varepsilon\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto {x}^{3} \cdot \color{blue}{\left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{\color{blue}{3}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(6 \cdot {\varepsilon}^{2} + x \cdot \left(\varepsilon + 4 \cdot \varepsilon\right)\right) - -4 \cdot {\varepsilon}^{2}\right) \cdot {x}^{\color{blue}{3}} \]

   7. Applied rewrites 93.4%

      \[\leadsto \left(\mathsf{fma}\left(5 \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(5 \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{x}\right) \]

      2. lift--.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(5 \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{fma}\left(5 \cdot \varepsilon, x, \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

      11. pow2 N/A

          \[\leadsto \left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot \left({x}^{2} \cdot x\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(\left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot {x}^{2}\right) \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(5 \cdot \varepsilon\right) \cdot x + \left(\varepsilon \cdot \varepsilon\right) \cdot 6\right) - \left(\varepsilon \cdot \varepsilon\right) \cdot -4\right) \cdot {x}^{2}\right) \cdot x \]

   9. Applied rewrites 93.4%

      \[\leadsto \left(\mathsf{fma}\left(5 \cdot \varepsilon, x, 10 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \left(x \cdot x\right)\right) \cdot x \]

   10. Taylor expanded in x around inf

       \[\leadsto \left(5 \cdot \left(\varepsilon \cdot {x}^{3}\right)\right) \cdot x \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\varepsilon \cdot {x}^{3}\right) \cdot 5\right) \cdot x \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\varepsilon \cdot {x}^{3}\right) \cdot 5\right) \cdot x \]

       3. *-commutative N/A

          \[\leadsto \left(\left({x}^{3} \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left({x}^{3} \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

       5. pow3 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

       6. lift-*.f64 N/A

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

       7. lift-*.f64 92.2

          \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

   12. Applied rewrites 92.2%

       \[\leadsto \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \varepsilon\right) \cdot 5\right) \cdot x \]

   if -1.65000000000000003e-44 < x < 9.4999999999999993e-50

   1. Initial program 99.4%

      \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

      2. pow-plus-rev N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      3. lower-*.f64 N/A

         \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

      4. metadata-eval N/A

         \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

      5. pow-prod-up N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      6. lower-*.f64 N/A

         \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      7. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

      9. unpow2 N/A

         \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

      10. lower-*.f64 99.0

          \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 88.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* (* eps eps) (* eps eps)) eps))

C

double code(double x, double eps) {
	return ((eps * eps) * (eps * eps)) * eps;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * eps) * (eps * eps)) * eps
end function

Java

public static double code(double x, double eps) {
	return ((eps * eps) * (eps * eps)) * eps;
}

Python

def code(x, eps):
	return ((eps * eps) * (eps * eps)) * eps

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * eps) * Float64(eps * eps)) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * eps) * (eps * eps)) * eps;
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * eps), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

   5. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   10. lower-*.f64 88.2

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 88.2%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. Add Preprocessing

Alternative 19: 88.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (* eps eps) (* (* eps eps) eps)))

C

double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * eps) * ((eps * eps) * eps)
end function

Java

public static double code(double x, double eps) {
	return (eps * eps) * ((eps * eps) * eps);
}

Python

def code(x, eps):
	return (eps * eps) * ((eps * eps) * eps)

Julia

function code(x, eps)
	return Float64(Float64(eps * eps) * Float64(Float64(eps * eps) * eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * eps) * ((eps * eps) * eps);
end

Wolfram

code[x_, eps_] := N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[{\left(x + \varepsilon\right)}^{5} - {x}^{5} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{5}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(4 + \color{blue}{1}\right)} \]

   2. pow-plus-rev N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   3. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{4} \cdot \color{blue}{\varepsilon} \]

   4. metadata-eval N/A

      \[\leadsto {\varepsilon}^{\left(2 + 2\right)} \cdot \varepsilon \]

   5. pow-prod-up N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   6. lower-*.f64 N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   7. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   9. unpow2 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   10. lower-*.f64 88.2

       \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

4. Applied rewrites 88.2%

   \[\leadsto \color{blue}{\left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \color{blue}{\varepsilon} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon \]

   3. pow2 N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{2} \cdot \varepsilon \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(\varepsilon \cdot \varepsilon\right)}^{2} \cdot \varepsilon \]

   5. unpow-prod-down N/A

      \[\leadsto \left({\varepsilon}^{2} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon \]

   6. associate-*l* N/A

      \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{\left({\varepsilon}^{2} \cdot \varepsilon\right)} \]

   7. pow2 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   8. unpow3 N/A

      \[\leadsto {\varepsilon}^{2} \cdot {\varepsilon}^{\color{blue}{3}} \]

   9. lower-*.f64 N/A

      \[\leadsto {\varepsilon}^{2} \cdot \color{blue}{{\varepsilon}^{3}} \]

   10. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot {\color{blue}{\varepsilon}}^{3} \]

   12. unpow3 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \color{blue}{\varepsilon}\right) \]

   13. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \varepsilon\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left({\varepsilon}^{2} \cdot \color{blue}{\varepsilon}\right) \]

   15. pow2 N/A

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

   16. lift-*.f64 88.1

       \[\leadsto \left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]

6. Applied rewrites 88.1%

   \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025114 
(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)))