ENA, Section 1.4, Exercise 4b, n=2

Percentage Accurate: 74.7% → 100.0%

Time: 4.7s

Alternatives: 4

Speedup: 23.2×

Specification

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

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

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) ** 2.0d0) - (x ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Initial Program: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

C

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

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) ** 2.0d0) - (x ** 2.0d0)
end function

Java

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

Python

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

Julia

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

MATLAB

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

Wolfram

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 17.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(2, x, \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (fma 2.0 x eps) eps))

C

double code(double x, double eps) {
	return fma(2.0, x, eps) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(2.0, x, eps) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(2.0 * x + eps), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 73.1%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right) \cdot \varepsilon} \]
6. Add Preprocessing

Alternative 2: 97.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(x + \varepsilon\right)}^{2} - {x}^{2} \leq 0:\\ \;\;\;\;\left(x + x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon + x\right) \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (pow (+ x eps) 2.0) (pow x 2.0)) 0.0)
   (* (+ x x) eps)
   (* (+ eps x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((pow((x + eps), 2.0) - pow(x, 2.0)) <= 0.0) {
		tmp = (x + x) * eps;
	} else {
		tmp = (eps + x) * eps;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((((x + eps) ** 2.0d0) - (x ** 2.0d0)) <= 0.0d0) then
        tmp = (x + x) * eps
    else
        tmp = (eps + x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.pow((x + eps), 2.0) - Math.pow(x, 2.0)) <= 0.0) {
		tmp = (x + x) * eps;
	} else {
		tmp = (eps + x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.pow((x + eps), 2.0) - math.pow(x, 2.0)) <= 0.0:
		tmp = (x + x) * eps
	else:
		tmp = (eps + x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0)) <= 0.0)
		tmp = Float64(Float64(x + x) * eps);
	else
		tmp = Float64(Float64(eps + x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((((x + eps) ^ 2.0) - (x ^ 2.0)) <= 0.0)
		tmp = (x + x) * eps;
	else
		tmp = (eps + x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(x + x), $MachinePrecision] * eps), $MachinePrecision], N[(N[(eps + x), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64))) < 0.0

   1. Initial program 58.5%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 58.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 58.5

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

   4. Applied rewrites 58.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 99.0%

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

   if 0.0 < (-.f64 (pow.f64 (+.f64 x eps) #s(literal 2 binary64)) (pow.f64 x #s(literal 2 binary64)))

   1. Initial program 98.1%

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

      1. lift--.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 98.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 90.0%

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

Alternative 3: 75.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

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

   1. lift--.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. difference-of-squares N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower--.f64 73.1

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 73.1

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

4. Applied rewrites 73.1%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 72.8%

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

Alternative 4: 72.2% accurate, 34.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \varepsilon \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.1%

   \[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{\varepsilon}^{2}} \]
4. Step-by-step derivation

5. Applied rewrites 69.2%

   \[\leadsto \color{blue}{\varepsilon \cdot \varepsilon} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))