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

Percentage Accurate: 84.3% → 96.9%

Time: 6.2s

Alternatives: 6

Speedup: 29.6×

Specification

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

Math

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

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

real(8) function code(x, eps)
    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: 84.3% 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

real(8) function code(x, eps)
    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: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{2} - {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot \left(x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -4.8e-91)
   (- (pow (+ x eps) 2.0) (pow x 2.0))
   (fma eps eps (* eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -4.8e-91) {
		tmp = pow((x + eps), 2.0) - pow(x, 2.0);
	} else {
		tmp = fma(eps, eps, (eps * (x * 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -4.8e-91)
		tmp = Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0));
	else
		tmp = fma(eps, eps, Float64(eps * Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -4.8e-91], N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision], N[(eps * eps + N[(eps * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000022e-91

   1. Initial program 90.7%

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

   if -4.80000000000000022e-91 < x

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

      2. sqr-pow 84.4%

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

      3. sqr-pow 84.4%

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

      4. unpow2 84.4%

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

      5. unpow2 84.4%

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

      6. difference-of-squares 84.4%

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

      7. associate--l+ 99.3%

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

      8. +-inverses 99.3%

         \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

      9. +-rgt-identity 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-+l+ 99.3%

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

      12. *-lft-identity 99.3%

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

      13. metadata-eval 99.3%

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

      14. distribute-rgt-out 99.3%

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

      15. distribute-lft-out 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. distribute-lft-in 99.3%

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

      2. fma-def 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-91}:\\ \;\;\;\;{\left(x + \varepsilon\right)}^{2} - {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot \left(x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-91}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot \left(x \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -9e-91)
   (* (+ x (+ x eps)) (- (+ x eps) x))
   (fma eps eps (* eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -9e-91) {
		tmp = (x + (x + eps)) * ((x + eps) - x);
	} else {
		tmp = fma(eps, eps, (eps * (x * 2.0)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -9e-91)
		tmp = Float64(Float64(x + Float64(x + eps)) * Float64(Float64(x + eps) - x));
	else
		tmp = fma(eps, eps, Float64(eps * Float64(x * 2.0)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -9e-91], N[(N[(x + N[(x + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(eps * eps + N[(eps * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.99999999999999952e-91

   1. Initial program 90.7%

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

      1. unpow2 90.7%

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

      2. unpow2 90.7%

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

      3. difference-of-squares 90.6%

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

      4. +-commutative 90.6%

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

      5. +-commutative 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -8.99999999999999952e-91 < x

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

      2. sqr-pow 84.4%

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

      3. sqr-pow 84.4%

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

      4. unpow2 84.4%

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

      5. unpow2 84.4%

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

      6. difference-of-squares 84.4%

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

      7. associate--l+ 99.3%

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

      8. +-inverses 99.3%

         \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

      9. +-rgt-identity 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-+l+ 99.3%

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

      12. *-lft-identity 99.3%

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

      13. metadata-eval 99.3%

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

      14. distribute-rgt-out 99.3%

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

      15. distribute-lft-out 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. distribute-lft-in 99.3%

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

      2. fma-def 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-91}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\varepsilon, \varepsilon, \varepsilon \cdot \left(x \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-91}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -5.5e-91)
   (* (+ x (+ x eps)) (- (+ x eps) x))
   (* eps (+ eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -5.5e-91) {
		tmp = (x + (x + eps)) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-5.5d-91)) then
        tmp = (x + (x + eps)) * ((x + eps) - x)
    else
        tmp = eps * (eps + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -5.5e-91) {
		tmp = (x + (x + eps)) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -5.5e-91:
		tmp = (x + (x + eps)) * ((x + eps) - x)
	else:
		tmp = eps * (eps + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -5.5e-91)
		tmp = Float64(Float64(x + Float64(x + eps)) * Float64(Float64(x + eps) - x));
	else
		tmp = Float64(eps * Float64(eps + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -5.5e-91)
		tmp = (x + (x + eps)) * ((x + eps) - x);
	else
		tmp = eps * (eps + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -5.5e-91], N[(N[(x + N[(x + eps), $MachinePrecision]), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999965e-91

   1. Initial program 90.7%

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

      1. unpow2 90.7%

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

      2. unpow2 90.7%

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

      3. difference-of-squares 90.6%

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

      4. +-commutative 90.6%

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

      5. +-commutative 90.6%

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

   4. Applied egg-rr 90.6%

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

   if -5.49999999999999965e-91 < x

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

      2. sqr-pow 84.4%

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

      3. sqr-pow 84.4%

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

      4. unpow2 84.4%

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

      5. unpow2 84.4%

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

      6. difference-of-squares 84.4%

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

      7. associate--l+ 99.3%

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

      8. +-inverses 99.3%

         \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

      9. +-rgt-identity 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-+l+ 99.3%

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

      12. *-lft-identity 99.3%

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

      13. metadata-eval 99.3%

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

      14. distribute-rgt-out 99.3%

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

      15. distribute-lft-out 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-91}:\\ \;\;\;\;\left(x + \left(x + \varepsilon\right)\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -1.35e-90) (* (* x 2.0) (- (+ x eps) x)) (* eps (+ eps (* x 2.0)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-90) {
		tmp = (x * 2.0) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-1.35d-90)) then
        tmp = (x * 2.0d0) * ((x + eps) - x)
    else
        tmp = eps * (eps + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -1.35e-90) {
		tmp = (x * 2.0) * ((x + eps) - x);
	} else {
		tmp = eps * (eps + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -1.35e-90:
		tmp = (x * 2.0) * ((x + eps) - x)
	else:
		tmp = eps * (eps + (x * 2.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -1.35e-90)
		tmp = Float64(Float64(x * 2.0) * Float64(Float64(x + eps) - x));
	else
		tmp = Float64(eps * Float64(eps + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -1.35e-90)
		tmp = (x * 2.0) * ((x + eps) - x);
	else
		tmp = eps * (eps + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -1.35e-90], N[(N[(x * 2.0), $MachinePrecision] * N[(N[(x + eps), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision], N[(eps * N[(eps + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.34999999999999998e-90

   1. Initial program 90.7%

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

      1. unpow2 90.7%

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

      2. unpow2 90.7%

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

      3. difference-of-squares 90.6%

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

      4. +-commutative 90.6%

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

      5. +-commutative 90.6%

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

   4. Applied egg-rr 90.6%

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

   5. Taylor expanded in eps around 0 75.6%

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

   if -1.34999999999999998e-90 < x

   1. Initial program 84.4%

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

      1. +-commutative 84.4%

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

      2. sqr-pow 84.4%

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

      3. sqr-pow 84.4%

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

      4. unpow2 84.4%

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

      5. unpow2 84.4%

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

      6. difference-of-squares 84.4%

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

      7. associate--l+ 99.3%

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

      8. +-inverses 99.3%

         \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

      9. +-rgt-identity 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-+l+ 99.3%

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

      12. *-lft-identity 99.3%

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

      13. metadata-eval 99.3%

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

      14. distribute-rgt-out 99.3%

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

      15. distribute-lft-out 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-90}:\\ \;\;\;\;\left(x \cdot 2\right) \cdot \left(\left(x + \varepsilon\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (eps + (x * 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return eps * (eps + (x * 2.0));
}

Python

def code(x, eps):
	return eps * (eps + (x * 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. sqr-pow 85.5%

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

   3. sqr-pow 85.5%

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

   4. unpow2 85.5%

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

   5. unpow2 85.5%

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

   6. difference-of-squares 85.5%

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

   7. associate--l+ 89.7%

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

   8. +-inverses 89.7%

      \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

   9. +-rgt-identity 89.7%

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

   10. *-commutative 89.7%

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

   11. associate-+l+ 89.7%

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

   12. *-lft-identity 89.7%

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

   13. metadata-eval 89.7%

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

   14. distribute-rgt-out 89.7%

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

   15. distribute-lft-out 89.7%

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

   16. metadata-eval 89.7%

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

   17. metadata-eval 89.7%

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

   18. metadata-eval 89.7%

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

3. Simplified 89.7%

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

5. Final simplification 89.7%

   \[\leadsto \varepsilon \cdot \left(\varepsilon + x \cdot 2\right) \]
6. Add Preprocessing

Alternative 6: 55.0% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * (x * eps)
end function

Java

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

Python

def code(x, eps):
	return 2.0 * (x * eps)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * (x * eps);
end

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. +-commutative 85.5%

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

   2. sqr-pow 85.5%

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

   3. sqr-pow 85.5%

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

   4. unpow2 85.5%

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

   5. unpow2 85.5%

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

   6. difference-of-squares 85.5%

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

   7. associate--l+ 89.7%

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

   8. +-inverses 89.7%

      \[\leadsto \left(\left(\varepsilon + x\right) + x\right) \cdot \left(\varepsilon + \color{blue}{0}\right) \]

   9. +-rgt-identity 89.7%

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

   10. *-commutative 89.7%

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

   11. associate-+l+ 89.7%

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

   12. *-lft-identity 89.7%

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

   13. metadata-eval 89.7%

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

   14. distribute-rgt-out 89.7%

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

   15. distribute-lft-out 89.7%

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

   16. metadata-eval 89.7%

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

   17. metadata-eval 89.7%

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

   18. metadata-eval 89.7%

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

3. Simplified 89.7%

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

5. Taylor expanded in eps around 0 52.6%

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

6. Final simplification 52.6%

   \[\leadsto 2 \cdot \left(x \cdot \varepsilon\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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)))