ENA, Section 1.4, Exercise 4d

Percentage Accurate: 87.9% → 89.3%

Time: 9.2s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = 3.087429202844854e-49
  2. eps = 1.5439980234394303e-142

Specification

\[\left(0 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

Math

\[\begin{array}{l} \\ x - \sqrt{x \cdot x - \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(((x * x) - eps))
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \sqrt{x \cdot x - \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(((x * x) - eps))
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.9 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\varepsilon + \left({x}^{2} - {x}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.9e-146)
   (/
    1.0
    (/ (+ x (hypot x (sqrt (- eps)))) (+ eps (- (pow x 2.0) (pow x 2.0)))))
   (- x (sqrt (- (* x x) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.9e-146) {
		tmp = 1.0 / ((x + hypot(x, sqrt(-eps))) / (eps + (pow(x, 2.0) - pow(x, 2.0))));
	} else {
		tmp = x - sqrt(((x * x) - eps));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -7.9e-146) {
		tmp = 1.0 / ((x + Math.hypot(x, Math.sqrt(-eps))) / (eps + (Math.pow(x, 2.0) - Math.pow(x, 2.0))));
	} else {
		tmp = x - Math.sqrt(((x * x) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -7.9e-146:
		tmp = 1.0 / ((x + math.hypot(x, math.sqrt(-eps))) / (eps + (math.pow(x, 2.0) - math.pow(x, 2.0))))
	else:
		tmp = x - math.sqrt(((x * x) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.9e-146)
		tmp = Float64(1.0 / Float64(Float64(x + hypot(x, sqrt(Float64(-eps)))) / Float64(eps + Float64((x ^ 2.0) - (x ^ 2.0)))));
	else
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -7.9e-146)
		tmp = 1.0 / ((x + hypot(x, sqrt(-eps))) / (eps + ((x ^ 2.0) - (x ^ 2.0))));
	else
		tmp = x - sqrt(((x * x) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.9e-146], N[(1.0 / N[(N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(eps + N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.90000000000000013e-146

   1. Initial program 82.3%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.1%

         \[\leadsto \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]

      2. clear-num 81.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}} \]

      3. sub-neg 81.9%

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

      4. add-sqr-sqrt 81.9%

         \[\leadsto \frac{1}{\frac{x + \sqrt{x \cdot x + \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

      5. hypot-define 81.9%

         \[\leadsto \frac{1}{\frac{x + \color{blue}{\mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

      6. add-sqr-sqrt 81.9%

         \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{x \cdot x - \color{blue}{\left(x \cdot x - \varepsilon\right)}}} \]

      7. associate--r- 98.3%

         \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\color{blue}{\left(x \cdot x - x \cdot x\right) + \varepsilon}}} \]

      8. pow2 98.3%

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

      9. pow2 98.3%

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

   4. Applied egg-rr 98.3%

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

   if -7.90000000000000013e-146 < eps

   1. Initial program 93.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.9 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\varepsilon + \left({x}^{2} - {x}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -6.3e-146)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (- x (sqrt (- (* x x) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -6.3e-146) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = x - sqrt(((x * x) - eps));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -6.3e-146) {
		tmp = eps / (x + Math.hypot(x, Math.sqrt(-eps)));
	} else {
		tmp = x - Math.sqrt(((x * x) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -6.3e-146:
		tmp = eps / (x + math.hypot(x, math.sqrt(-eps)))
	else:
		tmp = x - math.sqrt(((x * x) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -6.3e-146)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(x - sqrt(Float64(Float64(x * x) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -6.3e-146)
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	else
		tmp = x - sqrt(((x * x) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -6.3e-146], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -6.2999999999999997e-146

   1. Initial program 82.3%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.1%

         \[\leadsto \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]

      2. div-inv 81.8%

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

      3. add-sqr-sqrt 81.9%

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

      4. associate--r- 98.2%

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

      5. pow2 98.2%

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

      6. pow2 98.2%

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

      7. sub-neg 98.2%

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

      8. add-sqr-sqrt 98.2%

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

      9. hypot-define 98.2%

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

   4. Applied egg-rr 98.2%

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

      1. associate-*r/ 98.2%

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

      2. +-inverses 98.2%

         \[\leadsto \frac{\left(\color{blue}{0} + \varepsilon\right) \cdot 1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      3. +-lft-identity 98.2%

         \[\leadsto \frac{\color{blue}{\varepsilon} \cdot 1}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

      4. *-rgt-identity 98.2%

         \[\leadsto \frac{\color{blue}{\varepsilon}}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)} \]

   6. Simplified 98.2%

      \[\leadsto \color{blue}{\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}} \]

   if -6.2999999999999997e-146 < eps

   1. Initial program 93.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -6.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-107}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.7e-107) (- x (sqrt (- eps))) (- x (+ x (/ (* eps -0.5) x)))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.7e-107) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 2.7d-107) then
        tmp = x - sqrt(-eps)
    else
        tmp = x - (x + ((eps * (-0.5d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.7e-107) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.7e-107:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = x - (x + ((eps * -0.5) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.7e-107)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(x - Float64(x + Float64(Float64(eps * -0.5) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.7e-107)
		tmp = x - sqrt(-eps);
	else
		tmp = x - (x + ((eps * -0.5) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.7e-107], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(x - N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7e-107

   1. Initial program 98.0%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 94.7%

      \[\leadsto x - \sqrt{\color{blue}{-1 \cdot \varepsilon}} \]
   4. Step-by-step derivation

      1. neg-mul-1 94.7%

         \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   5. Simplified 94.7%

      \[\leadsto x - \sqrt{\color{blue}{-\varepsilon}} \]

   if 2.7e-107 < x

   1. Initial program 81.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.4%

      \[\leadsto x - \color{blue}{\left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.4%

         \[\leadsto x - \left(x + \color{blue}{\frac{-0.5 \cdot \varepsilon}{x}}\right) \]

   5. Simplified 71.4%

      \[\leadsto x - \color{blue}{\left(x + \frac{-0.5 \cdot \varepsilon}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-107}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \sqrt{x \cdot x - \varepsilon} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x - sqrt(((x * x) - eps))
end function

Java

public static double code(double x, double eps) {
	return x - Math.sqrt(((x * x) - eps));
}

Python

def code(x, eps):
	return x - math.sqrt(((x * x) - eps))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing

3. Final simplification 89.9%

   \[\leadsto x - \sqrt{x \cdot x - \varepsilon} \]
4. Add Preprocessing

Alternative 5: 35.1% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} + x \cdot \frac{2}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -3.5e-157)
   (/ 1.0 (+ (/ -0.5 x) (* x (/ 2.0 eps))))
   (- x (+ x (/ (* eps -0.5) x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-157) {
		tmp = 1.0 / ((-0.5 / x) + (x * (2.0 / eps)));
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-3.5d-157)) then
        tmp = 1.0d0 / (((-0.5d0) / x) + (x * (2.0d0 / eps)))
    else
        tmp = x - (x + ((eps * (-0.5d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -3.5e-157) {
		tmp = 1.0 / ((-0.5 / x) + (x * (2.0 / eps)));
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -3.5e-157:
		tmp = 1.0 / ((-0.5 / x) + (x * (2.0 / eps)))
	else:
		tmp = x - (x + ((eps * -0.5) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -3.5e-157)
		tmp = Float64(1.0 / Float64(Float64(-0.5 / x) + Float64(x * Float64(2.0 / eps))));
	else
		tmp = Float64(x - Float64(x + Float64(Float64(eps * -0.5) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -3.5e-157)
		tmp = 1.0 / ((-0.5 / x) + (x * (2.0 / eps)));
	else
		tmp = x - (x + ((eps * -0.5) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -3.5e-157], N[(1.0 / N[(N[(-0.5 / x), $MachinePrecision] + N[(x * N[(2.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.5000000000000002e-157

   1. Initial program 82.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip-- 82.2%

         \[\leadsto \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]

      2. clear-num 82.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}} \]

      3. sub-neg 82.0%

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

      4. add-sqr-sqrt 82.0%

         \[\leadsto \frac{1}{\frac{x + \sqrt{x \cdot x + \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

      5. hypot-define 82.0%

         \[\leadsto \frac{1}{\frac{x + \color{blue}{\mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

      6. add-sqr-sqrt 82.0%

         \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{x \cdot x - \color{blue}{\left(x \cdot x - \varepsilon\right)}}} \]

      7. associate--r- 97.3%

         \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\color{blue}{\left(x \cdot x - x \cdot x\right) + \varepsilon}}} \]

      8. pow2 97.3%

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

      9. pow2 97.3%

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

   4. Applied egg-rr 97.3%

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

   5. Taylor expanded in x around inf 0.0%

      \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot \frac{x}{\varepsilon}}} \]
   6. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}} \]

      2. fma-define 0.0%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

      3. associate-*r/ 0.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}\right)} \]

      4. unpow2 0.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{x}\right)} \]

      5. rem-square-sqrt 25.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{0.5 \cdot \color{blue}{-1}}{x}\right)} \]

      6. metadata-eval 25.0%

         \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{\color{blue}{-0.5}}{x}\right)} \]

   7. Simplified 25.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)}} \]
   8. Step-by-step derivation

      1. fma-undefine 25.0%

         \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + \frac{-0.5}{x}}} \]

      2. clear-num 25.1%

         \[\leadsto \frac{1}{2 \cdot \color{blue}{\frac{1}{\frac{\varepsilon}{x}}} + \frac{-0.5}{x}} \]

      3. un-div-inv 25.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{2}{\frac{\varepsilon}{x}}} + \frac{-0.5}{x}} \]

      4. clear-num 25.0%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{\frac{1}{\frac{x}{\varepsilon}}}} + \frac{-0.5}{x}} \]

      5. associate-/r/ 25.0%

         \[\leadsto \frac{1}{\frac{2}{\color{blue}{\frac{1}{x} \cdot \varepsilon}} + \frac{-0.5}{x}} \]

      6. associate-/l/ 24.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{2}{\varepsilon}}{\frac{1}{x}}} + \frac{-0.5}{x}} \]

      7. +-commutative 24.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{x} + \frac{\frac{2}{\varepsilon}}{\frac{1}{x}}}} \]

      8. clear-num 25.0%

         \[\leadsto \frac{1}{\frac{-0.5}{x} + \color{blue}{\frac{1}{\frac{\frac{1}{x}}{\frac{2}{\varepsilon}}}}} \]

      9. associate-/r* 25.1%

         \[\leadsto \frac{1}{\frac{-0.5}{x} + \frac{1}{\color{blue}{\frac{1}{x \cdot \frac{2}{\varepsilon}}}}} \]

      10. clear-num 25.0%

          \[\leadsto \frac{1}{\frac{-0.5}{x} + \color{blue}{\frac{x \cdot \frac{2}{\varepsilon}}{1}}} \]

      11. /-rgt-identity 25.0%

          \[\leadsto \frac{1}{\frac{-0.5}{x} + \color{blue}{x \cdot \frac{2}{\varepsilon}}} \]

   9. Applied egg-rr 25.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{-0.5}{x} + x \cdot \frac{2}{\varepsilon}}} \]

   if -3.5000000000000002e-157 < eps

   1. Initial program 93.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 53.5%

      \[\leadsto x - \color{blue}{\left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 53.5%

         \[\leadsto x - \left(x + \color{blue}{\frac{-0.5 \cdot \varepsilon}{x}}\right) \]

   5. Simplified 53.5%

      \[\leadsto x - \color{blue}{\left(x + \frac{-0.5 \cdot \varepsilon}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{1}{\frac{-0.5}{x} + x \cdot \frac{2}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-156}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.7e-156) (* 0.5 (/ eps x)) (- x (+ x (/ (* eps -0.5) x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.7e-156) {
		tmp = 0.5 * (eps / x);
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-1.7d-156)) then
        tmp = 0.5d0 * (eps / x)
    else
        tmp = x - (x + ((eps * (-0.5d0)) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -1.7e-156) {
		tmp = 0.5 * (eps / x);
	} else {
		tmp = x - (x + ((eps * -0.5) / x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.7e-156:
		tmp = 0.5 * (eps / x)
	else:
		tmp = x - (x + ((eps * -0.5) / x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.7e-156)
		tmp = Float64(0.5 * Float64(eps / x));
	else
		tmp = Float64(x - Float64(x + Float64(Float64(eps * -0.5) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.7e-156)
		tmp = 0.5 * (eps / x);
	else
		tmp = x - (x + ((eps * -0.5) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.7e-156], N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision], N[(x - N[(x + N[(N[(eps * -0.5), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.69999999999999995e-156

   1. Initial program 82.4%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 24.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]

   if -1.69999999999999995e-156 < eps

   1. Initial program 93.8%

      \[x - \sqrt{x \cdot x - \varepsilon} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 53.5%

      \[\leadsto x - \color{blue}{\left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ 53.5%

         \[\leadsto x - \left(x + \color{blue}{\frac{-0.5 \cdot \varepsilon}{x}}\right) \]

   5. Simplified 53.5%

      \[\leadsto x - \color{blue}{\left(x + \frac{-0.5 \cdot \varepsilon}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.7 \cdot 10^{-156}:\\ \;\;\;\;0.5 \cdot \frac{\varepsilon}{x}\\ \mathbf{else}:\\ \;\;\;\;x - \left(x + \frac{\varepsilon \cdot -0.5}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 18.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \frac{\varepsilon}{x} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 0.5 (/ eps x)))

C

double code(double x, double eps) {
	return 0.5 * (eps / x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.5d0 * (eps / x)
end function

Java

public static double code(double x, double eps) {
	return 0.5 * (eps / x);
}

Python

def code(x, eps):
	return 0.5 * (eps / x)

Julia

function code(x, eps)
	return Float64(0.5 * Float64(eps / x))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.5 * (eps / x);
end

Wolfram

code[x_, eps_] := N[(0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 16.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]

4. Final simplification 16.4%

   \[\leadsto 0.5 \cdot \frac{\varepsilon}{x} \]
5. Add Preprocessing

Alternative 8: 5.2% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -2 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 89.8%

      \[\leadsto \color{blue}{\frac{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}{x + \sqrt{x \cdot x - \varepsilon}}} \]

   2. clear-num 89.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x + \sqrt{x \cdot x - \varepsilon}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}}} \]

   3. sub-neg 89.6%

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

   4. add-sqr-sqrt 68.8%

      \[\leadsto \frac{1}{\frac{x + \sqrt{x \cdot x + \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

   5. hypot-define 68.8%

      \[\leadsto \frac{1}{\frac{x + \color{blue}{\mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}}{x \cdot x - \sqrt{x \cdot x - \varepsilon} \cdot \sqrt{x \cdot x - \varepsilon}}} \]

   6. add-sqr-sqrt 68.8%

      \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{x \cdot x - \color{blue}{\left(x \cdot x - \varepsilon\right)}}} \]

   7. associate--r- 63.2%

      \[\leadsto \frac{1}{\frac{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}{\color{blue}{\left(x \cdot x - x \cdot x\right) + \varepsilon}}} \]

   8. pow2 63.2%

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

   9. pow2 63.2%

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

4. Applied egg-rr 63.2%

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

5. Taylor expanded in x around inf 0.0%

   \[\leadsto \frac{1}{\color{blue}{0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x} + 2 \cdot \frac{x}{\varepsilon}}} \]
6. Step-by-step derivation

   1. +-commutative 0.0%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{x}{\varepsilon} + 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}}} \]

   2. fma-define 0.0%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, 0.5 \cdot \frac{{\left(\sqrt{-1}\right)}^{2}}{x}\right)}} \]

   3. associate-*r/ 0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \color{blue}{\frac{0.5 \cdot {\left(\sqrt{-1}\right)}^{2}}{x}}\right)} \]

   4. unpow2 0.0%

      \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{0.5 \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}}{x}\right)} \]

   5. rem-square-sqrt 17.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{0.5 \cdot \color{blue}{-1}}{x}\right)} \]

   6. metadata-eval 17.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{\color{blue}{-0.5}}{x}\right)} \]

7. Simplified 17.3%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(2, \frac{x}{\varepsilon}, \frac{-0.5}{x}\right)}} \]

8. Taylor expanded in x around 0 5.1%

   \[\leadsto \color{blue}{-2 \cdot x} \]

9. Final simplification 5.1%

   \[\leadsto x \cdot -2 \]
10. Add Preprocessing

Alternative 9: 3.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return x

Julia

function code(x, eps)
	return x
end

MATLAB

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

Wolfram

code[x_, eps_] := x

Derivation

1. Initial program 89.9%

   \[x - \sqrt{x \cdot x - \varepsilon} \]
2. Add Preprocessing

3. Taylor expanded in eps around inf 3.3%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 3.3%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{x \cdot x - \varepsilon}} \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (/ eps (+ x (sqrt (- (* x x) eps)))))

C

double code(double x, double eps) {
	return eps / (x + sqrt(((x * x) - eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps / (x + sqrt(((x * x) - eps)))
end function

Java

public static double code(double x, double eps) {
	return eps / (x + Math.sqrt(((x * x) - eps)));
}

Python

def code(x, eps):
	return eps / (x + math.sqrt(((x * x) - eps)))

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4d"
  :precision binary64
  :pre (and (and (<= 0.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))

  :herbie-target
  (/ eps (+ x (sqrt (- (* x x) eps))))

  (- x (sqrt (- (* x x) eps))))