ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.4% → 99.5%

Time: 9.6s

Alternatives: 8

Speedup: 1.0×

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: 62.4% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. flip-- 58.9%

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

   2. div-inv 58.7%

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

   3. add-sqr-sqrt 58.7%

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

   4. associate--r- 99.4%

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

   5. +-commutative 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. distribute-rgt-out-- 99.4%

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

   9. pow2 99.4%

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

   10. metadata-eval 99.4%

       \[\leadsto \left(\varepsilon + {x}^{2} \cdot \color{blue}{0}\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]

   11. pow2 99.4%

       \[\leadsto \left(\varepsilon + {x}^{2} \cdot 0\right) \cdot \frac{1}{x + \sqrt{\color{blue}{{x}^{2}} - \varepsilon}} \]

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(\varepsilon + {x}^{2} \cdot 0\right) \cdot \frac{1}{x + \sqrt{{x}^{2} - \varepsilon}}} \]
4. Step-by-step derivation

   1. mul0-rgt 99.4%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \frac{1}{x + \sqrt{{x}^{2} - \varepsilon}} \]

   2. +-rgt-identity 99.4%

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

   3. associate-*r/ 99.6%

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

   4. *-rgt-identity 99.6%

      \[\leadsto \frac{\color{blue}{\varepsilon}}{x + \sqrt{{x}^{2} - \varepsilon}} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}}} \]

6. Final simplification 99.6%

   \[\leadsto \frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}} \]

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(\frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}\right) + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152)
     t_0
     (+ (* 0.125 (* (/ eps (pow x 2.0)) (/ eps x))) (* (/ eps x) 0.5)))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = (0.125 * ((eps / pow(x, 2.0)) * (eps / x))) + ((eps / x) * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-152)) then
        tmp = t_0
    else
        tmp = (0.125d0 * ((eps / (x ** 2.0d0)) * (eps / x))) + ((eps / x) * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = (0.125 * ((eps / Math.pow(x, 2.0)) * (eps / x))) + ((eps / x) * 0.5);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-152:
		tmp = t_0
	else:
		tmp = (0.125 * ((eps / math.pow(x, 2.0)) * (eps / x))) + ((eps / x) * 0.5)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = Float64(Float64(0.125 * Float64(Float64(eps / (x ^ 2.0)) * Float64(eps / x))) + Float64(Float64(eps / x) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = (0.125 * ((eps / (x ^ 2.0)) * (eps / x))) + ((eps / x) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-152], t$95$0, N[(N[(0.125 * N[(N[(eps / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 98.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 90.2%

      \[\leadsto \color{blue}{0.125 \cdot \frac{{\varepsilon}^{2}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. unpow2 90.2%

         \[\leadsto 0.125 \cdot \frac{\color{blue}{\varepsilon \cdot \varepsilon}}{{x}^{3}} + 0.5 \cdot \frac{\varepsilon}{x} \]

      2. unpow3 90.2%

         \[\leadsto 0.125 \cdot \frac{\varepsilon \cdot \varepsilon}{\color{blue}{\left(x \cdot x\right) \cdot x}} + 0.5 \cdot \frac{\varepsilon}{x} \]

      3. unpow2 90.2%

         \[\leadsto 0.125 \cdot \frac{\varepsilon \cdot \varepsilon}{\color{blue}{{x}^{2}} \cdot x} + 0.5 \cdot \frac{\varepsilon}{x} \]

      4. times-frac 98.7%

         \[\leadsto 0.125 \cdot \color{blue}{\left(\frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}\right)} + 0.5 \cdot \frac{\varepsilon}{x} \]

   4. Applied egg-rr 98.7%

      \[\leadsto 0.125 \cdot \color{blue}{\left(\frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}\right)} + 0.5 \cdot \frac{\varepsilon}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \left(\frac{\varepsilon}{{x}^{2}} \cdot \frac{\varepsilon}{x}\right) + \frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]

Alternative 3: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-152) t_0 (/ (* eps 0.5) x))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = (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) :: t_0
    real(8) :: tmp
    t_0 = x - sqrt(((x * x) - eps))
    if (t_0 <= (-1d-152)) then
        tmp = t_0
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = x - Math.sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -1e-152) {
		tmp = t_0;
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -1e-152:
		tmp = t_0
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = x - sqrt(((x * x) - eps));
	tmp = 0.0;
	if (t_0 <= -1e-152)
		tmp = t_0;
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-152], t$95$0, N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -1.00000000000000007e-152

   1. Initial program 98.5%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   if -1.00000000000000007e-152 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.6%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 97.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. associate-*r/ 97.0%

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

      2. *-commutative 97.0%

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

   4. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot 0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \]

Alternative 4: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-116}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 5.1e-116) (- x (sqrt (- eps))) (/ (* eps 0.5) x)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 5.1e-116) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (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 <= 5.1d-116) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps * 0.5d0) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 5.1e-116) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps * 0.5) / x;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 5.1e-116:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps * 0.5) / x
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 5.1e-116)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps * 0.5) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 5.1e-116)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps * 0.5) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 5.1e-116], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps * 0.5), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.1000000000000002e-116

   1. Initial program 97.3%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around 0 95.9%

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

      1. mul-1-neg 95.9%

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

   4. Simplified 95.9%

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

   if 5.1000000000000002e-116 < x

   1. Initial program 24.1%

      \[x - \sqrt{x \cdot x - \varepsilon} \]

   2. Taylor expanded in x around inf 83.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
   3. Step-by-step derivation

      1. associate-*r/ 83.2%

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

      2. *-commutative 83.2%

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

   4. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{\varepsilon \cdot 0.5}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.1 \cdot 10^{-116}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot 0.5}{x}\\ \end{array} \]

Alternative 5: 43.8% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

   1. flip-- 58.9%

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

   2. div-inv 58.7%

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

   3. add-sqr-sqrt 58.7%

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

   4. associate--r- 99.4%

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

   5. +-commutative 99.4%

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

   6. *-un-lft-identity 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. distribute-rgt-out-- 99.4%

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

   9. pow2 99.4%

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

   10. metadata-eval 99.4%

       \[\leadsto \left(\varepsilon + {x}^{2} \cdot \color{blue}{0}\right) \cdot \frac{1}{x + \sqrt{x \cdot x - \varepsilon}} \]

   11. pow2 99.4%

       \[\leadsto \left(\varepsilon + {x}^{2} \cdot 0\right) \cdot \frac{1}{x + \sqrt{\color{blue}{{x}^{2}} - \varepsilon}} \]

3. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(\varepsilon + {x}^{2} \cdot 0\right) \cdot \frac{1}{x + \sqrt{{x}^{2} - \varepsilon}}} \]
4. Step-by-step derivation

   1. mul0-rgt 99.4%

      \[\leadsto \left(\varepsilon + \color{blue}{0}\right) \cdot \frac{1}{x + \sqrt{{x}^{2} - \varepsilon}} \]

   2. +-rgt-identity 99.4%

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

   3. associate-*r/ 99.6%

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

   4. *-rgt-identity 99.6%

      \[\leadsto \frac{\color{blue}{\varepsilon}}{x + \sqrt{{x}^{2} - \varepsilon}} \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\varepsilon}{x + \sqrt{{x}^{2} - \varepsilon}}} \]

6. Taylor expanded in eps around 0 47.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 47.4%

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

   2. associate-/l* 47.3%

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

8. Simplified 47.3%

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

9. Final simplification 47.3%

   \[\leadsto \frac{0.5}{\frac{x}{\varepsilon}} \]

Alternative 6: 44.0% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around inf 47.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\varepsilon}{x}} \]
3. Step-by-step derivation

   1. associate-*r/ 47.4%

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

   2. *-commutative 47.4%

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

4. Simplified 47.4%

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

5. Final simplification 47.4%

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

Alternative 7: 4.3% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around inf 4.3%

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

3. Final simplification 4.3%

   \[\leadsto x - x \]

Alternative 8: 3.5% 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 59.0%

   \[x - \sqrt{x \cdot x - \varepsilon} \]

2. Taylor expanded in x around 0 53.9%

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

   1. mul-1-neg 53.9%

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

4. Simplified 53.9%

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

5. Taylor expanded in x around inf 3.8%

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

6. Final simplification 3.8%

   \[\leadsto x \]

Developer target: 99.5% 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 2023301 
(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))))