ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.9% → 99.5%

Time: 7.0s

Alternatives: 7

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: 61.9% 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: 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 64.9%

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

   1. flip-- 65.0%

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

   2. div-inv 64.8%

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

   3. add-sqr-sqrt 64.7%

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

   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/ 99.5%

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

   4. *-rgt-identity 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 97.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-150) t_0 (/ eps (+ (* x 2.0) (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-150) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / 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 <= (-5d-150)) then
        tmp = t_0
    else
        tmp = eps / ((x * 2.0d0) + ((-0.5d0) * (eps / 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 <= -5e-150) {
		tmp = t_0;
	} else {
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = x - math.sqrt(((x * x) - eps))
	tmp = 0
	if t_0 <= -5e-150:
		tmp = t_0
	else:
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(x - sqrt(Float64(Float64(x * x) - eps)))
	tmp = 0.0
	if (t_0 <= -5e-150)
		tmp = t_0;
	else
		tmp = Float64(eps / Float64(Float64(x * 2.0) + Float64(-0.5 * Float64(eps / 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 <= -5e-150)
		tmp = t_0;
	else
		tmp = eps / ((x * 2.0) + (-0.5 * (eps / 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, -5e-150], t$95$0, N[(eps / N[(N[(x * 2.0), $MachinePrecision] + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -4.9999999999999999e-150

   1. Initial program 98.9%

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

   if -4.9999999999999999e-150 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.4%

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

      1. flip-- 7.5%

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

      2. div-inv 7.5%

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

      3. add-sqr-sqrt 7.5%

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

      4. associate--r- 99.7%

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

      5. +-commutative 99.7%

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

      6. *-un-lft-identity 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. distribute-rgt-out-- 99.7%

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

      9. pow2 99.7%

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

      10. metadata-eval 99.7%

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

      11. pow2 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. mul0-rgt 99.7%

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

      2. +-rgt-identity 99.7%

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

      3. associate-*r/ 100.0%

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

      4. *-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around 0 99.3%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -5 \cdot 10^{-150}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x \cdot 2 + -0.5 \cdot \frac{\varepsilon}{x}}\\ \end{array} \]

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 4e-108)
   (- x (sqrt (- eps)))
   (/ eps (+ x (+ x (* -0.5 (/ eps x)))))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 4e-108) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / 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 <= 4d-108) then
        tmp = x - sqrt(-eps)
    else
        tmp = eps / (x + (x + ((-0.5d0) * (eps / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 4e-108) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 4e-108:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = eps / (x + (x + (-0.5 * (eps / x))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 4e-108)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(eps / Float64(x + Float64(x + Float64(-0.5 * Float64(eps / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 4e-108)
		tmp = x - sqrt(-eps);
	else
		tmp = eps / (x + (x + (-0.5 * (eps / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 4e-108], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(eps / N[(x + N[(x + N[(-0.5 * N[(eps / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000016e-108

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 96.8%

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

      1. mul-1-neg 96.8%

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

   4. Simplified 96.8%

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

   if 4.00000000000000016e-108 < x

   1. Initial program 29.4%

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

      1. flip-- 29.6%

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

      2. div-inv 29.5%

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

      3. add-sqr-sqrt 29.6%

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

      4. associate--r- 99.5%

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

      5. +-commutative 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. distribute-rgt-out-- 99.5%

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

      9. pow2 99.5%

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

      10. metadata-eval 99.5%

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

      11. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. mul0-rgt 99.5%

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

      2. +-rgt-identity 99.5%

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

      3. associate-*r/ 99.8%

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

      4. *-rgt-identity 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in x around inf 78.4%

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

      1. *-commutative 78.4%

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

   8. Simplified 78.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)}\\ \end{array} \]

Alternative 4: 45.2% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\varepsilon}{x + \left(x + -0.5 \cdot \frac{\varepsilon}{x}\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. flip-- 65.0%

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

   2. div-inv 64.8%

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

   3. add-sqr-sqrt 64.7%

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

   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/ 99.5%

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

   4. *-rgt-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around inf 41.8%

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

   1. *-commutative 41.8%

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

8. Simplified 41.8%

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

9. Final simplification 41.8%

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

Alternative 5: 44.4% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

2. Taylor expanded in x around inf 41.2%

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

3. Final simplification 41.2%

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

Alternative 6: 5.3% 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 64.9%

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

   1. flip-- 65.0%

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

   2. div-inv 64.8%

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

   3. add-sqr-sqrt 64.7%

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

   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/ 99.5%

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

   4. *-rgt-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around inf 41.8%

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

   1. *-commutative 41.8%

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

8. Simplified 41.8%

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

9. Taylor expanded in eps around inf 5.5%

   \[\leadsto \color{blue}{-2 \cdot x} \]
10. Step-by-step derivation

    1. *-commutative 5.5%

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

11. Simplified 5.5%

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

12. Final simplification 5.5%

    \[\leadsto x \cdot -2 \]

Alternative 7: 11.4% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

   1. flip-- 65.0%

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

   2. div-inv 64.8%

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

   3. add-sqr-sqrt 64.7%

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

   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/ 99.5%

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

   4. *-rgt-identity 99.5%

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0 63.2%

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

   1. mul-1-neg 63.2%

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

8. Simplified 63.2%

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

9. Taylor expanded in eps around 0 11.0%

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

10. Final simplification 11.0%

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

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 2023305 
(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))))