ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.8% → 99.1%

Time: 7.4s

Alternatives: 6

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: 62.8% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- x (sqrt (- (* x x) eps))) -1e-154)
   (/ eps (+ x (hypot x (sqrt (- eps)))))
   (/ eps (fma x 2.0 (* (/ eps x) -0.5)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x - sqrt(((x * x) - eps))) <= -1e-154) {
		tmp = eps / (x + hypot(x, sqrt(-eps)));
	} else {
		tmp = eps / fma(x, 2.0, ((eps / x) * -0.5));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(x - sqrt(Float64(Float64(x * x) - eps))) <= -1e-154)
		tmp = Float64(eps / Float64(x + hypot(x, sqrt(Float64(-eps)))));
	else
		tmp = Float64(eps / fma(x, 2.0, Float64(Float64(eps / x) * -0.5)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(x - N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1e-154], N[(eps / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[(-eps)], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps / N[(x * 2.0 + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.6%

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

      1. flip-- 98.5%

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

      2. div-inv 98.2%

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

      3. add-sqr-sqrt 98.0%

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

      4. associate--r- 99.3%

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

      5. pow2 99.3%

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

      6. pow2 99.3%

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

      7. sub-neg 99.3%

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

      8. add-sqr-sqrt 99.3%

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

      9. hypot-def 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. +-inverses 99.3%

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

      2. +-lft-identity 99.3%

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

      3. associate-*r/ 99.4%

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

      4. associate-/l* 99.4%

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

      5. /-rgt-identity 99.4%

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

   6. Simplified 99.4%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 7.9%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 51.3%

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

      9. hypot-def 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. +-inverses 51.3%

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

      2. +-lft-identity 51.3%

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

      3. associate-*r/ 51.6%

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

      4. associate-/l* 51.6%

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

      5. /-rgt-identity 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;\frac{\varepsilon}{x + \mathsf{hypot}\left(x, \sqrt{-\varepsilon}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (/ eps (fma x 2.0 (* (/ 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-154) {
		tmp = t_0;
	} else {
		tmp = eps / fma(x, 2.0, ((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-154)
		tmp = t_0;
	else
		tmp = Float64(eps / fma(x, 2.0, Float64(Float64(eps / x) * -0.5)));
	end
	return 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-154], t$95$0, N[(eps / N[(x * 2.0 + N[(N[(eps / x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.6%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

      1. flip-- 7.8%

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

      2. div-inv 7.8%

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

      3. add-sqr-sqrt 7.9%

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

      4. associate--r- 99.6%

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

      5. pow2 99.6%

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

      6. pow2 99.6%

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

      7. sub-neg 99.6%

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

      8. add-sqr-sqrt 51.3%

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

      9. hypot-def 51.3%

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

   4. Applied egg-rr 51.3%

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

      1. +-inverses 51.3%

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

      2. +-lft-identity 51.3%

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

      3. associate-*r/ 51.6%

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

      4. associate-/l* 51.6%

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

      5. /-rgt-identity 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in x around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. *-commutative 0.0%

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

      3. fma-def 0.0%

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

      4. associate-*r/ 0.0%

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

      5. *-commutative 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 98.0%

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

      8. associate-*r* 98.0%

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

      9. metadata-eval 98.0%

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

      10. associate-*r/ 98.0%

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

      11. *-commutative 98.0%

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

   9. Simplified 98.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\mathsf{fma}\left(x, 2, \frac{\varepsilon}{x} \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-154}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -1e-154) t_0 (* (/ 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-154) {
		tmp = t_0;
	} else {
		tmp = (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-154)) then
        tmp = t_0
    else
        tmp = (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-154) {
		tmp = t_0;
	} else {
		tmp = (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-154:
		tmp = t_0
	else:
		tmp = (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-154)
		tmp = t_0;
	else
		tmp = 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-154)
		tmp = t_0;
	else
		tmp = (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-154], t$95$0, N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -9.9999999999999997e-155

   1. Initial program 98.6%

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

   if -9.9999999999999997e-155 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 7.8%

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

   3. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \sqrt{x \cdot x - \varepsilon} \leq -1 \cdot 10^{-154}:\\ \;\;\;\;x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 2.5e-79) (- x (sqrt (- eps))) (* (/ eps x) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-79) {
		tmp = x - sqrt(-eps);
	} else {
		tmp = (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) :: tmp
    if (x <= 2.5d-79) then
        tmp = x - sqrt(-eps)
    else
        tmp = (eps / x) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= 2.5e-79) {
		tmp = x - Math.sqrt(-eps);
	} else {
		tmp = (eps / x) * 0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= 2.5e-79:
		tmp = x - math.sqrt(-eps)
	else:
		tmp = (eps / x) * 0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 2.5e-79)
		tmp = Float64(x - sqrt(Float64(-eps)));
	else
		tmp = Float64(Float64(eps / x) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 2.5e-79)
		tmp = x - sqrt(-eps);
	else
		tmp = (eps / x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 2.5e-79], N[(x - N[Sqrt[(-eps)], $MachinePrecision]), $MachinePrecision], N[(N[(eps / x), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5e-79

   1. Initial program 88.9%

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

   3. Taylor expanded in x around 0 87.2%

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

      1. neg-mul-1 87.2%

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

   5. Simplified 87.2%

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

   if 2.5e-79 < x

   1. Initial program 24.4%

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

   3. Taylor expanded in x around inf 82.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-79}:\\ \;\;\;\;x - \sqrt{-\varepsilon}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{x} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 43.6% 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. Add Preprocessing

3. Taylor expanded in x around inf 41.2%

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

4. Final simplification 41.2%

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

Alternative 6: 4.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 64.9%

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

   1. sub-neg 64.9%

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

   2. +-commutative 64.9%

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

   3. add-sqr-sqrt 64.5%

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

   4. distribute-rgt-neg-in 64.5%

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

   5. fma-def 64.5%

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

   6. pow1/2 64.5%

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

   7. sqrt-pow1 64.6%

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

   8. pow2 64.6%

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

   9. metadata-eval 64.6%

      \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.25}}, -\sqrt{\sqrt{x \cdot x - \varepsilon}}, x\right) \]

   10. pow1/2 64.6%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -\sqrt{\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{0.5}}}, x\right) \]

   11. sqrt-pow1 64.5%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -\color{blue}{{\left(x \cdot x - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}}, x\right) \]

   12. pow2 64.5%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left(\color{blue}{{x}^{2}} - \varepsilon\right)}^{\left(\frac{0.5}{2}\right)}, x\right) \]

   13. metadata-eval 64.5%

       \[\leadsto \mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left({x}^{2} - \varepsilon\right)}^{\color{blue}{0.25}}, x\right) \]

4. Applied egg-rr 64.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left({x}^{2} - \varepsilon\right)}^{0.25}, -{\left({x}^{2} - \varepsilon\right)}^{0.25}, x\right)} \]

5. Taylor expanded in x around inf 4.1%

   \[\leadsto \color{blue}{x + -1 \cdot x} \]
6. Step-by-step derivation

   1. distribute-rgt1-in 4.1%

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

   2. metadata-eval 4.1%

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 4.1%

      \[\leadsto \color{blue}{0} \]

7. Simplified 4.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 4.1%

   \[\leadsto 0 \]
9. Add Preprocessing

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 2024011 
(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))))