ENA, Section 1.4, Exercise 4d

Percentage Accurate: 62.4% → 99.7%

Time: 13.0s

Alternatives: 7

Speedup: 0.6×

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(-x, x, \varepsilon\right), \frac{1}{\sqrt{x \cdot x - \varepsilon}}, -x\right)} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/ (- eps) (fma (fma (- x) x eps) (/ 1.0 (sqrt (- (* x x) eps))) (- x))))

C

double code(double x, double eps) {
	return -eps / fma(fma(-x, x, eps), (1.0 / sqrt(((x * x) - eps))), -x);
}

Julia

function code(x, eps)
	return Float64(Float64(-eps) / fma(fma(Float64(-x), x, eps), Float64(1.0 / sqrt(Float64(Float64(x * x) - eps))), Float64(-x)))
end

Wolfram

code[x_, eps_] := N[((-eps) / N[(N[((-x) * x + eps), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(x * x), $MachinePrecision] - eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. sqr-neg N/A

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

   6. lower-/.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-neg.f64 62.9

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

4. Applied rewrites 62.9%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.5

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

7. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. neg-sub0 N/A

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

   8. flip-- N/A

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

   9. div-inv N/A

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

   10. lower-fma.f64 N/A

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{-\varepsilon}{\color{blue}{\mathsf{fma}\left(0 - \left(x \cdot x - \varepsilon\right), \frac{1}{0 + \sqrt{x \cdot x - \varepsilon}}, -x\right)}} \]
10. Step-by-step derivation

    1. lift--.f64 N/A

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

    2. lift--.f64 N/A

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

    3. associate--r- N/A

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

    4. neg-sub0 N/A

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

    5. lift-*.f64 N/A

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

    6. distribute-lft-neg-in N/A

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

    7. lift-neg.f64 N/A

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

    8. lower-fma.f64 99.7

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

11. Applied rewrites 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{-\varepsilon}{\mathsf{fma}\left(\mathsf{fma}\left(-x, x, \varepsilon\right), \frac{1}{\sqrt{x \cdot x - \varepsilon}}, -x\right)} \]
13. Add Preprocessing

Alternative 2: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sqrt{x \cdot x - \varepsilon}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-\varepsilon}{\left(-x\right) - \mathsf{fma}\left(\varepsilon, \frac{-0.5}{x}, x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- x (sqrt (- (* x x) eps)))))
   (if (<= t_0 -5e-154) t_0 (/ (- eps) (- (- x) (fma eps (/ -0.5 x) x))))))

C

double code(double x, double eps) {
	double t_0 = x - sqrt(((x * x) - eps));
	double tmp;
	if (t_0 <= -5e-154) {
		tmp = t_0;
	} else {
		tmp = -eps / (-x - fma(eps, (-0.5 / x), 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-154)
		tmp = t_0;
	else
		tmp = Float64(Float64(-eps) / Float64(Float64(-x) - fma(eps, Float64(-0.5 / x), x)));
	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, -5e-154], t$95$0, N[((-eps) / N[((-x) - N[(eps * N[(-0.5 / x), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps))) < -5.0000000000000002e-154

   1. Initial program 98.5%

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

   if -5.0000000000000002e-154 < (-.f64 x (sqrt.f64 (-.f64 (*.f64 x x) eps)))

   1. Initial program 9.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. sqr-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-neg.f64 9.1

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

   4. Applied rewrites 9.1%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in eps around 0

      \[\leadsto \frac{\mathsf{neg}\left(\varepsilon\right)}{\left(\mathsf{neg}\left(\color{blue}{\left(x + \frac{-1}{2} \cdot \frac{\varepsilon}{x}\right)}\right)\right) - x} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 98.4

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

   10. Applied rewrites 98.4%

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

4. Final simplification 98.5%

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

Developer Target 1: 99.5% accurate, 0.7× 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 2024219 
(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)))

  :alt
  (! :herbie-platform default (/ eps (+ x (sqrt (- (* x x) eps)))))

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