ENA, Section 1.4, Exercise 4d

Percentage Accurate: 61.7% → 45.5%

Time: 8.7s

Alternatives: 6

Speedup: N/A×

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: 61.7% 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: 45.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

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

   1. flip-- 58.5%

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

   2. div-inv 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   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. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 77.2%

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

   9. hypot-def 77.2%

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

4. Applied egg-rr 77.2%

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

   1. +-inverses 77.2%

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

   2. +-lft-identity 77.2%

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

   3. associate-*r/ 77.4%

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

   4. associate-/l* 77.4%

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

   5. /-rgt-identity 77.4%

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

6. Simplified 77.4%

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

7. Taylor expanded in x around inf 0.0%

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

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 50.0%

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

   5. associate-*r* 50.0%

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

   6. metadata-eval 50.0%

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

   7. *-commutative 50.0%

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

9. Simplified 50.0%

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

10. Final simplification 50.0%

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

Alternative 2: 44.5% accurate, 21.4× speedup

Math

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

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(eps * Float64(0.5 / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

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

   1. sub-neg 58.6%

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

   2. +-commutative 58.6%

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

   3. add-sqr-sqrt 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-def 58.0%

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

   6. pow1/2 58.0%

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

   7. sqrt-pow1 58.1%

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

   8. pow2 58.1%

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

   9. metadata-eval 58.1%

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

   10. pow1/2 58.1%

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

   11. sqrt-pow1 58.0%

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

   12. pow2 58.0%

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

   13. metadata-eval 58.0%

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

4. Applied egg-rr 58.0%

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

5. Taylor expanded in x around inf 8.5%

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

   1. associate-+r+ 48.3%

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

   2. distribute-rgt1-in 48.3%

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

   3. metadata-eval 48.3%

      \[\leadsto \color{blue}{0} \cdot x + 0.5 \cdot \frac{\varepsilon}{x} \]

   4. mul0-lft 48.3%

      \[\leadsto \color{blue}{0} + 0.5 \cdot \frac{\varepsilon}{x} \]

   5. +-lft-identity 48.3%

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

   6. metadata-eval 48.3%

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

   7. times-frac 48.3%

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

   8. *-commutative 48.3%

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

   9. times-frac 48.1%

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

   10. /-rgt-identity 48.1%

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

7. Simplified 48.1%

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

8. Final simplification 48.1%

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

Alternative 3: 44.5% 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 58.6%

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

   1. flip-- 58.5%

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

   2. div-inv 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   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. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 77.2%

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

   9. hypot-def 77.2%

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

4. Applied egg-rr 77.2%

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

   1. +-inverses 77.2%

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

   2. +-lft-identity 77.2%

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

   3. associate-*r/ 77.4%

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

   4. associate-/l* 77.4%

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

   5. /-rgt-identity 77.4%

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

6. Simplified 77.4%

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

7. Taylor expanded in eps around 0 48.3%

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

   1. associate-*r/ 48.3%

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

   2. associate-/l* 48.2%

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

9. Simplified 48.2%

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

10. Final simplification 48.2%

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

Alternative 4: 44.7% 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 58.6%

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

3. Taylor expanded in x around inf 48.3%

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

   1. *-commutative 48.3%

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

   2. associate-*l/ 48.3%

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

5. Simplified 48.3%

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

6. Final simplification 48.3%

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

Alternative 5: 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 58.6%

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

   1. flip-- 58.5%

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

   2. div-inv 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   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. pow2 99.4%

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

   6. pow2 99.4%

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

   7. sub-neg 99.4%

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

   8. add-sqr-sqrt 77.2%

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

   9. hypot-def 77.2%

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

4. Applied egg-rr 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-/r/ 77.3%

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

   3. +-inverses 77.3%

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

   4. +-commutative 77.3%

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

6. Simplified 77.3%

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

7. Taylor expanded in x around inf 0.0%

   \[\leadsto \frac{1}{\frac{x + \color{blue}{\left(x + 0.5 \cdot \frac{\varepsilon \cdot {\left(\sqrt{-1}\right)}^{2}}{x}\right)}}{\varepsilon + 0}} \]
8. Step-by-step derivation

   1. associate-*r/ 0.0%

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

   2. *-commutative 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 50.0%

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

   5. associate-*r* 50.0%

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

   6. metadata-eval 50.0%

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

   7. *-commutative 50.0%

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

9. Simplified 49.9%

   \[\leadsto \frac{1}{\frac{x + \color{blue}{\left(x + \frac{\varepsilon \cdot -0.5}{x}\right)}}{\varepsilon + 0}} \]

10. Taylor expanded in x around 0 5.5%

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

    1. *-commutative 5.5%

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

12. Simplified 5.5%

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

13. Final simplification 5.5%

    \[\leadsto x \cdot -2 \]
14. 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 58.6%

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

   1. sub-neg 58.6%

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

   2. +-commutative 58.6%

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

   3. add-sqr-sqrt 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-def 58.0%

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

   6. pow1/2 58.0%

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

   7. sqrt-pow1 58.1%

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

   8. pow2 58.1%

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

   9. metadata-eval 58.1%

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

   10. pow1/2 58.1%

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

   11. sqrt-pow1 58.0%

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

   12. pow2 58.0%

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

   13. metadata-eval 58.0%

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

4. Applied egg-rr 58.0%

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

5. Taylor expanded in x around inf 4.3%

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

   1. distribute-rgt1-in 4.3%

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

   2. metadata-eval 4.3%

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

   3. mul0-lft 4.3%

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

7. Simplified 4.3%

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

8. Final simplification 4.3%

   \[\leadsto 0 \]
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 2024033 
(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))))