2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.6%

Time: 13.4s

Alternatives: 4

Speedup: 205.0×

• could not determine a ground truth

  1. x = 1.005584046301334e+52

Specification

\[x > 1 \land x < 10^{+308}\]

Math

\[\begin{array}{l} \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

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

Initial Program: 6.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x + 1} - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (sqrt (+ x 1.0)) (sqrt x)))

C

double code(double x) {
	return sqrt((x + 1.0)) - sqrt(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((x + 1.0d0)) - sqrt(x)
end function

Java

public static double code(double x) {
	return Math.sqrt((x + 1.0)) - Math.sqrt(x);
}

Python

def code(x):
	return math.sqrt((x + 1.0)) - math.sqrt(x)

Julia

function code(x)
	return Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt((x + 1.0)) - sqrt(x);
end

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{1 + x} + \sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ 1.0 x)) (sqrt x))))

C

double code(double x) {
	return 1.0 / (sqrt((1.0 + x)) + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (sqrt((1.0d0 + x)) + sqrt(x))
end function

Java

public static double code(double x) {
	return 1.0 / (Math.sqrt((1.0 + x)) + Math.sqrt(x));
}

Python

def code(x):
	return 1.0 / (math.sqrt((1.0 + x)) + math.sqrt(x))

Julia

function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(1.0 + x)) + sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (sqrt((1.0 + x)) + sqrt(x));
end

Wolfram

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

Derivation

1. Initial program 6.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 7.3%

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

   2. div-inv 7.3%

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

   3. add-sqr-sqrt 7.4%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. associate-+r- 8.4%

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

   4. remove-double-neg 8.4%

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

   5. sub-neg 8.4%

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

   6. div-sub 6.8%

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

   7. rem-square-sqrt 6.6%

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

   8. sqr-neg 6.6%

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

   9. div-sub 7.4%

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

   10. +-commutative 7.4%

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

   11. sqr-neg 7.4%

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

   12. rem-square-sqrt 8.4%

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

   13. associate--l+ 99.6%

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

   14. +-inverses 99.6%

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

   15. metadata-eval 99.6%

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

   16. sub-neg 99.6%

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

6. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{1}{\sqrt{1 + x} + \sqrt{x}} \]
8. Add Preprocessing

Alternative 2: 98.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* 0.5 (pow x -0.5)))

C

double code(double x) {
	return 0.5 * pow(x, -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -0.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -0.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -0.5);
end

Wolfram

code[x_] := N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 7.3%

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

   2. div-inv 7.3%

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

   3. add-sqr-sqrt 7.4%

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

   4. add-sqr-sqrt 8.4%

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

   5. associate--l+ 8.4%

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

4. Applied egg-rr 8.4%

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

   1. associate-*r/ 8.4%

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

   2. *-rgt-identity 8.4%

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

   3. associate-+r- 8.4%

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

   4. remove-double-neg 8.4%

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

   5. sub-neg 8.4%

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

   6. div-sub 6.8%

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

   7. rem-square-sqrt 6.6%

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

   8. sqr-neg 6.6%

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

   9. div-sub 7.4%

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

   10. +-commutative 7.4%

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

   11. sqr-neg 7.4%

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

   12. rem-square-sqrt 8.4%

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

   13. associate--l+ 99.6%

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

   14. +-inverses 99.6%

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

   15. metadata-eval 99.6%

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

   16. sub-neg 99.6%

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

6. Simplified 99.6%

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

   1. flip3-+ 66.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}}{\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)}}} \]

   2. associate-/r/ 66.7%

      \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{1 + x}\right)}^{3} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right)} \]

   3. sqrt-pow2 66.7%

      \[\leadsto \frac{1}{\color{blue}{{\left(1 + x\right)}^{\left(\frac{3}{2}\right)}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   4. metadata-eval 66.7%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{\color{blue}{1.5}} + {\left(\sqrt{x}\right)}^{3}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   5. sqrt-pow2 66.5%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   6. metadata-eval 66.5%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{\color{blue}{1.5}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   7. add-sqr-sqrt 66.7%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(1 + x\right)} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   8. add-sqr-sqrt 66.5%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(1 + x\right) + \left(\color{blue}{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

   9. associate-+r- 66.5%

      \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(\left(1 + x\right) + x\right) - \sqrt{1 + x} \cdot \sqrt{x}\right)} \]

   10. sqrt-unprod 47.5%

       \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(1 + x\right) + x\right) - \color{blue}{\sqrt{\left(1 + x\right) \cdot x}}\right) \]

8. Applied egg-rr 47.5%

   \[\leadsto \color{blue}{\frac{1}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)} \]
9. Step-by-step derivation

   1. associate-*l/ 47.7%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}} \]

   2. *-lft-identity 47.7%

      \[\leadsto \frac{\color{blue}{\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   3. associate-+l+ 47.6%

      \[\leadsto \frac{\color{blue}{\left(1 + \left(x + x\right)\right)} - \sqrt{\left(1 + x\right) \cdot x}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   4. associate--l+ 47.6%

      \[\leadsto \frac{\color{blue}{1 + \left(\left(x + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   5. count-2 47.6%

      \[\leadsto \frac{1 + \left(\color{blue}{2 \cdot x} - \sqrt{\left(1 + x\right) \cdot x}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   6. *-commutative 47.6%

      \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot \left(1 + x\right)}}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

10. Simplified 47.6%

    \[\leadsto \color{blue}{\frac{1 + \left(2 \cdot x - \sqrt{x \cdot \left(1 + x\right)}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}}} \]

11. Taylor expanded in x around inf 66.0%

    \[\leadsto \frac{1 + \left(2 \cdot x - \color{blue}{\left(0.5 + x\right)}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
12. Step-by-step derivation

    1. +-commutative 66.0%

       \[\leadsto \frac{1 + \left(2 \cdot x - \color{blue}{\left(x + 0.5\right)}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

13. Simplified 66.0%

    \[\leadsto \frac{1 + \left(2 \cdot x - \color{blue}{\left(x + 0.5\right)}\right)}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

14. Taylor expanded in x around inf 97.8%

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
15. Step-by-step derivation

    1. unpow1/2 97.8%

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

    2. rem-exp-log 90.4%

       \[\leadsto 0.5 \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \]

    3. exp-neg 90.4%

       \[\leadsto 0.5 \cdot {\color{blue}{\left(e^{-\log x}\right)}}^{0.5} \]

    4. exp-prod 90.4%

       \[\leadsto 0.5 \cdot \color{blue}{e^{\left(-\log x\right) \cdot 0.5}} \]

    5. distribute-lft-neg-out 90.4%

       \[\leadsto 0.5 \cdot e^{\color{blue}{-\log x \cdot 0.5}} \]

    6. distribute-rgt-neg-in 90.4%

       \[\leadsto 0.5 \cdot e^{\color{blue}{\log x \cdot \left(-0.5\right)}} \]

    7. metadata-eval 90.4%

       \[\leadsto 0.5 \cdot e^{\log x \cdot \color{blue}{-0.5}} \]

    8. exp-to-pow 98.0%

       \[\leadsto 0.5 \cdot \color{blue}{{x}^{-0.5}} \]

16. Simplified 98.0%

    \[\leadsto \color{blue}{0.5 \cdot {x}^{-0.5}} \]

17. Final simplification 98.0%

    \[\leadsto 0.5 \cdot {x}^{-0.5} \]
18. Add Preprocessing

Alternative 3: 3.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 6.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 3.9%

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

4. Final simplification 3.9%

   \[\leadsto 0 \]
5. Add Preprocessing

Alternative 4: 6.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 6.8%

   \[\sqrt{x + 1} - \sqrt{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 6.7%

   \[\leadsto \color{blue}{1} \]

4. Final simplification 6.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{x + 1} + \sqrt{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))

C

double code(double x) {
	return 1.0 / (sqrt((x + 1.0)) + sqrt(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
end function

Java

public static double code(double x) {
	return 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
}

Python

def code(x):
	return 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))

Julia

function code(x)
	return Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
end

Wolfram

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

Reproduce

herbie shell --seed 2024046 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :herbie-target
  (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))

  (- (sqrt (+ x 1.0)) (sqrt x)))