2sqrt (example 3.1)

Percentage Accurate: 6.8% → 99.0%

Time: 7.0s

Alternatives: 6

Speedup: 1.0×

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: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   -0.125
   (sqrt (/ 1.0 x))
   (fma 0.0625 (sqrt (/ 1.0 (* x (* x x)))) (* (sqrt x) 0.5)))
  x))

C

double code(double x) {
	return fma(-0.125, sqrt((1.0 / x)), fma(0.0625, sqrt((1.0 / (x * (x * x)))), (sqrt(x) * 0.5))) / x;
}

Julia

function code(x)
	return Float64(fma(-0.125, sqrt(Float64(1.0 / x)), fma(0.0625, sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), Float64(sqrt(x) * 0.5))) / x)
end

Wolfram

code[x_] := N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)}\right)}{x} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   8. cube-mult N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   11. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

   13. lower-sqrt.f64 99.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x}} \cdot 0.5\right)\right)}{x} \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x}} \]
6. Add Preprocessing

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.125, \sqrt{\frac{1}{x}} \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (sqrt (/ 1.0 (* x (* x x)))) -0.125 (* (sqrt (/ 1.0 x)) 0.5)))

C

double code(double x) {
	return fma(sqrt((1.0 / (x * (x * x)))), -0.125, (sqrt((1.0 / x)) * 0.5));
}

Julia

function code(x)
	return fma(sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), -0.125, Float64(sqrt(Float64(1.0 / x)) * 0.5))
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.125 + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)}\right)}{x} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   8. cube-mult N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

   11. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

   13. lower-sqrt.f64 99.1

       \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x}} \cdot 0.5\right)\right)}{x} \]

5. Simplified 99.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x}} \]

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{\frac{1}{{x}^{3}}} \cdot \frac{-1}{8}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{{x}^{3}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

   4. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{-1}{8}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \frac{-1}{8}, \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}}\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \frac{-1}{8}, \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right) \]

   10. lower-/.f64 98.7

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.125, 0.5 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right) \]

8. Simplified 98.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.125, 0.5 \cdot \sqrt{\frac{1}{x}}\right)} \]

9. Final simplification 98.7%

   \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, -0.125, \sqrt{\frac{1}{x}} \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, -0.125 \cdot \sqrt{\frac{1}{x}}\right)}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (sqrt x) 0.5 (* -0.125 (sqrt (/ 1.0 x)))) x))

C

double code(double x) {
	return fma(sqrt(x), 0.5, (-0.125 * sqrt((1.0 / x)))) / x;
}

Julia

function code(x)
	return Float64(fma(sqrt(x), 0.5, Float64(-0.125 * sqrt(Float64(1.0 / x)))) / x)
end

Wolfram

code[x_] := N[(N[(N[Sqrt[x], $MachinePrecision] * 0.5 + N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \sqrt{x} + \frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}}{x} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{x} \cdot \frac{1}{2}} + \frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}{x} \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{x}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right)}}{x} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\sqrt{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{x}}\right)}{x} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, \frac{1}{2}, \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}}}\right)}{x} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, \frac{1}{2}, \frac{-1}{8} \cdot \color{blue}{\sqrt{\frac{1}{x}}}\right)}{x} \]

   8. lower-/.f64 98.6

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{x}, 0.5, -0.125 \cdot \sqrt{\color{blue}{\frac{1}{x}}}\right)}{x} \]

5. Simplified 98.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{x}, 0.5, -0.125 \cdot \sqrt{\frac{1}{x}}\right)}{x}} \]
6. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{x}} \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt (/ 1.0 x)) 0.5))

C

double code(double x) {
	return sqrt((1.0 / x)) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((1.0d0 / x)) * 0.5d0
end function

Java

public static double code(double x) {
	return Math.sqrt((1.0 / x)) * 0.5;
}

Python

def code(x):
	return math.sqrt((1.0 / x)) * 0.5

Julia

function code(x)
	return Float64(sqrt(Float64(1.0 / x)) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = sqrt((1.0 / x)) * 0.5;
end

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\sqrt{\frac{1}{x}}} \]

   3. lower-/.f64 97.6

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

5. Simplified 97.6%

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

6. Final simplification 97.6%

   \[\leadsto \sqrt{\frac{1}{x}} \cdot 0.5 \]
7. Add Preprocessing

Alternative 5: 4.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.3%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sqrt.f64 4.5

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

5. Simplified 4.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot x} - \sqrt{x} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 4.5

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

8. Simplified 4.5%

   \[\leadsto \color{blue}{x \cdot 0.5} - \sqrt{x} \]
9. Add Preprocessing

Alternative 6: 3.8% accurate, 27.0× 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 7.3%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 3.8

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

5. Simplified 3.8%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 3.8%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.7× 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]

Developer Target 2: 98.0% accurate, 0.3× 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]

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

  :alt
  (! :herbie-platform default (* 1/2 (pow x -1/2)))

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