2sqrt (example 3.1)

Percentage Accurate: 6.7% → 98.6%

Time: 9.1s

Alternatives: 5

Speedup: 1.2×

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: 6.7% 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: 98.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. /-lowering-/.f64 N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-commutative N/A

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

   6. *-lowering-*.f64 N/A

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

   7. sqrt-lowering-sqrt.f64 98.9

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

5. Simplified 98.9%

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

   1. /-lowering-/.f64 N/A

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

   2. +-commutative N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. un-div-inv N/A

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

   8. /-lowering-/.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 98.9

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

7. Applied egg-rr 98.9%

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

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 98.3

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

5. Simplified 98.3%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
6. Add Preprocessing

Alternative 3: 97.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.2%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. /-lowering-/.f64 98.3

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

5. Simplified 98.3%

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

   1. sqrt-div N/A

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

   2. metadata-eval N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 98.1

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

7. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{\frac{0.5}{\sqrt{x}}} \]
8. Add Preprocessing

Alternative 4: 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 6.2%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. +-commutative N/A

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

   2. *-commutative N/A

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

   3. accelerator-lowering-fma.f64 4.4

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

5. Simplified 4.4%

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

6. Taylor expanded in x around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x + 0\right)} - \sqrt{x} \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot \frac{1}{2}} + 0\right) - \sqrt{x} \]

   3. accelerator-lowering-fma.f64 4.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} - \sqrt{x} \]

8. Simplified 4.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 0\right)} - \sqrt{x} \]
9. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-lowering-*.f64 4.4

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

10. Applied egg-rr 4.4%

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

Alternative 5: 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 6.2%

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

3. Taylor expanded in x around inf

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

   1. sqrt-lowering-sqrt.f64 3.8

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

5. Simplified 3.8%

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

   1. +-inverses 3.8

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

7. Applied egg-rr 3.8%

   \[\leadsto \color{blue}{0} \]
8. 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: 98.1% 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 2024194 
(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)))