Numeric.Log:$clog1p from log-domain-0.10.2.1, B

Percentage Accurate: 99.7% → 99.7%

Time: 2.5s

Alternatives: 8

Speedup: 0.4×

Specification

Math

\[\frac{x}{1 + \sqrt{x + 1}} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	x / ((1) + (sqrt((x + (1)))))
END code

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\frac{x}{1 + \sqrt{x + 1}} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	x / ((1) + (sqrt((x + (1)))))
END code

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-0.125, x \cdot x, 0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-6)
  (fma -0.125 (* x x) (* 0.5 x))
  (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-6) {
		tmp = fma(-0.125, (x * x), (0.5 * x));
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-6)
		tmp = fma(-0.125, Float64(x * x), Float64(0.5 * x));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(-0.125 * N[(x * x), $MachinePrecision] + N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (50000000000000004090152695701565477293115691281855106353759765625e-70)) THEN (((-125e-3) * (x * x)) + ((5e-1) * x)) ELSE ((sqrt(((1) + x))) + (-1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.0000000000000004e-6

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 65.8%

      \[\leadsto x \cdot \left(0.5 + -0.125 \cdot x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 65.8%

      \[\leadsto \mathsf{fma}\left(-0.125, x \cdot x, 0.5 \cdot x\right) \]

   if 5.0000000000000004e-6 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 39.3%

      \[\leadsto \sqrt{1 + x} + -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(0.5, x, 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-6)
  (/ x (fma 0.5 x 2.0))
  (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-6) {
		tmp = x / fma(0.5, x, 2.0);
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-6)
		tmp = Float64(x / fma(0.5, x, 2.0));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(x / N[(0.5 * x + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (50000000000000004090152695701565477293115691281855106353759765625e-70)) THEN (x / (((5e-1) * x) + (2))) ELSE ((sqrt(((1) + x))) + (-1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.0000000000000004e-6

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{x}{2 + \frac{1}{2} \cdot x} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.9%

      \[\leadsto \frac{x}{2 + 0.5 \cdot x} \]
   5. Step-by-step derivation

   6. Applied rewrites 67.9%

      \[\leadsto \frac{x}{\mathsf{fma}\left(0.5, x, 2\right)} \]

   if 5.0000000000000004e-6 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 39.3%

      \[\leadsto \sqrt{1 + x} + -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(-0.125, x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 5e-6)
  (* x (fma -0.125 x 0.5))
  (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 5e-6) {
		tmp = x * fma(-0.125, x, 0.5);
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 5e-6)
		tmp = Float64(x * fma(-0.125, x, 0.5));
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(x * N[(-0.125 * x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (50000000000000004090152695701565477293115691281855106353759765625e-70)) THEN (x * (((-125e-3) * x) + (5e-1))) ELSE ((sqrt(((1) + x))) + (-1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 5.0000000000000004e-6

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 65.8%

      \[\leadsto x \cdot \left(0.5 + -0.125 \cdot x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 65.8%

      \[\leadsto x \cdot \mathsf{fma}\left(-0.125, x, 0.5\right) \]

   if 5.0000000000000004e-6 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 39.3%

      \[\leadsto \sqrt{1 + x} + -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 10^{-12}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} + -1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 1e-12)
  (* 0.5 x)
  (+ (sqrt (+ 1.0 x)) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 1e-12) {
		tmp = 0.5 * x;
	} else {
		tmp = sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 + sqrt((x + 1.0d0)))) <= 1d-12) then
        tmp = 0.5d0 * x
    else
        tmp = sqrt((1.0d0 + x)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 + Math.sqrt((x + 1.0)))) <= 1e-12) {
		tmp = 0.5 * x;
	} else {
		tmp = Math.sqrt((1.0 + x)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 1e-12:
		tmp = 0.5 * x
	else:
		tmp = math.sqrt((1.0 + x)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 1e-12)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(sqrt(Float64(1.0 + x)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 1e-12)
		tmp = 0.5 * x;
	else
		tmp = sqrt((1.0 + x)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-12], N[(0.5 * x), $MachinePrecision], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (99999999999999997988664762925561536725284350612952266601496376097202301025390625e-92)) THEN ((5e-1) * x) ELSE ((sqrt(((1) + x))) + (-1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 9.9999999999999998e-13

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot x \]
   3. Step-by-step derivation

   4. Applied rewrites 67.3%

      \[\leadsto 0.5 \cdot x \]

   if 9.9999999999999998e-13 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

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

   3. Applied rewrites 99.7%

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

   5. Applied rewrites 39.3%

      \[\leadsto \sqrt{1 + x} + -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 0.05:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 0.05) (* 0.5 x) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.05) {
		tmp = 0.5 * x;
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 + sqrt((x + 1.0d0)))) <= 0.05d0) then
        tmp = 0.5d0 * x
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 + Math.sqrt((x + 1.0)))) <= 0.05) {
		tmp = 0.5 * x;
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 0.05:
		tmp = 0.5 * x
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 0.05)
		tmp = Float64(0.5 * x);
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 0.05)
		tmp = 0.5 * x;
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.05], N[(0.5 * x), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (5000000000000000277555756156289135105907917022705078125e-56)) THEN ((5e-1) * x) ELSE (sqrt(x)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 0.050000000000000003

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{1}{2} \cdot x \]
   3. Step-by-step derivation

   4. Applied rewrites 67.3%

      \[\leadsto 0.5 \cdot x \]

   if 0.050000000000000003 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 34.5%

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 34.6%

      \[\leadsto \sqrt{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 36.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{1 + \sqrt{x + 1}} \leq 4 \cdot 10^{-210}:\\ \;\;\;\;\sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (if (<= (/ x (+ 1.0 (sqrt (+ x 1.0)))) 4e-210) (sqrt 0.0) (sqrt x)))

C

double code(double x) {
	double tmp;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 4e-210) {
		tmp = sqrt(0.0);
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x / (1.0d0 + sqrt((x + 1.0d0)))) <= 4d-210) then
        tmp = sqrt(0.0d0)
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x / (1.0 + Math.sqrt((x + 1.0)))) <= 4e-210) {
		tmp = Math.sqrt(0.0);
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x / (1.0 + math.sqrt((x + 1.0)))) <= 4e-210:
		tmp = math.sqrt(0.0)
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x / Float64(1.0 + sqrt(Float64(x + 1.0)))) <= 4e-210)
		tmp = sqrt(0.0);
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x / (1.0 + sqrt((x + 1.0)))) <= 4e-210)
		tmp = sqrt(0.0);
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x / N[(1.0 + N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e-210], N[Sqrt[0.0], $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	LET tmp = IF ((x / ((1) + (sqrt((x + (1)))))) <= (40000000000000001754955922235892998006218353004534480350476593552086812312828762824609665667893470382148702525598865546093318435031640624626745900808864392768698828487063690213161547567176664403997963093444535205751393006391340634994537348728463689202749036892606246670413469482542891968475704029388092629395892275133681299899143423556785033464480203676385451229669899140176458712009917313966594515003167803012359338524600134372869624583143557517732978270358050059486718115895288627009090148714358370352073279718752019107341766357421875e-745)) THEN (sqrt((0))) ELSE (sqrt(x)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 4.0000000000000002e-210

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 34.5%

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 34.6%

      \[\leadsto \sqrt{x} \]

   7. Taylor expanded in undef-var around zero

      \[\leadsto \sqrt{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.5%

      \[\leadsto \sqrt{0} \]

   if 4.0000000000000002e-210 < (/.f64 x (+.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 99.7%

      \[\frac{x}{1 + \sqrt{x + 1}} \]

   2. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 34.5%

      \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

   6. Applied rewrites 34.6%

      \[\leadsto \sqrt{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 34.6% accurate, 4.0× speedup

Math

\[\sqrt{x} \]

FPCore

(FPCore (x)
  :precision binary64
  :pre TRUE
  (sqrt x))

C

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

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = sqrt(x)
end function

Java

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

Python

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

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[x], $MachinePrecision]

ReFlow

f(x):
	x in [-inf, +inf]
code: THEORY
BEGIN
f(x: real): real =
	sqrt(x)
END code

Derivation

1. Initial program 99.7%

   \[\frac{x}{1 + \sqrt{x + 1}} \]

2. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
3. Step-by-step derivation

4. Applied rewrites 34.5%

   \[\leadsto \frac{1}{\sqrt{\frac{1}{x}}} \]
5. Step-by-step derivation

6. Applied rewrites 34.6%

   \[\leadsto \sqrt{x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))