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

Percentage Accurate: 99.7% → 99.7%

Time: 5.8s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 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)
    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]

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 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)
    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]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 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)
    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]

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00025:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00025)
   (/ x (+ 2.0 (* x (+ 0.5 (* x -0.125)))))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00025) {
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))));
	} else {
		tmp = sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00025d0) then
        tmp = x / (2.0d0 + (x * (0.5d0 + (x * (-0.125d0)))))
    else
        tmp = sqrt((x + 1.0d0)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00025) {
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00025:
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00025)
		tmp = Float64(x / Float64(2.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))));
	else
		tmp = Float64(sqrt(Float64(x + 1.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00025)
		tmp = x / (2.0 + (x * (0.5 + (x * -0.125))));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00025], N[(x / N[(2.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

      \[\leadsto \frac{x}{\color{blue}{2 + \left(-0.125 \cdot {x}^{2} + 0.5 \cdot x\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto \frac{x}{2 + \color{blue}{\left(0.5 \cdot x + -0.125 \cdot {x}^{2}\right)}} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{x}{2 + \left(\color{blue}{x \cdot 0.5} + -0.125 \cdot {x}^{2}\right)} \]

      3. *-commutative 99.7%

         \[\leadsto \frac{x}{2 + \left(x \cdot 0.5 + \color{blue}{{x}^{2} \cdot -0.125}\right)} \]

      4. unpow2 99.7%

         \[\leadsto \frac{x}{2 + \left(x \cdot 0.5 + \color{blue}{\left(x \cdot x\right)} \cdot -0.125\right)} \]

      5. associate-*l* 99.7%

         \[\leadsto \frac{x}{2 + \left(x \cdot 0.5 + \color{blue}{x \cdot \left(x \cdot -0.125\right)}\right)} \]

      6. distribute-lft-out 99.7%

         \[\leadsto \frac{x}{2 + \color{blue}{x \cdot \left(0.5 + x \cdot -0.125\right)}} \]

   4. Simplified 99.7%

      \[\leadsto \frac{x}{\color{blue}{2 + x \cdot \left(0.5 + x \cdot -0.125\right)}} \]

   if 2.5000000000000001e-4 < x

   1. Initial program 99.3%

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

      1. flip-+ 99.2%

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

      2. associate-/r/ 99.1%

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

      3. sub-neg 99.1%

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

      4. distribute-lft-in 99.0%

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

      5. metadata-eval 99.0%

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

      6. add-sqr-sqrt 99.0%

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

      7. +-commutative 99.0%

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

      8. associate--r+ 98.9%

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

      9. metadata-eval 98.9%

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

      10. neg-sub0 98.9%

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

   3. Applied egg-rr 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. sub-neg 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-*r/ 99.9%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. *-inverses 99.9%

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

      8. metadata-eval 99.9%

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

      9. metadata-eval 99.9%

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

      10. neg-mul-1 99.9%

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

      11. neg-sub0 99.9%

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

      12. associate--r- 99.9%

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

      13. metadata-eval 99.9%

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

      14. +-commutative 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(1, \sqrt{x}\right) + -1} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 92.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)\right)} + -1 \]

      2. expm1-udef 92.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)} - 1\right)} + -1 \]

      3. associate-+l- 92.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)} - \left(1 - -1\right)} \]

      4. metadata-eval 92.2%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)} - \color{blue}{2} \]

      5. metadata-eval 92.2%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)} - \color{blue}{\left(1 + 1\right)} \]

      6. associate--l- 92.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)} - 1\right) - 1} \]

      7. expm1-udef 92.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)\right)} - 1 \]

      8. expm1-log1p-u 99.9%

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

      9. hypot-udef 99.8%

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

      10. metadata-eval 99.8%

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

      11. add-sqr-sqrt 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\sqrt{1 + x} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00025:\\ \;\;\;\;\frac{x}{2 + x \cdot \left(0.5 + x \cdot -0.125\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 3: 68.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2 + x \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 2.0 (* x 0.5))))

C

double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (2.0d0 + (x * 0.5d0))
end function

Java

public static double code(double x) {
	return x / (2.0 + (x * 0.5));
}

Python

def code(x):
	return x / (2.0 + (x * 0.5))

Julia

function code(x)
	return Float64(x / Float64(2.0 + Float64(x * 0.5)))
end

MATLAB

function tmp = code(x)
	tmp = x / (2.0 + (x * 0.5));
end

Wolfram

code[x_] := N[(x / N[(2.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 70.1%

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

   1. +-commutative 70.1%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

4. Simplified 70.1%

   \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

5. Final simplification 70.1%

   \[\leadsto \frac{x}{2 + x \cdot 0.5} \]

Alternative 4: 67.4% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x 2.0))

C

double code(double x) {
	return x / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / 2.0d0
end function

Java

public static double code(double x) {
	return x / 2.0;
}

Python

def code(x):
	return x / 2.0

Julia

function code(x)
	return Float64(x / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x / 2.0;
end

Wolfram

code[x_] := N[(x / 2.0), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 69.5%

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

3. Final simplification 69.5%

   \[\leadsto \frac{x}{2} \]

Alternative 5: 4.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 70.1%

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

   1. +-commutative 70.1%

      \[\leadsto \frac{x}{\color{blue}{0.5 \cdot x + 2}} \]

4. Simplified 70.1%

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

   1. frac-2neg 70.1%

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

   2. clear-num 69.9%

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

   3. associate-/r/ 70.1%

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

   4. neg-sub0 70.1%

      \[\leadsto \frac{1}{\color{blue}{0 - \left(0.5 \cdot x + 2\right)}} \cdot \left(-x\right) \]

   5. +-commutative 70.1%

      \[\leadsto \frac{1}{0 - \color{blue}{\left(2 + 0.5 \cdot x\right)}} \cdot \left(-x\right) \]

   6. associate--r+ 70.1%

      \[\leadsto \frac{1}{\color{blue}{\left(0 - 2\right) - 0.5 \cdot x}} \cdot \left(-x\right) \]

   7. metadata-eval 70.1%

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

   8. *-commutative 70.1%

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

6. Applied egg-rr 70.1%

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

7. Taylor expanded in x around inf 4.8%

   \[\leadsto \color{blue}{2} \]

8. Final simplification 4.8%

   \[\leadsto 2 \]

Alternative 6: 14.8% accurate, 107.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 99.8%

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

   1. +-commutative 99.8%

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

   2. add-sqr-sqrt 99.7%

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

   3. fma-def 99.7%

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

   4. pow1/2 99.7%

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

   5. sqrt-pow1 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{{\left(x + 1\right)}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x + 1}}, 1\right)} \]

   6. metadata-eval 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{x + 1}}, 1\right)} \]

   7. pow1/2 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, \sqrt{\color{blue}{{\left(x + 1\right)}^{0.5}}}, 1\right)} \]

   8. sqrt-pow1 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, \color{blue}{{\left(x + 1\right)}^{\left(\frac{0.5}{2}\right)}}, 1\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \frac{x}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{\color{blue}{0.25}}, 1\right)} \]

3. Applied egg-rr 99.7%

   \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left({\left(x + 1\right)}^{0.25}, {\left(x + 1\right)}^{0.25}, 1\right)}} \]

4. Taylor expanded in x around inf 37.6%

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

5. Taylor expanded in x around 0 15.0%

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

6. Final simplification 15.0%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))