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

Percentage Accurate: 99.7% → 99.7%

Time: 4.1s

Alternatives: 5

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 99.7%

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

2. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{2 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.2e-6) (/ x (+ 2.0 (* x 0.5))) (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 7.2e-6) {
		tmp = x / (2.0 + (x * 0.5));
	} 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 <= 7.2d-6) then
        tmp = x / (2.0d0 + (x * 0.5d0))
    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 <= 7.2e-6) {
		tmp = x / (2.0 + (x * 0.5));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.2e-6:
		tmp = x / (2.0 + (x * 0.5))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.2e-6)
		tmp = Float64(x / Float64(2.0 + Float64(x * 0.5)));
	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 <= 7.2e-6)
		tmp = x / (2.0 + (x * 0.5));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.2e-6], N[(x / N[(2.0 + N[(x * 0.5), $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 < 7.19999999999999967e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.6%

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

      1. +-commutative 99.6%

         \[\leadsto \color{blue}{0.5 \cdot x + -0.125 \cdot {x}^{2}} \]

      2. unpow2 99.6%

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

      3. associate-*r* 99.6%

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

      4. distribute-rgt-out 99.5%

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

      5. *-commutative 99.5%

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

   4. Simplified 99.5%

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

      1. flip-+ 99.5%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

         \[\leadsto \frac{x \cdot \left(\color{blue}{0.25} - \left(x \cdot -0.125\right) \cdot \left(x \cdot -0.125\right)\right)}{0.5 - x \cdot -0.125} \]

      4. swap-sqr 99.6%

         \[\leadsto \frac{x \cdot \left(0.25 - \color{blue}{\left(x \cdot x\right) \cdot \left(-0.125 \cdot -0.125\right)}\right)}{0.5 - x \cdot -0.125} \]

      5. metadata-eval 99.6%

         \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot \color{blue}{0.015625}\right)}{0.5 - x \cdot -0.125} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 - \color{blue}{-0.125 \cdot x}} \]

      7. cancel-sign-sub-inv 99.6%

         \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{\color{blue}{0.5 + \left(--0.125\right) \cdot x}} \]

      8. metadata-eval 99.6%

         \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + \color{blue}{0.125} \cdot x} \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + 0.125 \cdot x}} \]
   7. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + 0.125 \cdot x}{0.25 - \left(x \cdot x\right) \cdot 0.015625}}} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{x}{\frac{0.5 + \color{blue}{x \cdot 0.125}}{0.25 - \left(x \cdot x\right) \cdot 0.015625}} \]

      3. associate-*l* 99.6%

         \[\leadsto \frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - \color{blue}{x \cdot \left(x \cdot 0.015625\right)}}} \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - x \cdot \left(x \cdot 0.015625\right)}}} \]

   9. Taylor expanded in x around 0 99.6%

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

   if 7.19999999999999967e-6 < x

   1. Initial program 99.3%

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

      1. flip-+ 99.1%

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

      2. metadata-eval 99.1%

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

      3. add-sqr-sqrt 99.8%

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

      4. +-commutative 99.8%

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

      5. associate--r+ 99.8%

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

      6. metadata-eval 99.8%

         \[\leadsto \frac{x}{\frac{\color{blue}{0} - x}{1 - \sqrt{x + 1}}} \]

      7. neg-sub0 99.8%

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

      8. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-in 99.8%

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

      4. distribute-lft-neg-out 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-/r* 99.8%

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

      9. *-inverses 99.8%

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

      10. metadata-eval 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-rgt-identity 99.8%

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

      13. associate-*r/ 99.8%

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

      14. neg-mul-1 99.8%

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

      15. times-frac 99.8%

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

      16. metadata-eval 99.8%

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

      17. mul-1-neg 99.8%

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

      18. *-inverses 99.8%

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

      19. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{2 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 3: 68.5% 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 99.7%

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

2. Taylor expanded in x around 0 67.3%

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

   1. +-commutative 67.3%

      \[\leadsto \color{blue}{0.5 \cdot x + -0.125 \cdot {x}^{2}} \]

   2. unpow2 67.3%

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

   3. associate-*r* 67.3%

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

   4. distribute-rgt-out 67.3%

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

   5. *-commutative 67.3%

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

4. Simplified 67.3%

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

   1. flip-+ 67.3%

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

   2. associate-*r/ 67.3%

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

   3. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(\color{blue}{0.25} - \left(x \cdot -0.125\right) \cdot \left(x \cdot -0.125\right)\right)}{0.5 - x \cdot -0.125} \]

   4. swap-sqr 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \color{blue}{\left(x \cdot x\right) \cdot \left(-0.125 \cdot -0.125\right)}\right)}{0.5 - x \cdot -0.125} \]

   5. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot \color{blue}{0.015625}\right)}{0.5 - x \cdot -0.125} \]

   6. *-commutative 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 - \color{blue}{-0.125 \cdot x}} \]

   7. cancel-sign-sub-inv 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{\color{blue}{0.5 + \left(--0.125\right) \cdot x}} \]

   8. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + \color{blue}{0.125} \cdot x} \]

6. Applied egg-rr 67.3%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + 0.125 \cdot x}} \]
7. Step-by-step derivation

   1. associate-/l* 67.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + 0.125 \cdot x}{0.25 - \left(x \cdot x\right) \cdot 0.015625}}} \]

   2. *-commutative 67.3%

      \[\leadsto \frac{x}{\frac{0.5 + \color{blue}{x \cdot 0.125}}{0.25 - \left(x \cdot x\right) \cdot 0.015625}} \]

   3. associate-*l* 67.3%

      \[\leadsto \frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - \color{blue}{x \cdot \left(x \cdot 0.015625\right)}}} \]

8. Simplified 67.3%

   \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - x \cdot \left(x \cdot 0.015625\right)}}} \]

9. Taylor expanded in x around 0 69.3%

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

10. Final simplification 69.3%

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

Alternative 4: 67.9% accurate, 35.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 68.5%

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

3. Final simplification 68.5%

   \[\leadsto x \cdot 0.5 \]

Alternative 5: 4.8% 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 99.7%

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

2. Taylor expanded in x around 0 67.3%

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

   1. +-commutative 67.3%

      \[\leadsto \color{blue}{0.5 \cdot x + -0.125 \cdot {x}^{2}} \]

   2. unpow2 67.3%

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

   3. associate-*r* 67.3%

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

   4. distribute-rgt-out 67.3%

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

   5. *-commutative 67.3%

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

4. Simplified 67.3%

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

   1. flip-+ 67.3%

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

   2. associate-*r/ 67.3%

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

   3. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(\color{blue}{0.25} - \left(x \cdot -0.125\right) \cdot \left(x \cdot -0.125\right)\right)}{0.5 - x \cdot -0.125} \]

   4. swap-sqr 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \color{blue}{\left(x \cdot x\right) \cdot \left(-0.125 \cdot -0.125\right)}\right)}{0.5 - x \cdot -0.125} \]

   5. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot \color{blue}{0.015625}\right)}{0.5 - x \cdot -0.125} \]

   6. *-commutative 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 - \color{blue}{-0.125 \cdot x}} \]

   7. cancel-sign-sub-inv 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{\color{blue}{0.5 + \left(--0.125\right) \cdot x}} \]

   8. metadata-eval 67.3%

      \[\leadsto \frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + \color{blue}{0.125} \cdot x} \]

6. Applied egg-rr 67.3%

   \[\leadsto \color{blue}{\frac{x \cdot \left(0.25 - \left(x \cdot x\right) \cdot 0.015625\right)}{0.5 + 0.125 \cdot x}} \]
7. Step-by-step derivation

   1. associate-/l* 67.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + 0.125 \cdot x}{0.25 - \left(x \cdot x\right) \cdot 0.015625}}} \]

   2. *-commutative 67.3%

      \[\leadsto \frac{x}{\frac{0.5 + \color{blue}{x \cdot 0.125}}{0.25 - \left(x \cdot x\right) \cdot 0.015625}} \]

   3. associate-*l* 67.3%

      \[\leadsto \frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - \color{blue}{x \cdot \left(x \cdot 0.015625\right)}}} \]

8. Simplified 67.3%

   \[\leadsto \color{blue}{\frac{x}{\frac{0.5 + x \cdot 0.125}{0.25 - x \cdot \left(x \cdot 0.015625\right)}}} \]

9. Taylor expanded in x around 0 69.3%

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

10. Taylor expanded in x around inf 4.8%

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

11. Final simplification 4.8%

    \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023287 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))