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

Percentage Accurate: 99.7% → 99.7%

Time: 5.4s

Alternatives: 7

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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;\frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \left(x \cdot 0.125 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000205)
   (/ x (* (- 4.0 (* x (* x 0.25))) (+ (* x 0.125) 0.5)))
   (+ (sqrt (+ x 1.0)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000205) {
		tmp = x / ((4.0 - (x * (x * 0.25))) * ((x * 0.125) + 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 <= 0.000205d0) then
        tmp = x / ((4.0d0 - (x * (x * 0.25d0))) * ((x * 0.125d0) + 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 <= 0.000205) {
		tmp = x / ((4.0 - (x * (x * 0.25))) * ((x * 0.125) + 0.5));
	} else {
		tmp = Math.sqrt((x + 1.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000205:
		tmp = x / ((4.0 - (x * (x * 0.25))) * ((x * 0.125) + 0.5))
	else:
		tmp = math.sqrt((x + 1.0)) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000205)
		tmp = Float64(x / Float64(Float64(4.0 - Float64(x * Float64(x * 0.25))) * Float64(Float64(x * 0.125) + 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 <= 0.000205)
		tmp = x / ((4.0 - (x * (x * 0.25))) * ((x * 0.125) + 0.5));
	else
		tmp = sqrt((x + 1.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000205], N[(x / N[(N[(4.0 - N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * 0.125), $MachinePrecision] + 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 < 2.05e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

      \[\leadsto \frac{x}{1 + \color{blue}{\left(0.5 \cdot x + 1\right)}} \]
   3. Step-by-step derivation

      1. +-commutative 99.3%

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

      2. associate-+r+ 99.3%

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

      3. metadata-eval 99.3%

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

      4. flip-+ 99.3%

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

      5. metadata-eval 99.3%

         \[\leadsto \frac{x}{\frac{\color{blue}{4} - \left(0.5 \cdot x\right) \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}} \]

      6. *-commutative 99.3%

         \[\leadsto \frac{x}{\frac{4 - \color{blue}{\left(x \cdot 0.5\right)} \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}} \]

      7. *-commutative 99.3%

         \[\leadsto \frac{x}{\frac{4 - \left(x \cdot 0.5\right) \cdot \color{blue}{\left(x \cdot 0.5\right)}}{2 - 0.5 \cdot x}} \]

      8. swap-sqr 99.3%

         \[\leadsto \frac{x}{\frac{4 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot 0.5\right)}}{2 - 0.5 \cdot x}} \]

      9. metadata-eval 99.3%

         \[\leadsto \frac{x}{\frac{4 - \left(x \cdot x\right) \cdot \color{blue}{0.25}}{2 - 0.5 \cdot x}} \]

      10. *-commutative 99.3%

          \[\leadsto \frac{x}{\frac{4 - \left(x \cdot x\right) \cdot 0.25}{2 - \color{blue}{x \cdot 0.5}}} \]

   4. Applied egg-rr 99.3%

      \[\leadsto \frac{x}{\color{blue}{\frac{4 - \left(x \cdot x\right) \cdot 0.25}{2 - x \cdot 0.5}}} \]
   5. Step-by-step derivation

      1. div-inv 99.3%

         \[\leadsto \frac{x}{\color{blue}{\left(4 - \left(x \cdot x\right) \cdot 0.25\right) \cdot \frac{1}{2 - x \cdot 0.5}}} \]

      2. associate-*l* 99.3%

         \[\leadsto \frac{x}{\left(4 - \color{blue}{x \cdot \left(x \cdot 0.25\right)}\right) \cdot \frac{1}{2 - x \cdot 0.5}} \]

      3. sub-neg 99.3%

         \[\leadsto \frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \frac{1}{\color{blue}{2 + \left(-x \cdot 0.5\right)}}} \]

      4. distribute-rgt-neg-in 99.3%

         \[\leadsto \frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \frac{1}{2 + \color{blue}{x \cdot \left(-0.5\right)}}} \]

      5. metadata-eval 99.3%

         \[\leadsto \frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \frac{1}{2 + x \cdot \color{blue}{-0.5}}} \]

   6. Applied egg-rr 99.3%

      \[\leadsto \frac{x}{\color{blue}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \frac{1}{2 + x \cdot -0.5}}} \]

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \color{blue}{\left(0.125 \cdot x + 0.5\right)}} \]

   if 2.05e-4 < x

   1. Initial program 99.3%

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

      1. flip-+ 98.9%

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

      2. metadata-eval 98.9%

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

      3. add-sqr-sqrt 99.6%

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

      4. +-commutative 99.6%

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

      5. associate--r+ 99.6%

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

      6. metadata-eval 99.6%

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

      7. neg-sub0 99.6%

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

      8. associate-/r/ 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. remove-double-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. *-inverses 99.6%

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

      4. metadata-eval 99.6%

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

      5. neg-mul-1 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-neg-in 99.6%

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

      9. remove-double-neg 99.6%

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

      10. metadata-eval 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000205:\\ \;\;\;\;\frac{x}{\left(4 - x \cdot \left(x \cdot 0.25\right)\right) \cdot \left(x \cdot 0.125 + 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} + -1\\ \end{array} \]

Alternative 3: 69.1% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - x \cdot 0.25} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+
  (* (/ x (- 4.0 (* x (* x 0.25)))) 2.0)
  (* (/ x (- (/ 4.0 x) (* x 0.25))) -0.5)))

C

double code(double x) {
	return ((x / (4.0 - (x * (x * 0.25)))) * 2.0) + ((x / ((4.0 / x) - (x * 0.25))) * -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x / (4.0d0 - (x * (x * 0.25d0)))) * 2.0d0) + ((x / ((4.0d0 / x) - (x * 0.25d0))) * (-0.5d0))
end function

Java

public static double code(double x) {
	return ((x / (4.0 - (x * (x * 0.25)))) * 2.0) + ((x / ((4.0 / x) - (x * 0.25))) * -0.5);
}

Python

def code(x):
	return ((x / (4.0 - (x * (x * 0.25)))) * 2.0) + ((x / ((4.0 / x) - (x * 0.25))) * -0.5)

Julia

function code(x)
	return Float64(Float64(Float64(x / Float64(4.0 - Float64(x * Float64(x * 0.25)))) * 2.0) + Float64(Float64(x / Float64(Float64(4.0 / x) - Float64(x * 0.25))) * -0.5))
end

MATLAB

function tmp = code(x)
	tmp = ((x / (4.0 - (x * (x * 0.25)))) * 2.0) + ((x / ((4.0 / x) - (x * 0.25))) * -0.5);
end

Wolfram

code[x_] := N[(N[(N[(x / N[(4.0 - N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(x / N[(N[(4.0 / x), $MachinePrecision] - N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 72.2%

   \[\leadsto \frac{x}{1 + \color{blue}{\left(0.5 \cdot x + 1\right)}} \]
3. Step-by-step derivation

   1. +-commutative 72.2%

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

   2. associate-+r+ 72.2%

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

   3. metadata-eval 72.2%

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

   4. flip-+ 71.8%

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

   5. metadata-eval 71.8%

      \[\leadsto \frac{x}{\frac{\color{blue}{4} - \left(0.5 \cdot x\right) \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}} \]

   6. *-commutative 71.8%

      \[\leadsto \frac{x}{\frac{4 - \color{blue}{\left(x \cdot 0.5\right)} \cdot \left(0.5 \cdot x\right)}{2 - 0.5 \cdot x}} \]

   7. *-commutative 71.8%

      \[\leadsto \frac{x}{\frac{4 - \left(x \cdot 0.5\right) \cdot \color{blue}{\left(x \cdot 0.5\right)}}{2 - 0.5 \cdot x}} \]

   8. swap-sqr 71.8%

      \[\leadsto \frac{x}{\frac{4 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.5 \cdot 0.5\right)}}{2 - 0.5 \cdot x}} \]

   9. metadata-eval 71.8%

      \[\leadsto \frac{x}{\frac{4 - \left(x \cdot x\right) \cdot \color{blue}{0.25}}{2 - 0.5 \cdot x}} \]

   10. *-commutative 71.8%

       \[\leadsto \frac{x}{\frac{4 - \left(x \cdot x\right) \cdot 0.25}{2 - \color{blue}{x \cdot 0.5}}} \]

4. Applied egg-rr 71.8%

   \[\leadsto \frac{x}{\color{blue}{\frac{4 - \left(x \cdot x\right) \cdot 0.25}{2 - x \cdot 0.5}}} \]
5. Step-by-step derivation

   1. associate-/r/ 71.8%

      \[\leadsto \color{blue}{\frac{x}{4 - \left(x \cdot x\right) \cdot 0.25} \cdot \left(2 - x \cdot 0.5\right)} \]

   2. sub-neg 71.8%

      \[\leadsto \frac{x}{4 - \left(x \cdot x\right) \cdot 0.25} \cdot \color{blue}{\left(2 + \left(-x \cdot 0.5\right)\right)} \]

   3. distribute-lft-in 71.8%

      \[\leadsto \color{blue}{\frac{x}{4 - \left(x \cdot x\right) \cdot 0.25} \cdot 2 + \frac{x}{4 - \left(x \cdot x\right) \cdot 0.25} \cdot \left(-x \cdot 0.5\right)} \]

   4. associate-*l* 71.8%

      \[\leadsto \frac{x}{4 - \color{blue}{x \cdot \left(x \cdot 0.25\right)}} \cdot 2 + \frac{x}{4 - \left(x \cdot x\right) \cdot 0.25} \cdot \left(-x \cdot 0.5\right) \]

   5. associate-*l* 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{4 - \color{blue}{x \cdot \left(x \cdot 0.25\right)}} \cdot \left(-x \cdot 0.5\right) \]

   6. distribute-rgt-neg-in 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot \color{blue}{\left(x \cdot \left(-0.5\right)\right)} \]

   7. metadata-eval 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot \left(x \cdot \color{blue}{-0.5}\right) \]

6. Applied egg-rr 71.8%

   \[\leadsto \color{blue}{\frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot \left(x \cdot -0.5\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot \left(x \cdot -0.5\right)\right)\right)} \]

   2. expm1-udef 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot \left(x \cdot -0.5\right)\right)} - 1\right)} \]

   3. associate-*l/ 71.0%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(x \cdot -0.5\right)}{4 - x \cdot \left(x \cdot 0.25\right)}}\right)} - 1\right) \]

8. Applied egg-rr 71.0%

   \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x \cdot \left(x \cdot -0.5\right)}{4 - x \cdot \left(x \cdot 0.25\right)}\right)} - 1\right)} \]
9. Step-by-step derivation

   1. expm1-def 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x \cdot \left(x \cdot -0.5\right)}{4 - x \cdot \left(x \cdot 0.25\right)}\right)\right)} \]

   2. expm1-log1p 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\frac{x \cdot \left(x \cdot -0.5\right)}{4 - x \cdot \left(x \cdot 0.25\right)}} \]

   3. associate-*r* 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{\color{blue}{\left(x \cdot x\right) \cdot -0.5}}{4 - x \cdot \left(x \cdot 0.25\right)} \]

   4. unpow2 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{\color{blue}{{x}^{2}} \cdot -0.5}{4 - x \cdot \left(x \cdot 0.25\right)} \]

   5. associate-*l/ 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\frac{{x}^{2}}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot -0.5} \]

   6. unpow2 71.4%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{\color{blue}{x \cdot x}}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot -0.5 \]

   7. associate-/l* 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\frac{x}{\frac{4 - x \cdot \left(x \cdot 0.25\right)}{x}}} \cdot -0.5 \]

   8. div-sub 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\color{blue}{\frac{4}{x} - \frac{x \cdot \left(x \cdot 0.25\right)}{x}}} \cdot -0.5 \]

   9. *-commutative 71.8%

      \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - \frac{\color{blue}{\left(x \cdot 0.25\right) \cdot x}}{x}} \cdot -0.5 \]

   10. associate-/l* 72.2%

       \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - \color{blue}{\frac{x \cdot 0.25}{\frac{x}{x}}}} \cdot -0.5 \]

   11. *-commutative 72.2%

       \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - \frac{\color{blue}{0.25 \cdot x}}{\frac{x}{x}}} \cdot -0.5 \]

   12. *-inverses 72.2%

       \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - \frac{0.25 \cdot x}{\color{blue}{1}}} \cdot -0.5 \]

   13. associate-*r/ 72.2%

       \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - \color{blue}{0.25 \cdot \frac{x}{1}}} \cdot -0.5 \]

   14. /-rgt-identity 72.2%

       \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - 0.25 \cdot \color{blue}{x}} \cdot -0.5 \]

10. Simplified 72.2%

    \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \color{blue}{\frac{x}{\frac{4}{x} - 0.25 \cdot x} \cdot -0.5} \]

11. Final simplification 72.2%

    \[\leadsto \frac{x}{4 - x \cdot \left(x \cdot 0.25\right)} \cdot 2 + \frac{x}{\frac{4}{x} - x \cdot 0.25} \cdot -0.5 \]

Alternative 4: 69.1% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \left(1 + x \cdot 0.5\right)} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x / (1.0 + (1.0 + (x * 0.5)));
}

Python

def code(x):
	return x / (1.0 + (1.0 + (x * 0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 72.2%

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

3. Final simplification 72.2%

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

Alternative 5: 69.1% 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 72.2%

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

3. Final simplification 72.2%

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

Alternative 6: 68.4% 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 71.4%

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

3. Final simplification 71.4%

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

Alternative 7: 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 72.2%

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

3. Taylor expanded in x around inf 4.6%

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

4. Final simplification 4.6%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023252 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))