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

Percentage Accurate: 99.7% → 99.8%

Time: 11.9s

Alternatives: 9

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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00058:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot \left(0.0625 + x \cdot -0.0390625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{0.5} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00058)
   (* x (+ 0.5 (* x (+ -0.125 (* x (+ 0.0625 (* x -0.0390625)))))))
   (+ (pow (+ 1.0 x) 0.5) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00058) {
		tmp = x * (0.5 + (x * (-0.125 + (x * (0.0625 + (x * -0.0390625))))));
	} else {
		tmp = pow((1.0 + x), 0.5) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00058d0) then
        tmp = x * (0.5d0 + (x * ((-0.125d0) + (x * (0.0625d0 + (x * (-0.0390625d0)))))))
    else
        tmp = ((1.0d0 + x) ** 0.5d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00058) {
		tmp = x * (0.5 + (x * (-0.125 + (x * (0.0625 + (x * -0.0390625))))));
	} else {
		tmp = Math.pow((1.0 + x), 0.5) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00058:
		tmp = x * (0.5 + (x * (-0.125 + (x * (0.0625 + (x * -0.0390625))))))
	else:
		tmp = math.pow((1.0 + x), 0.5) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00058)
		tmp = Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * Float64(0.0625 + Float64(x * -0.0390625)))))));
	else
		tmp = Float64((Float64(1.0 + x) ^ 0.5) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00058)
		tmp = x * (0.5 + (x * (-0.125 + (x * (0.0625 + (x * -0.0390625))))));
	else
		tmp = ((1.0 + x) ^ 0.5) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00058], N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * N[(0.0625 + N[(x * -0.0390625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], 0.5], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e-4

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)} \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} + x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) - \frac{1}{8}\right)}\right)\right)\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right) + \frac{-1}{8}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \color{blue}{x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right)}\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \color{blue}{\left(x \cdot \left(\frac{1}{16} + \frac{-5}{128} \cdot x\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{16} + \frac{-5}{128} \cdot x\right)}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{16}, \color{blue}{\left(\frac{-5}{128} \cdot x\right)}\right)\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{16}, \left(x \cdot \color{blue}{\frac{-5}{128}}\right)\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 99.9%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-5}{128}}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot \left(0.0625 + x \cdot -0.0390625\right)\right)\right)} \]

   if 5.8e-4 < x

   1. Initial program 99.1%

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

      1. frac-2neg N/A

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

      2. neg-sub0 N/A

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

      3. metadata-eval N/A

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

      4. associate--r+ N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. rem-square-sqrt N/A

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

      8. distribute-neg-frac2 N/A

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

      9. flip-- N/A

         \[\leadsto \mathsf{neg}\left(\left(1 - \sqrt{x + 1}\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(1 - \sqrt{x + 1}\right)\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{\_.f64}\left(1, \left(\sqrt{x + 1}\right)\right)\right) \]

      12. pow1/2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(x + 1\right)}^{\frac{1}{2}}\right)\right)\right) \]

      13. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{2}\right)\right)\right) \]

      14. +-lowering-+.f64 99.8%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{-\left(1 - {\left(x + 1\right)}^{0.5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00058:\\ \;\;\;\;x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot \left(0.0625 + x \cdot -0.0390625\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{0.5} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * (1.0 / (1.0 + math.pow((1.0 + x), 0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. div-inv N/A

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

   2. flip-+ N/A

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

   3. clear-num N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1 - \sqrt{x + 1}}{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}\right), \color{blue}{x}\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1 \cdot 1 - \sqrt{x + 1} \cdot \sqrt{x + 1}}{1 - \sqrt{x + 1}}}\right), x\right) \]

   7. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{1 + \sqrt{x + 1}}\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \sqrt{x + 1}\right)\right), x\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(\sqrt{x + 1}\right)\right)\right), x\right) \]

   10. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left({\left(x + 1\right)}^{\frac{1}{2}}\right)\right)\right), x\right) \]

   11. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\left(x + 1\right), \frac{1}{2}\right)\right)\right), x\right) \]

   12. +-lowering-+.f64 99.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x, 1\right), \frac{1}{2}\right)\right)\right), x\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

   \[\leadsto x \cdot \frac{1}{1 + {\left(1 + x\right)}^{0.5}} \]
6. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.0)
   (/ x (+ (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))) 2.0))
   (+ (sqrt x) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = sqrt(x) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.0d0) then
        tmp = x / ((x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0))))) + 2.0d0)
    else
        tmp = sqrt(x) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.0) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = Math.sqrt(x) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.0:
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0)
	else:
		tmp = math.sqrt(x) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.0)
		tmp = Float64(x / Float64(Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))) + 2.0));
	else
		tmp = Float64(sqrt(x) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.0)
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	else
		tmp = sqrt(x) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.0], N[(x / N[(N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right), 2\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right), 2\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \frac{x}{\color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}} \]

   if 3 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\sqrt{x} - 1} \]
   4. Step-by-step derivation

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\sqrt{x}\right), \color{blue}{-1}\right) \]

      4. sqrt-lowering-sqrt.f64 100.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(x\right), -1\right) \]

   5. Simplified 100.0%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

   \[\leadsto \frac{x}{1 + \sqrt{1 + x}} \]
4. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7:\\ \;\;\;\;\frac{x}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.7)
   (/ x (+ (* x (+ 0.5 (* x (+ -0.125 (* x 0.0625))))) 2.0))
   (sqrt x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.7d0) then
        tmp = x / ((x * (0.5d0 + (x * ((-0.125d0) + (x * 0.0625d0))))) + 2.0d0)
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.7) {
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.7:
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0)
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.7)
		tmp = Float64(x / Float64(Float64(x * Float64(0.5 + Float64(x * Float64(-0.125 + Float64(x * 0.0625))))) + 2.0));
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.7)
		tmp = x / ((x * (0.5 + (x * (-0.125 + (x * 0.0625))))) + 2.0);
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.7], N[(x / N[(N[(x * N[(0.5 + N[(x * N[(-0.125 + N[(x * 0.0625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right) + \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(x \cdot \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), \color{blue}{2}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{2} + x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x - \frac{1}{8}\right)\right)\right)\right), 2\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \left(\mathsf{neg}\left(\frac{1}{8}\right)\right)\right)\right)\right)\right), 2\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{1}{16} \cdot x + \frac{-1}{8}\right)\right)\right)\right), 2\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \left(\frac{-1}{8} + \frac{1}{16} \cdot x\right)\right)\right)\right), 2\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(\frac{1}{16} \cdot x\right)\right)\right)\right)\right), 2\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \left(x \cdot \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

      11. *-lowering-*.f64 99.3%

          \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \frac{1}{16}\right)\right)\right)\right)\right), 2\right)\right) \]

   5. Simplified 99.3%

      \[\leadsto \frac{x}{\color{blue}{x \cdot \left(0.5 + x \cdot \left(-0.125 + x \cdot 0.0625\right)\right) + 2}} \]

   if 3.7000000000000002 < x

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

      1. sqrt-lowering-sqrt.f64 97.4%

         \[\leadsto \mathsf{sqrt.f64}\left(x\right) \]

   5. Simplified 97.4%

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

Alternative 6: 68.3% 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. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 69.6%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), 2\right)\right) \]

5. Simplified 69.6%

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

6. Final simplification 69.6%

   \[\leadsto \frac{x}{2 + x \cdot 0.5} \]
7. Add Preprocessing

Alternative 7: 68.1% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{0.5 + \frac{2}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 0.5 (/ 2.0 x))))

C

double code(double x) {
	return 1.0 / (0.5 + (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (0.5d0 + (2.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (0.5 + (2.0 / x));
}

Python

def code(x):
	return 1.0 / (0.5 + (2.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(0.5 + Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (0.5 + (2.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(0.5 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 69.6%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), 2\right)\right) \]

5. Simplified 69.6%

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

   1. clear-num N/A

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

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

      \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} \cdot x + 2}{x}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot x + 2\right), \color{blue}{x}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 + \frac{1}{2} \cdot x\right), x\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{2} \cdot x\right)\right), x\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \frac{1}{2}\right)\right), x\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(x \cdot \frac{1}{2}\right)\right), x\right)\right) \]

   8. div-inv N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{x}{2}\right)\right), x\right)\right) \]

   9. /-lowering-/.f64 69.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{/.f64}\left(x, 2\right)\right), x\right)\right) \]

7. Applied egg-rr 69.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{2} + 2 \cdot \frac{1}{x}\right)}\right) \]
9. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(2 \cdot \frac{1}{x}\right)}\right)\right) \]

   2. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{2 \cdot 1}{\color{blue}{x}}\right)\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{2}{x}\right)\right)\right) \]

   4. /-lowering-/.f64 69.5%

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(2, \color{blue}{x}\right)\right)\right) \]

10. Simplified 69.5%

    \[\leadsto \frac{1}{\color{blue}{0.5 + \frac{2}{x}}} \]
11. Add Preprocessing

Alternative 8: 67.6% 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 99.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{2}\right) \]
4. Step-by-step derivation

5. Simplified 68.7%

   \[\leadsto \frac{x}{\color{blue}{2}} \]
6. Add Preprocessing

Alternative 9: 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 99.7%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(2 + \frac{1}{2} \cdot x\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{2} \cdot x + \color{blue}{2}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{1}{2} \cdot x\right), \color{blue}{2}\right)\right) \]

   3. *-lowering-*.f64 69.6%

      \[\leadsto \mathsf{/.f64}\left(x, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, x\right), 2\right)\right) \]

5. Simplified 69.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{2} \]
7. Step-by-step derivation

8. Simplified 5.0%

   \[\leadsto \color{blue}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Numeric.Log:$clog1p from log-domain-0.10.2.1, B"
  :precision binary64
  (/ x (+ 1.0 (sqrt (+ x 1.0)))))