ln(1 + x)

Percentage Accurate: 38.7% → 100.0%

Time: 6.7s

Alternatives: 5

Speedup: 17.1×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ 1.0 x)))

C

double code(double x) {
	return log((1.0 + x));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log((1.0 + x))

Julia

function code(x)
	return log(Float64(1.0 + x))
end

MATLAB

function tmp = code(x)
	tmp = log((1.0 + x));
end

Wolfram

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

Initial Program: 38.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ 1.0 x)))

C

double code(double x) {
	return log((1.0 + x));
}

Fortran

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

Java

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

Python

def code(x):
	return math.log((1.0 + x))

Julia

function code(x)
	return log(Float64(1.0 + x))
end

MATLAB

function tmp = code(x)
	tmp = log((1.0 + x));
end

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p x))

C

double code(double x) {
	return log1p(x);
}

Java

public static double code(double x) {
	return Math.log1p(x);
}

Python

def code(x):
	return math.log1p(x)

Julia

function code(x)
	return log1p(x)
end

Wolfram

code[x_] := N[Log[1 + x], $MachinePrecision]

Derivation

1. Initial program 39.6%

   \[\log \left(1 + x\right) \]
2. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{log1p}\left(x\right) \]

   2. log1p-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 71.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;x \cdot \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.45) (* x (+ 1.0 (* x -0.5))) 2.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = x * (1.0 + (x * -0.5));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = x * (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = x * (1.0 + (x * -0.5));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = x * (1.0 + (x * -0.5))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(x * Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = x * (1.0 + (x * -0.5));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 8.5%

      \[\log \left(1 + x\right) \]
   2. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{log1p}\left(x\right) \]

      2. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{log1p}\left(x\right) \]

      2. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 0.9%

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

   7. Simplified 0.9%

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

      1. flip3-+ N/A

         \[\leadsto x \cdot \frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}}}\right)\right) \]

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 0.9%

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

   9. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + x \cdot -0.5}}} \]

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 14.5%

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

   12. Simplified 14.5%

       \[\leadsto \frac{x}{\color{blue}{1 + x \cdot 0.5}} \]

   13. Taylor expanded in x around inf

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

   15. Simplified 14.5%

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

Alternative 3: 71.0% accurate, 14.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.6%

   \[\log \left(1 + x\right) \]
2. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{log1p}\left(x\right) \]

   2. log1p-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 66.0%

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

7. Simplified 66.0%

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

   1. flip3-+ N/A

      \[\leadsto x \cdot \frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}} \]

   2. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}\right)}\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}}}\right)\right) \]

   6. flip3-+ N/A

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

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

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

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

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

   9. *-lowering-*.f64 66.0%

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

9. Applied egg-rr 66.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + x \cdot -0.5}}} \]

10. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 70.6%

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

12. Simplified 70.6%

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

Alternative 4: 70.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) x 2.0))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = x
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = x;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = x
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = x;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], x, 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 8.5%

      \[\log \left(1 + x\right) \]
   2. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{log1p}\left(x\right) \]

      2. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

   7. Simplified 98.0%

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

   if 2 < x

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{log1p}\left(x\right) \]

      2. log1p-lowering-log1p.f64 100.0%

         \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 0.9%

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

   7. Simplified 0.9%

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

      1. flip3-+ N/A

         \[\leadsto x \cdot \frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}\right)}\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}}}\right)\right) \]

      6. flip3-+ N/A

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

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

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

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

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

      9. *-lowering-*.f64 0.9%

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

   9. Applied egg-rr 0.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + x \cdot -0.5}}} \]

   10. Taylor expanded in x around 0

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

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

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

       2. *-commutative N/A

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

       3. *-lowering-*.f64 14.5%

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

   12. Simplified 14.5%

       \[\leadsto \frac{x}{\color{blue}{1 + x \cdot 0.5}} \]

   13. Taylor expanded in x around inf

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

   15. Simplified 14.5%

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

Alternative 5: 7.4% accurate, 103.0× 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 39.6%

   \[\log \left(1 + x\right) \]
2. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{log1p}\left(x\right) \]

   2. log1p-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0

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

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

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 66.0%

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

7. Simplified 66.0%

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

   1. flip3-+ N/A

      \[\leadsto x \cdot \frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}} \]

   2. clear-num N/A

      \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}}} \]

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

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}\right)}\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{{1}^{3} + {\left(x \cdot \frac{-1}{2}\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \frac{-1}{2}\right) \cdot \left(x \cdot \frac{-1}{2}\right) - 1 \cdot \left(x \cdot \frac{-1}{2}\right)\right)}}}\right)\right) \]

   6. flip3-+ N/A

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

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

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

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

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

   9. *-lowering-*.f64 66.0%

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

9. Applied egg-rr 66.0%

   \[\leadsto \color{blue}{\frac{x}{\frac{1}{1 + x \cdot -0.5}}} \]

10. Taylor expanded in x around 0

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

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

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

    2. *-commutative N/A

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

    3. *-lowering-*.f64 70.6%

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

12. Simplified 70.6%

    \[\leadsto \frac{x}{\color{blue}{1 + x \cdot 0.5}} \]

13. Taylor expanded in x around inf

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

15. Simplified 7.4%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + x = 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \log \left(1 + x\right)}{\left(1 + x\right) - 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0))))

C

double code(double x) {
	double tmp;
	if ((1.0 + x) == 1.0) {
		tmp = x;
	} else {
		tmp = (x * log((1.0 + x))) / ((1.0 + x) - 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((1.0d0 + x) == 1.0d0) then
        tmp = x
    else
        tmp = (x * log((1.0d0 + x))) / ((1.0d0 + x) - 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 + x) == 1.0) {
		tmp = x;
	} else {
		tmp = (x * Math.log((1.0 + x))) / ((1.0 + x) - 1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 + x) == 1.0:
		tmp = x
	else:
		tmp = (x * math.log((1.0 + x))) / ((1.0 + x) - 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 + x) == 1.0)
		tmp = x;
	else
		tmp = Float64(Float64(x * log(Float64(1.0 + x))) / Float64(Float64(1.0 + x) - 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 + x) == 1.0)
		tmp = x;
	else
		tmp = (x * log((1.0 + x))) / ((1.0 + x) - 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Equal[N[(1.0 + x), $MachinePrecision], 1.0], x, N[(N[(x * N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :alt
  (! :herbie-platform default (if (== (+ 1 x) 1) x (/ (* x (log (+ 1 x))) (- (+ 1 x) 1))))

  (log (+ 1.0 x)))