ln(1 + x)

Percentage Accurate: 39.8% → 100.0%

Time: 5.1s

Alternatives: 8

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: 39.8% 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 40.9%

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

   1. log1p-define 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 70.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.56)
   (* x (+ 1.0 (* x (- (* x (+ 0.3333333333333333 (* x -0.25))) 0.5))))
   9.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.56d0) then
        tmp = x * (1.0d0 + (x * ((x * (0.3333333333333333d0 + (x * (-0.25d0)))) - 0.5d0)))
    else
        tmp = 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.56:
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))
	else:
		tmp = 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.56)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.3333333333333333 + Float64(x * -0.25))) - 0.5))));
	else
		tmp = 9.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.56)
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	else
		tmp = 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.56], N[(x * N[(1.0 + N[(x * N[(N[(x * N[(0.3333333333333333 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 9.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 8.8%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + -0.25 \cdot x\right) - 0.5\right)\right)} \]

   if 1.5600000000000001 < x

   1. Initial program 100.0%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in x around inf 0.9%

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

   9. Taylor expanded in x around 0 5.4%

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

   11. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2) (* x (+ 1.0 (* x (- (* x 0.3333333333333333) 0.5)))) 9.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.2d0) then
        tmp = x * (1.0d0 + (x * ((x * 0.3333333333333333d0) - 0.5d0)))
    else
        tmp = 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.2:
		tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))
	else:
		tmp = 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.3333333333333333) - 0.5))));
	else
		tmp = 9.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.2)
		tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
	else
		tmp = 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[(x * N[(1.0 + N[(x * N[(N[(x * 0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 9.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.2000000000000002

   1. Initial program 8.8%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.3333333333333333 \cdot x - 0.5\right)\right)} \]

   if 3.2000000000000002 < x

   1. Initial program 100.0%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in x around inf 0.9%

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

   9. Taylor expanded in x around 0 5.4%

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

   11. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{9} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.4% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.9:\\ \;\;\;\;x \cdot \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.9) (* x (+ 1.0 (* x -0.5))) 9.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = x * (1.0 + (x * -0.5));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.9d0) then
        tmp = x * (1.0d0 + (x * (-0.5d0)))
    else
        tmp = 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.9) {
		tmp = x * (1.0 + (x * -0.5));
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.9:
		tmp = x * (1.0 + (x * -0.5))
	else:
		tmp = 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.9)
		tmp = Float64(x * Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = 9.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.9)
		tmp = x * (1.0 + (x * -0.5));
	else
		tmp = 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.9], N[(x * N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 9.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.8999999999999999

   1. Initial program 8.8%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   if 1.8999999999999999 < x

   1. Initial program 100.0%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in x around inf 0.9%

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

   9. Taylor expanded in x around 0 5.4%

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

   11. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{9} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.7% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;9\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 9.0) x 9.0))

C

double code(double x) {
	double tmp;
	if (x <= 9.0) {
		tmp = x;
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 9.0d0) then
        tmp = x
    else
        tmp = 9.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 9.0) {
		tmp = x;
	} else {
		tmp = 9.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 9.0:
		tmp = x
	else:
		tmp = 9.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 9.0)
		tmp = x;
	else
		tmp = 9.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 9.0)
		tmp = x;
	else
		tmp = 9.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 9.0], x, 9.0]

Derivation

1. Split input into 2 regimes

2. if x < 9

   1. Initial program 8.8%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 97.5%

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

   if 9 < x

   1. Initial program 100.0%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 0.9%

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

      1. *-commutative 0.9%

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

   7. Simplified 0.9%

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

   8. Taylor expanded in x around inf 0.9%

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

   9. Taylor expanded in x around 0 5.4%

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

   11. Applied egg-rr 15.4%

       \[\leadsto \color{blue}{9} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 7.8% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 9 \end{array} \]

FPCore

(FPCore (x) :precision binary64 9.0)

C

double code(double x) {
	return 9.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 9.0d0
end function

Java

public static double code(double x) {
	return 9.0;
}

Python

def code(x):
	return 9.0

Julia

function code(x)
	return 9.0
end

MATLAB

function tmp = code(x)
	tmp = 9.0;
end

Wolfram

code[x_] := 9.0

Derivation

1. Initial program 40.9%

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

   1. log1p-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 64.1%

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

   1. *-commutative 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in x around inf 64.1%

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

9. Taylor expanded in x around 0 65.0%

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

11. Applied egg-rr 7.8%

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

Alternative 7: 7.7% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (x) :precision binary64 8.0)

C

double code(double x) {
	return 8.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 8.0d0
end function

Java

public static double code(double x) {
	return 8.0;
}

Python

def code(x):
	return 8.0

Julia

function code(x)
	return 8.0
end

MATLAB

function tmp = code(x)
	tmp = 8.0;
end

Wolfram

code[x_] := 8.0

Derivation

1. Initial program 40.9%

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

   1. log1p-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 64.1%

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

   1. *-commutative 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in x around inf 64.1%

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

9. Taylor expanded in x around 0 65.0%

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

11. Applied egg-rr 7.8%

    \[\leadsto \color{blue}{8} \]
12. Add Preprocessing

Alternative 8: 7.2% accurate, 103.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.3333333333333333)

C

double code(double x) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.3333333333333333d0
end function

Java

public static double code(double x) {
	return 0.3333333333333333;
}

Python

def code(x):
	return 0.3333333333333333

Julia

function code(x)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_] := 0.3333333333333333

Derivation

1. Initial program 40.9%

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

   1. log1p-define 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0 64.1%

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

   1. *-commutative 64.1%

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

7. Simplified 64.1%

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

8. Taylor expanded in x around inf 64.1%

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

9. Taylor expanded in x around 0 65.0%

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

11. Applied egg-rr 7.2%

    \[\leadsto \color{blue}{0.3333333333333333} \]
12. Add Preprocessing

Developer Target 1: 99.6% 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 2024130 
(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)))