ln(1 + x)

Percentage Accurate: 38.3% → 100.0%

Time: 3.9s

Alternatives: 4

Speedup: 103.0×

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: 38.3% 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 41.2%

   \[\log \left(1 + x\right) \]
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. Final simplification 100.0%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (* x (+ 1.0 (* x (- (* x (+ 0.3333333333333333 (* x -0.25))) 0.5))))
   (log x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	} else {
		tmp = log(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = x * (1.0d0 + (x * ((x * (0.3333333333333333d0 + (x * (-0.25d0)))) - 0.5d0)))
    else
        tmp = log(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)))
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.3333333333333333 + Float64(x * -0.25))) - 0.5))));
	else
		tmp = log(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = x * (1.0 + (x * ((x * (0.3333333333333333 + (x * -0.25))) - 0.5)));
	else
		tmp = log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], 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], N[Log[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 7.6%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

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

   if 1.3999999999999999 < x

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.0%

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

      1. mul-1-neg 96.0%

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

      2. log-rec 96.0%

         \[\leadsto -\color{blue}{\left(-\log x\right)} \]

      3. remove-double-neg 96.0%

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

   5. Simplified 96.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;x \cdot \left(1 + x \cdot \left(x \cdot \left(0.3333333333333333 + x \cdot -0.25\right) - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (- (* x 0.3333333333333333) 0.5)))))

C

double code(double x) {
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * ((x * 0.3333333333333333d0) - 0.5d0)))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
}

Python

def code(x):
	return x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(Float64(x * 0.3333333333333333) - 0.5))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * ((x * 0.3333333333333333) - 0.5)));
end

Wolfram

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

Derivation

1. Initial program 41.2%

   \[\log \left(1 + x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 64.9%

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

4. Final simplification 64.9%

   \[\leadsto x \cdot \left(1 + x \cdot \left(x \cdot 0.3333333333333333 - 0.5\right)\right) \]
5. Add Preprocessing

Alternative 4: 67.8% accurate, 103.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 41.2%

   \[\log \left(1 + x\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 64.9%

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

4. Final simplification 64.9%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 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 2024060 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :alt
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))