ln(1 + x)

Percentage Accurate: 38.3% → 100.0%

Time: 4.1s

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 40.0%

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

   1. log1p-def 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot -0.25 + 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45)
   (+ x (* (* x x) (+ -0.5 (* x (+ (* x -0.25) 0.3333333333333333)))))
   (log x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = x + ((x * x) * (-0.5 + (x * ((x * -0.25) + 0.3333333333333333))));
	} else {
		tmp = log(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = x + ((x * x) * ((-0.5d0) + (x * ((x * (-0.25d0)) + 0.3333333333333333d0))))
    else
        tmp = log(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = x + ((x * x) * (-0.5 + (x * ((x * -0.25) + 0.3333333333333333))));
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = x + ((x * x) * (-0.5 + (x * ((x * -0.25) + 0.3333333333333333))))
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(-0.5 + Float64(x * Float64(Float64(x * -0.25) + 0.3333333333333333)))));
	else
		tmp = log(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = x + ((x * x) * (-0.5 + (x * ((x * -0.25) + 0.3333333333333333))));
	else
		tmp = log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(x + N[(N[(x * x), $MachinePrecision] * N[(-0.5 + N[(x * N[(N[(x * -0.25), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 8.5%

      \[\log \left(1 + x\right) \]

   2. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x + \left(-0.5 \cdot {x}^{2} + \left(-0.25 \cdot {x}^{4} + 0.3333333333333333 \cdot {x}^{3}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto x + \left(\color{blue}{{x}^{2} \cdot -0.5} + \left(-0.25 \cdot {x}^{4} + 0.3333333333333333 \cdot {x}^{3}\right)\right) \]

      2. +-commutative 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \color{blue}{\left(0.3333333333333333 \cdot {x}^{3} + -0.25 \cdot {x}^{4}\right)}\right) \]

      3. *-commutative 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{3} \cdot 0.3333333333333333} + -0.25 \cdot {x}^{4}\right)\right) \]

      4. unpow3 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.3333333333333333 + -0.25 \cdot {x}^{4}\right)\right) \]

      5. unpow2 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot 0.3333333333333333 + -0.25 \cdot {x}^{4}\right)\right) \]

      6. associate-*l* 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{2} \cdot \left(x \cdot 0.3333333333333333\right)} + -0.25 \cdot {x}^{4}\right)\right) \]

      7. *-commutative 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{{x}^{4} \cdot -0.25}\right)\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot -0.25\right)\right) \]

      9. pow-sqr 99.7%

         \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot -0.25\right)\right) \]

      10. associate-*l* 99.7%

          \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) \]

      11. distribute-lft-out 99.7%

          \[\leadsto x + \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)}\right) \]

      12. distribute-lft-out 99.7%

          \[\leadsto x + \color{blue}{{x}^{2} \cdot \left(-0.5 + \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)\right)} \]

      13. unpow2 99.7%

          \[\leadsto x + \color{blue}{\left(x \cdot x\right)} \cdot \left(-0.5 + \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)\right) \]

      14. +-commutative 99.7%

          \[\leadsto x + \left(x \cdot x\right) \cdot \left(-0.5 + \color{blue}{\left({x}^{2} \cdot -0.25 + x \cdot 0.3333333333333333\right)}\right) \]

      15. unpow2 99.7%

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

      16. associate-*l* 99.7%

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

      17. distribute-lft-out 99.7%

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

   4. Simplified 99.7%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]

   2. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. log-rec 99.2%

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

      3. remove-double-neg 99.2%

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

   4. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(-0.5 + x \cdot \left(x \cdot -0.25 + 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]

Alternative 3: 67.9% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.0%

   \[\log \left(1 + x\right) \]

2. Taylor expanded in x around 0 65.6%

   \[\leadsto \color{blue}{x + \left(-0.5 \cdot {x}^{2} + \left(-0.25 \cdot {x}^{4} + 0.3333333333333333 \cdot {x}^{3}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative 65.6%

      \[\leadsto x + \left(\color{blue}{{x}^{2} \cdot -0.5} + \left(-0.25 \cdot {x}^{4} + 0.3333333333333333 \cdot {x}^{3}\right)\right) \]

   2. +-commutative 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \color{blue}{\left(0.3333333333333333 \cdot {x}^{3} + -0.25 \cdot {x}^{4}\right)}\right) \]

   3. *-commutative 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{3} \cdot 0.3333333333333333} + -0.25 \cdot {x}^{4}\right)\right) \]

   4. unpow3 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot 0.3333333333333333 + -0.25 \cdot {x}^{4}\right)\right) \]

   5. unpow2 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\left(\color{blue}{{x}^{2}} \cdot x\right) \cdot 0.3333333333333333 + -0.25 \cdot {x}^{4}\right)\right) \]

   6. associate-*l* 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left(\color{blue}{{x}^{2} \cdot \left(x \cdot 0.3333333333333333\right)} + -0.25 \cdot {x}^{4}\right)\right) \]

   7. *-commutative 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{{x}^{4} \cdot -0.25}\right)\right) \]

   8. metadata-eval 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + {x}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot -0.25\right)\right) \]

   9. pow-sqr 65.6%

      \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)} \cdot -0.25\right)\right) \]

   10. associate-*l* 65.6%

       \[\leadsto x + \left({x}^{2} \cdot -0.5 + \left({x}^{2} \cdot \left(x \cdot 0.3333333333333333\right) + \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot -0.25\right)}\right)\right) \]

   11. distribute-lft-out 65.7%

       \[\leadsto x + \left({x}^{2} \cdot -0.5 + \color{blue}{{x}^{2} \cdot \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)}\right) \]

   12. distribute-lft-out 65.7%

       \[\leadsto x + \color{blue}{{x}^{2} \cdot \left(-0.5 + \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)\right)} \]

   13. unpow2 65.7%

       \[\leadsto x + \color{blue}{\left(x \cdot x\right)} \cdot \left(-0.5 + \left(x \cdot 0.3333333333333333 + {x}^{2} \cdot -0.25\right)\right) \]

   14. +-commutative 65.7%

       \[\leadsto x + \left(x \cdot x\right) \cdot \left(-0.5 + \color{blue}{\left({x}^{2} \cdot -0.25 + x \cdot 0.3333333333333333\right)}\right) \]

   15. unpow2 65.7%

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

   16. associate-*l* 65.7%

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

   17. distribute-lft-out 65.7%

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

4. Simplified 65.7%

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

5. Taylor expanded in x around 0 66.7%

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

6. Final simplification 66.7%

   \[\leadsto x + \left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333 - 0.5\right) \]

Alternative 4: 67.7% 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 40.0%

   \[\log \left(1 + x\right) \]

2. Taylor expanded in x around 0 66.2%

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

3. Final simplification 66.2%

   \[\leadsto x \]

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 2023271 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

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

  (log (+ 1.0 x)))