ln(1 + x)

Percentage Accurate: 39.0% → 100.0%

Time: 9.2s

Alternatives: 6

Speedup: 8.7×

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: 39.0% 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 36.1%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lower-log1p.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 68.0% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x \cdot -0.25}{\mathsf{fma}\left(x, 0.3333333333333333, 0.5\right)}, x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (/ (* x -0.25) (fma x 0.3333333333333333 0.5)) x x))

C

double code(double x) {
	return fma(((x * -0.25) / fma(x, 0.3333333333333333, 0.5)), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * -0.25) / fma(x, 0.3333333333333333, 0.5)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(x * -0.25), $MachinePrecision] / N[(x * 0.3333333333333333 + 0.5), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. sub-neg N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right)}, {x}^{2}, x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{3}, \frac{-1}{2}\right), \color{blue}{x \cdot x}, x\right) \]

   13. lower-*.f64 67.4

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

5. Applied rewrites 67.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), x \cdot x, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 67.4%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, 0.1111111111111111, -0.25\right)}{\mathsf{fma}\left(x, 0.3333333333333333, 0.5\right)}, \color{blue}{x} \cdot x, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{4}}{\mathsf{fma}\left(x, \frac{1}{3}, \frac{1}{2}\right)}, x \cdot x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 67.5%

    \[\leadsto \mathsf{fma}\left(\frac{-0.25}{\mathsf{fma}\left(x, 0.3333333333333333, 0.5\right)}, x \cdot x, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 68.0%

    \[\leadsto \mathsf{fma}\left(\frac{-0.25 \cdot x}{\mathsf{fma}\left(x, 0.3333333333333333, 0.5\right)}, \color{blue}{x}, x\right) \]

13. Final simplification 68.0%

    \[\leadsto \mathsf{fma}\left(\frac{x \cdot -0.25}{\mathsf{fma}\left(x, 0.3333333333333333, 0.5\right)}, x, x\right) \]
14. Add Preprocessing

Developer Target 1: 99.6% accurate, 0.8× 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 2024225 
(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)))