ln(1 + x)

Percentage Accurate: 39.6% → 100.0%

Time: 6.5s

Alternatives: 5

Speedup: 17.3×

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: 39.6% 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 40.5%

   \[\log \left(1 + x\right) \]
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: 68.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, -0.25, -0.3333333333333333\right)}{\mathsf{fma}\left(-0.25, x, -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, x, 1\right)} \cdot x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/
   (fma x -0.25 -0.3333333333333333)
   (* (fma -0.25 x -0.3333333333333333) (fma 0.5 x 1.0)))
  x))

C

double code(double x) {
	return (fma(x, -0.25, -0.3333333333333333) / (fma(-0.25, x, -0.3333333333333333) * fma(0.5, x, 1.0))) * x;
}

Julia

function code(x)
	return Float64(Float64(fma(x, -0.25, -0.3333333333333333) / Float64(fma(-0.25, x, -0.3333333333333333) * fma(0.5, x, 1.0))) * x)
end

Wolfram

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

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, x, 1\right)} \cdot x \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x, 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x, 1\right) \cdot x \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) \cdot x + \color{blue}{\frac{-1}{2}}, x, 1\right) \cdot x \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{4} \cdot x, x, \frac{-1}{2}\right)}, x, 1\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, x, \frac{-1}{2}\right), x, 1\right) \cdot x \]

   11. lower-fma.f64 66.0

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

5. Applied rewrites 66.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right), x, -0.5\right), x, 1\right) \cdot x} \]
6. Step-by-step derivation

7. Applied rewrites 66.0%

   \[\leadsto \mathsf{fma}\left(-\mathsf{fma}\left(0.25, x \cdot x, -1\right), \frac{1}{\mathsf{fma}\left(0.5, x, 1\right)}, \left(\mathsf{fma}\left(-0.25, x, 0.3333333333333333\right) \cdot x\right) \cdot x\right) \cdot x \]
8. Step-by-step derivation

9. Applied rewrites 65.9%

   \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right), \mathsf{fma}\left(-0.25, x, -0.3333333333333333\right), \mathsf{fma}\left(0.5, x, 1\right) \cdot \left(\mathsf{fma}\left(0.0625, x \cdot x, -0.1111111111111111\right) \cdot \left(x \cdot x\right)\right)\right)}{\mathsf{fma}\left(0.5, x, 1\right) \cdot \mathsf{fma}\left(-0.25, x, -0.3333333333333333\right)} \cdot x \]

10. Taylor expanded in x around 0

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

12. Applied rewrites 68.5%

    \[\leadsto \frac{\mathsf{fma}\left(x, -0.25, -0.3333333333333333\right)}{\mathsf{fma}\left(0.5, x, 1\right) \cdot \mathsf{fma}\left(-0.25, x, -0.3333333333333333\right)} \cdot x \]

13. Final simplification 68.5%

    \[\leadsto \frac{\mathsf{fma}\left(x, -0.25, -0.3333333333333333\right)}{\mathsf{fma}\left(-0.25, x, -0.3333333333333333\right) \cdot \mathsf{fma}\left(0.5, x, 1\right)} \cdot x \]
14. Add Preprocessing

Alternative 3: 67.0% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 67.0%

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

Alternative 4: 66.9% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.5%

   \[\log \left(1 + x\right) \]
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. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 67.0

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

5. Applied rewrites 67.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x, 1\right) \cdot x} \]
6. Add Preprocessing

Alternative 5: 66.5% accurate, 17.3× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* 1.0 x))

C

double code(double x) {
	return 1.0 * x;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0 * x

Julia

function code(x)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 * x;
end

Wolfram

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

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) - \frac{1}{2}, x, 1\right)} \cdot x \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, x, 1\right) \cdot x \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), x, 1\right) \cdot x \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} + \frac{-1}{4} \cdot x\right) \cdot x + \color{blue}{\frac{-1}{2}}, x, 1\right) \cdot x \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{-1}{4} \cdot x, x, \frac{-1}{2}\right)}, x, 1\right) \cdot x \]

   10. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{4} \cdot x + \frac{1}{3}}, x, \frac{-1}{2}\right), x, 1\right) \cdot x \]

   11. lower-fma.f64 66.0

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

5. Applied rewrites 66.0%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \cdot x \]
7. Step-by-step derivation

8. Applied rewrites 66.1%

   \[\leadsto 1 \cdot x \]
9. Add Preprocessing

Developer Target 1: 99.7% 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 2024270 
(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)))