ln(1 + x)

Percentage Accurate: 38.7% → 100.0%

Time: 11.6s

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: 38.7% 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 38.1%

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

   1. accelerator-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

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

Alternative 2: 67.7% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. accelerator-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{3} \cdot x}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{3} \cdot x\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 68.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

7. Simplified 68.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(-0.5 + x \cdot 0.3333333333333333\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{3}\right)\right)\right), \color{blue}{x}\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(x \cdot \left(\frac{-1}{2} + x \cdot \frac{1}{3}\right)\right) \cdot x\right), x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{-1}{2} + x \cdot \frac{1}{3}\right) \cdot x\right) \cdot x\right), x\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{-1}{2} + x \cdot \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right), x\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{-1}{2} + x \cdot \frac{1}{3}\right), \left(x \cdot x\right)\right), x\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \frac{1}{3}\right)\right), \left(x \cdot x\right)\right), x\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right), \left(x \cdot x\right)\right), x\right) \]

   11. *-lowering-*.f64 68.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \frac{1}{3}\right)\right), \mathsf{*.f64}\left(x, x\right)\right), x\right) \]

9. Applied egg-rr 68.3%

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

10. Final simplification 68.3%

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

Alternative 3: 67.7% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.1%

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

   1. accelerator-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(x \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)\right)}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)}\right)\right)\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{1}{3} \cdot x + \frac{-1}{2}\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\frac{-1}{2} + \color{blue}{\frac{1}{3} \cdot x}\right)\right)\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{3} \cdot x\right)}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \left(x \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 68.3%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(x, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

7. Simplified 68.3%

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

Alternative 4: 67.3% 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 38.1%

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

   1. accelerator-lowering-log1p.f64 100.0%

      \[\leadsto \mathsf{log1p.f64}\left(x\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 68.0%

   \[\leadsto \color{blue}{x} \]
8. 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 2024191 
(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)))