ln(1 + x)

Percentage Accurate: 39.5% → 100.0%

Time: 8.2s

Alternatives: 7

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: 39.5% 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 39.0%

   \[\log \left(1 + x\right) \]
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: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0)
   (fma (fma x (fma x -0.25 0.3333333333333333) -0.5) (* x x) x)
   (log x)))

C

double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = fma(fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), (x * x), x);
	} else {
		tmp = log(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 2.0)
		tmp = fma(fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), Float64(x * x), x);
	else
		tmp = log(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 2.0], N[(N[(x * N[(x * -0.25 + 0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[Log[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 8.7%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2 < (+.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f64 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0)
   (fma (fma x (fma x -0.25 0.3333333333333333) -0.5) (* x x) x)
   (/ 1.0 0.5)))

C

double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = fma(fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), (x * x), x);
	} else {
		tmp = 1.0 / 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 2.0)
		tmp = fma(fma(x, fma(x, -0.25, 0.3333333333333333), -0.5), Float64(x * x), x);
	else
		tmp = Float64(1.0 / 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 2.0], N[(N[(x * N[(x * -0.25 + 0.3333333333333333), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(1.0 / 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 8.7%

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

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. unpow2 N/A

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

      6. *-lft-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if 2 < (+.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   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 4.0

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

   5. Applied rewrites 4.0%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{1 + \frac{1}{2} \cdot x}{\color{blue}{x}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{\frac{1}{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 14.5%

       \[\leadsto \frac{1}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.25, 0.3333333333333333\right), -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.1% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (/ (fma x 0.5 1.0) x)))

C

double code(double x) {
	return 1.0 / (fma(x, 0.5, 1.0) / x);
}

Julia

function code(x)
	return Float64(1.0 / Float64(fma(x, 0.5, 1.0) / x))
end

Wolfram

code[x_] := N[(1.0 / N[(N[(x * 0.5 + 1.0), $MachinePrecision] / x), $MachinePrecision]), $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 67.6

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

5. Applied rewrites 67.6%

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

7. Applied rewrites 67.4%

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

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\frac{1 + \frac{1}{2} \cdot x}{\color{blue}{x}}} \]
9. Step-by-step derivation

10. Applied rewrites 70.6%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x, 0.5, 1\right)}{\color{blue}{x}}} \]
11. Add Preprocessing

Alternative 5: 70.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0)
   (fma (fma x 0.3333333333333333 -0.5) (* x x) x)
   (/ 1.0 0.5)))

C

double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = fma(fma(x, 0.3333333333333333, -0.5), (x * x), x);
	} else {
		tmp = 1.0 / 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 2.0)
		tmp = fma(fma(x, 0.3333333333333333, -0.5), Float64(x * x), x);
	else
		tmp = Float64(1.0 / 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 2.0], N[(N[(x * 0.3333333333333333 + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(1.0 / 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 8.7%

      \[\log \left(1 + x\right) \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 2 < (+.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   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 4.0

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

   5. Applied rewrites 4.0%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{1 + \frac{1}{2} \cdot x}{\color{blue}{x}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{\frac{1}{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 14.5%

       \[\leadsto \frac{1}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x, 0.3333333333333333, -0.5\right), x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x 1.0) 2.0) (* x (fma x -0.5 1.0)) (/ 1.0 0.5)))

C

double code(double x) {
	double tmp;
	if ((x + 1.0) <= 2.0) {
		tmp = x * fma(x, -0.5, 1.0);
	} else {
		tmp = 1.0 / 0.5;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + 1.0) <= 2.0)
		tmp = Float64(x * fma(x, -0.5, 1.0));
	else
		tmp = Float64(1.0 / 0.5);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + 1.0), $MachinePrecision], 2.0], N[(x * N[(x * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 #s(literal 1 binary64) x) < 2

   1. Initial program 8.7%

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

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

   if 2 < (+.f64 #s(literal 1 binary64) x)

   1. Initial program 100.0%

      \[\log \left(1 + x\right) \]
   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 4.0

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

   5. Applied rewrites 4.0%

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

   7. Applied rewrites 4.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\frac{1 + \frac{1}{2} \cdot x}{\color{blue}{x}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 14.5%

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

   11. Taylor expanded in x around inf

       \[\leadsto \frac{1}{\frac{1}{2}} \]
   12. Step-by-step derivation

   13. Applied rewrites 14.5%

       \[\leadsto \frac{1}{0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + 1 \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.5% accurate, 17.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 66.3

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

5. Applied rewrites 66.3%

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

7. Applied rewrites 66.3%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 67.2%

    \[\leadsto 1 \cdot x \]

11. Final simplification 67.2%

    \[\leadsto x \cdot 1 \]
12. 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 2024223 
(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)))