ln(1 + x)

Percentage Accurate: 39.5% → 100.0%

Time: 7.9s

Alternatives: 5

Speedup: 104.0×

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: 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 42.0%

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

   1. lower-log1p.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 67.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\log \left(1 + x\right) \]
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 64.6

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

5. Simplified 64.6%

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

   1. flip-+ N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. swap-sqr N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{9}}, \frac{-1}{4}\right)}{x \cdot \frac{1}{3} - \frac{-1}{2}}, x \cdot x, x\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{9}, \frac{-1}{4}\right)}{\color{blue}{x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}, x \cdot x, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{9}, \frac{-1}{4}\right)}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{3}, \mathsf{neg}\left(\frac{-1}{2}\right)\right)}}, x \cdot x, x\right) \]

   12. metadata-eval 64.6

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

7. Applied egg-rr 64.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 64.5%

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

    1. lift-fma.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. lift-/.f64 N/A

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

    7. frac-2neg N/A

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

    8. associate-*l/ N/A

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

    9. associate-*l/ N/A

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

    10. associate-/l* N/A

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

    11. lower-fma.f64 N/A

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

12. Applied egg-rr 65.1%

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

Alternative 3: 67.6% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\log \left(1 + x\right) \]
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 64.6

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

5. Simplified 64.6%

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

   1. flip-+ N/A

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

   2. lower-/.f64 N/A

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

   3. sub-neg N/A

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

   4. swap-sqr N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{9}}, \frac{-1}{4}\right)}{x \cdot \frac{1}{3} - \frac{-1}{2}}, x \cdot x, x\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{9}, \frac{-1}{4}\right)}{\color{blue}{x \cdot \frac{1}{3} + \left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}, x \cdot x, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{9}, \frac{-1}{4}\right)}{\color{blue}{\mathsf{fma}\left(x, \frac{1}{3}, \mathsf{neg}\left(\frac{-1}{2}\right)\right)}}, x \cdot x, x\right) \]

   12. metadata-eval 64.6

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

7. Applied egg-rr 64.6%

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

8. Taylor expanded in x around 0

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

10. Simplified 64.5%

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

    1. lift-fma.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*r* N/A

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

    6. distribute-lft1-in N/A

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

    7. lower-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-fma.f64 65.1

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

12. Applied egg-rr 65.1%

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

13. Final simplification 65.1%

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

Alternative 4: 67.0% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.0%

   \[\log \left(1 + x\right) \]
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 64.6

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

5. Simplified 64.6%

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

Alternative 5: 66.6% accurate, 104.0× 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 42.0%

   \[\log \left(1 + x\right) \]
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 63.2

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

5. Simplified 63.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft1-in N/A

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

   4. lower-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 63.2

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

7. Applied egg-rr 63.2%

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

8. Taylor expanded in x around 0

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

10. Simplified 64.1%

    \[\leadsto \color{blue}{1} \cdot x \]

11. Final simplification 64.1%

    \[\leadsto x \]
12. 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 2024207 
(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)))