ln(1 + x)

Percentage Accurate: 96.7% → 98.6%

Time: 13.3s

Alternatives: 7

Speedup: 1.0×

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: 96.7% 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: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26} \lor \neg \left(x \leq 5.2 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.6e-26) (not (<= x 5.2e-26))) (log1p x) (+ (+ x 1.0) -1.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.6e-26) || !(x <= 5.2e-26)) {
		tmp = log1p(x);
	} else {
		tmp = (x + 1.0) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.6e-26) || !(x <= 5.2e-26)) {
		tmp = Math.log1p(x);
	} else {
		tmp = (x + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.6e-26) or not (x <= 5.2e-26):
		tmp = math.log1p(x)
	else:
		tmp = (x + 1.0) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.6e-26) || !(x <= 5.2e-26))
		tmp = log1p(x);
	else
		tmp = Float64(Float64(x + 1.0) + -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.6e-26], N[Not[LessEqual[x, 5.2e-26]], $MachinePrecision]], N[Log[1 + x], $MachinePrecision], N[(N[(x + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001e-26 or 5.2000000000000002e-26 < x

   1. Initial program 89.4%

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

      1. log1p-def 95.3%

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

   3. Simplified 95.3%

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

   if -2.6000000000000001e-26 < x < 5.2000000000000002e-26

   1. Initial program 100.0%

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

      1. add-cbrt-cube 100.0%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}\right)} \]

      2. pow1/3 100.0%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 100.0%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)} \]

      4. pow3 100.0%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(1 + x\right)}^{3}\right)} \]

      5. log-pow 100.0%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(1 + x\right)\right)} \]

      6. log1p-def 5.4%

         \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

   4. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \mathsf{log1p}\left(x\right)\right)} \]

   5. Taylor expanded in x around 0 5.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)} \]
   6. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)} \]

   7. Simplified 5.4%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)} \]
   8. Step-by-step derivation

      1. *-commutative 5.4%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)} \]

      2. associate-*r* 5.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x} \]

      3. metadata-eval 5.4%

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

      4. expm1-log1p-u 5.4%

         \[\leadsto 1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

      5. *-un-lft-identity 5.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

      6. expm1-udef 100.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]

      7. log1p-udef 100.0%

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

      8. rem-exp-log 100.0%

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

      9. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x + 1\right) - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26} \lor \neg \left(x \leq 5.2 \cdot 10^{-26}\right):\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \left(\left(1 + \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right) + -1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow (pow (log1p x) 16.0) 0.05555555555555555)
  (+ (+ 1.0 (cbrt (cbrt (log1p x)))) -1.0)))

C

double code(double x) {
	return pow(pow(log1p(x), 16.0), 0.05555555555555555) * ((1.0 + cbrt(cbrt(log1p(x)))) + -1.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.log1p(x), 16.0), 0.05555555555555555) * ((1.0 + Math.cbrt(Math.cbrt(Math.log1p(x)))) + -1.0);
}

Julia

function code(x)
	return Float64(((log1p(x) ^ 16.0) ^ 0.05555555555555555) * Float64(Float64(1.0 + cbrt(cbrt(log1p(x)))) + -1.0))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[Log[1 + x], $MachinePrecision], 16.0], $MachinePrecision], 0.05555555555555555], $MachinePrecision] * N[(N[(1.0 + N[Power[N[Power[N[Log[1 + x], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. add-cbrt-cube 96.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}} \]

   2. pow1/3 94.4%

      \[\leadsto \color{blue}{{\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)\right)}^{0.3333333333333333}} \]

   3. add-cube-cbrt 94.4%

      \[\leadsto {\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) \cdot \sqrt[3]{\log \left(1 + x\right)}\right)}\right)}^{0.3333333333333333} \]

   4. associate-*r* 94.4%

      \[\leadsto {\color{blue}{\left(\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right)\right) \cdot \sqrt[3]{\log \left(1 + x\right)}\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 94.5%

      \[\leadsto \color{blue}{{\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right)\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\log \left(1 + x\right)}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 76.8%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}} \]
5. Step-by-step derivation

   1. unpow1/3 76.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. unpow2 76.9%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. rem-cube-cbrt 76.7%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-plus 76.7%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(3 + 1\right)}} \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. metadata-eval 76.7%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{4}} \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. rem-cube-cbrt 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot \left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. pow-plus 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(3 + 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   8. metadata-eval 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{4}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   9. pow-sqr 76.6%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(2 \cdot 4\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   10. metadata-eval 76.6%

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{8}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

6. Simplified 76.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}} \]
7. Step-by-step derivation

   1. pow1/3 76.6%

      \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. add-cbrt-cube 93.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. pow1/3 93.3%

      \[\leadsto {\color{blue}{\left({\left(\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{0.3333333333333333}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-pow 93.5%

      \[\leadsto \color{blue}{{\left(\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. pow3 93.5%

      \[\leadsto {\color{blue}{\left({\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{3}\right)}}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. pow1/3 57.8%

      \[\leadsto {\left({\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{0.3333333333333333}\right)}}^{8}\right)}^{3}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. pow-pow 57.8%

      \[\leadsto {\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(0.3333333333333333 \cdot 8\right)}\right)}}^{3}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   8. pow-pow 93.6%

      \[\leadsto {\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(\left(0.3333333333333333 \cdot 8\right) \cdot 3\right)}\right)}}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   9. metadata-eval 93.6%

      \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(\color{blue}{2.6666666666666665} \cdot 3\right)}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   10. metadata-eval 93.6%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{8}}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   11. metadata-eval 93.6%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\color{blue}{0.1111111111111111}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 93.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. expm1-udef 95.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. sqr-pow 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)} \cdot {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-prod-down 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. pow-prod-up 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(8 + 8\right)}\right)}}^{\left(\frac{0.1111111111111111}{2}\right)}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. metadata-eval 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{16}}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. metadata-eval 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{\color{blue}{0.05555555555555555}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

10. Applied egg-rr 95.8%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
11. Step-by-step derivation

    1. expm1-def 97.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

    2. expm1-log1p 97.3%

       \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

12. Simplified 97.3%

    \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
13. Step-by-step derivation

    1. expm1-log1p-u 97.4%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right)\right)} \]

    2. expm1-udef 97.4%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right)} - 1\right)} \]

    3. log1p-udef 97.4%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \left(e^{\color{blue}{\log \left(1 + \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right)}} - 1\right) \]

    4. rem-exp-log 97.4%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \left(\color{blue}{\left(1 + \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right)} - 1\right) \]

14. Applied egg-rr 97.4%

    \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \color{blue}{\left(\left(1 + \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right) - 1\right)} \]

15. Final simplification 97.4%

    \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \left(\left(1 + \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}\right) + -1\right) \]
16. Add Preprocessing

Alternative 3: 97.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (pow (log1p x) 16.0) 0.05555555555555555) (cbrt (cbrt (log1p x)))))

C

double code(double x) {
	return pow(pow(log1p(x), 16.0), 0.05555555555555555) * cbrt(cbrt(log1p(x)));
}

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.log1p(x), 16.0), 0.05555555555555555) * Math.cbrt(Math.cbrt(Math.log1p(x)));
}

Julia

function code(x)
	return Float64(((log1p(x) ^ 16.0) ^ 0.05555555555555555) * cbrt(cbrt(log1p(x))))
end

Wolfram

code[x_] := N[(N[Power[N[Power[N[Log[1 + x], $MachinePrecision], 16.0], $MachinePrecision], 0.05555555555555555], $MachinePrecision] * N[Power[N[Power[N[Log[1 + x], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.3%

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

   1. add-cbrt-cube 96.0%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)}} \]

   2. pow1/3 94.4%

      \[\leadsto \color{blue}{{\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \log \left(1 + x\right)\right)}^{0.3333333333333333}} \]

   3. add-cube-cbrt 94.4%

      \[\leadsto {\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right) \cdot \sqrt[3]{\log \left(1 + x\right)}\right)}\right)}^{0.3333333333333333} \]

   4. associate-*r* 94.4%

      \[\leadsto {\color{blue}{\left(\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right)\right) \cdot \sqrt[3]{\log \left(1 + x\right)}\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 94.5%

      \[\leadsto \color{blue}{{\left(\left(\log \left(1 + x\right) \cdot \log \left(1 + x\right)\right) \cdot \left(\sqrt[3]{\log \left(1 + x\right)} \cdot \sqrt[3]{\log \left(1 + x\right)}\right)\right)}^{0.3333333333333333} \cdot {\left(\sqrt[3]{\log \left(1 + x\right)}\right)}^{0.3333333333333333}} \]

4. Applied egg-rr 76.8%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{2}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}} \]
5. Step-by-step derivation

   1. unpow1/3 76.9%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. unpow2 76.9%

      \[\leadsto \sqrt[3]{\color{blue}{\left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. rem-cube-cbrt 76.7%

      \[\leadsto \sqrt[3]{\left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right) \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-plus 76.7%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(3 + 1\right)}} \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. metadata-eval 76.7%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{4}} \cdot \left(\mathsf{log1p}\left(x\right) \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. rem-cube-cbrt 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot \left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{3}} \cdot \sqrt[3]{\mathsf{log1p}\left(x\right)}\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. pow-plus 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(3 + 1\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   8. metadata-eval 76.6%

      \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{4} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{4}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   9. pow-sqr 76.6%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\left(2 \cdot 4\right)}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   10. metadata-eval 76.6%

       \[\leadsto \sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{\color{blue}{8}}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

6. Simplified 76.6%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}}} \]
7. Step-by-step derivation

   1. pow1/3 76.6%

      \[\leadsto \color{blue}{{\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. add-cbrt-cube 93.5%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. pow1/3 93.3%

      \[\leadsto {\color{blue}{\left({\left(\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{0.3333333333333333}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-pow 93.5%

      \[\leadsto \color{blue}{{\left(\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8} \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right) \cdot {\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. pow3 93.5%

      \[\leadsto {\color{blue}{\left({\left({\left(\sqrt[3]{\mathsf{log1p}\left(x\right)}\right)}^{8}\right)}^{3}\right)}}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. pow1/3 57.8%

      \[\leadsto {\left({\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{0.3333333333333333}\right)}}^{8}\right)}^{3}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. pow-pow 57.8%

      \[\leadsto {\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(0.3333333333333333 \cdot 8\right)}\right)}}^{3}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   8. pow-pow 93.6%

      \[\leadsto {\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(\left(0.3333333333333333 \cdot 8\right) \cdot 3\right)}\right)}}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   9. metadata-eval 93.6%

      \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(\color{blue}{2.6666666666666665} \cdot 3\right)}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   10. metadata-eval 93.6%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{8}}\right)}^{\left(0.3333333333333333 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   11. metadata-eval 93.6%

       \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\color{blue}{0.1111111111111111}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

8. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
9. Step-by-step derivation

   1. expm1-log1p-u 93.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   2. expm1-udef 95.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{0.1111111111111111}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   3. sqr-pow 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)} \cdot {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   4. pow-prod-down 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{8} \cdot {\left(\mathsf{log1p}\left(x\right)\right)}^{8}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   5. pow-prod-up 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\color{blue}{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\left(8 + 8\right)}\right)}}^{\left(\frac{0.1111111111111111}{2}\right)}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   6. metadata-eval 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{\color{blue}{16}}\right)}^{\left(\frac{0.1111111111111111}{2}\right)}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

   7. metadata-eval 95.8%

      \[\leadsto \left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{\color{blue}{0.05555555555555555}}\right)} - 1\right) \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

10. Applied egg-rr 95.8%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}\right)} - 1\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
11. Step-by-step derivation

    1. expm1-def 97.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}\right)\right)} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

    2. expm1-log1p 97.3%

       \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

12. Simplified 97.3%

    \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555}} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]

13. Final simplification 97.3%

    \[\leadsto {\left({\left(\mathsf{log1p}\left(x\right)\right)}^{16}\right)}^{0.05555555555555555} \cdot \sqrt[3]{\sqrt[3]{\mathsf{log1p}\left(x\right)}} \]
14. Add Preprocessing

Alternative 4: 97.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{log1p}\left(x\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.6e-26) (* 0.3333333333333333 (* (log1p x) 3.0)) (log (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6e-26) {
		tmp = 0.3333333333333333 * (log1p(x) * 3.0);
	} else {
		tmp = log((x + 1.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.6e-26) {
		tmp = 0.3333333333333333 * (Math.log1p(x) * 3.0);
	} else {
		tmp = Math.log((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.6e-26:
		tmp = 0.3333333333333333 * (math.log1p(x) * 3.0)
	else:
		tmp = math.log((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6e-26)
		tmp = Float64(0.3333333333333333 * Float64(log1p(x) * 3.0));
	else
		tmp = log(Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.6e-26], N[(0.3333333333333333 * N[(N[Log[1 + x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001e-26

   1. Initial program 29.1%

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

      1. add-cbrt-cube 27.5%

         \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}\right)} \]

      2. pow1/3 29.1%

         \[\leadsto \log \color{blue}{\left({\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right)} \]

      3. log-pow 29.1%

         \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)} \]

      4. pow3 28.9%

         \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(1 + x\right)}^{3}\right)} \]

      5. log-pow 29.1%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(1 + x\right)\right)} \]

      6. log1p-def 68.4%

         \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

   4. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \mathsf{log1p}\left(x\right)\right)} \]

   if -2.6000000000000001e-26 < x

   1. Initial program 99.3%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\mathsf{log1p}\left(x\right) \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -2.6e-26) (log1p x) (log (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2.6e-26) {
		tmp = log1p(x);
	} else {
		tmp = log((x + 1.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.6e-26) {
		tmp = Math.log1p(x);
	} else {
		tmp = Math.log((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.6e-26:
		tmp = math.log1p(x)
	else:
		tmp = math.log((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.6e-26)
		tmp = log1p(x);
	else
		tmp = log(Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -2.6e-26], N[Log[1 + x], $MachinePrecision], N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6000000000000001e-26

   1. Initial program 29.1%

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

      1. log1p-def 68.4%

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

   3. Simplified 68.4%

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

   if -2.6000000000000001e-26 < x

   1. Initial program 99.3%

      \[\log \left(1 + x\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.3% accurate, 20.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (+ (+ x 1.0) -1.0))

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x + 1.0) + -1.0

Julia

function code(x)
	return Float64(Float64(x + 1.0) + -1.0)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.3%

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

   1. add-cbrt-cube 79.9%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)}\right)} \]

   2. pow1/3 79.9%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 79.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(1 + x\right) \cdot \left(1 + x\right)\right) \cdot \left(1 + x\right)\right)} \]

   4. pow3 79.9%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(1 + x\right)}^{3}\right)} \]

   5. log-pow 96.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(1 + x\right)\right)} \]

   6. log1p-def 36.9%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \color{blue}{\mathsf{log1p}\left(x\right)}\right) \]

4. Applied egg-rr 36.9%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \mathsf{log1p}\left(x\right)\right)} \]

5. Taylor expanded in x around 0 8.1%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative 8.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)} \]

7. Simplified 8.1%

   \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(x \cdot 3\right)} \]
8. Step-by-step derivation

   1. *-commutative 8.1%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot x\right)} \]

   2. associate-*r* 8.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot x} \]

   3. metadata-eval 8.1%

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

   4. expm1-log1p-u 8.1%

      \[\leadsto 1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   5. *-un-lft-identity 8.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)} \]

   6. expm1-udef 67.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(x\right)} - 1} \]

   7. log1p-udef 67.6%

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

   8. rem-exp-log 67.6%

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

   9. +-commutative 67.6%

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

9. Applied egg-rr 67.6%

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

10. Final simplification 67.6%

    \[\leadsto \left(x + 1\right) + -1 \]
11. Add Preprocessing

Alternative 7: 8.0% 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 96.3%

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

3. Taylor expanded in x around 0 8.1%

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

4. Final simplification 8.1%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 40.7% 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 2024031 
(FPCore (x)
  :name "ln(1 + x)"
  :precision binary64

  :herbie-target
  (if (== (+ 1.0 x) 1.0) x (/ (* x (log (+ 1.0 x))) (- (+ 1.0 x) 1.0)))

  (log (+ 1.0 x)))