2log (problem 3.3.6)

Percentage Accurate: 53.9% → 99.8%

Time: 7.1s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (N) :precision binary64 (- (log (+ N 1.0)) (log N)))

C

double code(double N) {
	return log((N + 1.0)) - log(N);
}

Fortran

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

Java

public static double code(double N) {
	return Math.log((N + 1.0)) - Math.log(N);
}

Python

def code(N):
	return math.log((N + 1.0)) - math.log(N)

Julia

function code(N)
	return Float64(log(Float64(N + 1.0)) - log(N))
end

MATLAB

function tmp = code(N)
	tmp = log((N + 1.0)) - log(N);
end

Wolfram

code[N_] := N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{log1p}\left(N\right)\right)}^{0.6666666666666666}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-7)
   (+ (/ 1.0 N) (- (/ 0.3333333333333333 (pow N 3.0)) (/ 0.5 (* N N))))
   (fma (pow (log1p N) 0.6666666666666666) (cbrt (log1p N)) (- (log N)))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - (0.5 / (N * N)));
	} else {
		tmp = fma(pow(log1p(N), 0.6666666666666666), cbrt(log1p(N)), -log(N));
	}
	return tmp;
}

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 5e-7)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(0.5 / Float64(N * N))));
	else
		tmp = fma((log1p(N) ^ 0.6666666666666666), cbrt(log1p(N)), Float64(-log(N)));
	end
	return tmp
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Log[1 + N], $MachinePrecision], 0.6666666666666666], $MachinePrecision] * N[Power[N[Log[1 + N], $MachinePrecision], 1/3], $MachinePrecision] + (-N[Log[N], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 4.99999999999999977e-7

   1. Initial program 7.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{N} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{\color{blue}{0.5}}{{N}^{2}}\right) \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{\color{blue}{N \cdot N}}\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)} \]

   if 4.99999999999999977e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. add-cube-cbrt 99.9%

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

      2. fma-neg 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\log \left(N + 1\right)} \cdot \sqrt[3]{\log \left(N + 1\right)}, \sqrt[3]{\log \left(N + 1\right)}, -\log N\right)} \]

      3. cbrt-unprod 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt[3]{\log \left(N + 1\right) \cdot \log \left(N + 1\right)}}, \sqrt[3]{\log \left(N + 1\right)}, -\log N\right) \]

      4. pow2 99.9%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{\color{blue}{{\log \left(N + 1\right)}^{2}}}, \sqrt[3]{\log \left(N + 1\right)}, -\log N\right) \]

      5. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\log \color{blue}{\left(1 + N\right)}}^{2}}, \sqrt[3]{\log \left(N + 1\right)}, -\log N\right) \]

      6. log1p-udef 99.9%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\color{blue}{\left(\mathsf{log1p}\left(N\right)\right)}}^{2}}, \sqrt[3]{\log \left(N + 1\right)}, -\log N\right) \]

      7. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}, \sqrt[3]{\log \color{blue}{\left(1 + N\right)}}, -\log N\right) \]

      8. log1p-udef 99.9%

         \[\leadsto \mathsf{fma}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}, \sqrt[3]{\color{blue}{\mathsf{log1p}\left(N\right)}}, -\log N\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{\left(\mathsf{log1p}\left(N\right)\right)}^{2}}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right)} \]
   4. Step-by-step derivation

      1. pow1/3 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left({\left(\mathsf{log1p}\left(N\right)\right)}^{2}\right)}^{0.3333333333333333}}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right) \]

      2. pow-pow 99.9%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{log1p}\left(N\right)\right)}^{\left(2 \cdot 0.3333333333333333\right)}}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right) \]

      3. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left({\left(\mathsf{log1p}\left(N\right)\right)}^{\color{blue}{0.6666666666666666}}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\mathsf{log1p}\left(N\right)\right)}^{0.6666666666666666}}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{log1p}\left(N\right)\right)}^{0.6666666666666666}, \sqrt[3]{\mathsf{log1p}\left(N\right)}, -\log N\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= (- (log (+ N 1.0)) (log N)) 5e-7)
   (+ (/ 1.0 N) (- (/ 0.3333333333333333 (pow N 3.0)) (/ 0.5 (* N N))))
   (log (+ 1.0 (/ 1.0 N)))))

C

double code(double N) {
	double tmp;
	if ((log((N + 1.0)) - log(N)) <= 5e-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / pow(N, 3.0)) - (0.5 / (N * N)));
	} else {
		tmp = log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((log((n + 1.0d0)) - log(n)) <= 5d-7) then
        tmp = (1.0d0 / n) + ((0.3333333333333333d0 / (n ** 3.0d0)) - (0.5d0 / (n * n)))
    else
        tmp = log((1.0d0 + (1.0d0 / n)))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if ((Math.log((N + 1.0)) - Math.log(N)) <= 5e-7) {
		tmp = (1.0 / N) + ((0.3333333333333333 / Math.pow(N, 3.0)) - (0.5 / (N * N)));
	} else {
		tmp = Math.log((1.0 + (1.0 / N)));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if (math.log((N + 1.0)) - math.log(N)) <= 5e-7:
		tmp = (1.0 / N) + ((0.3333333333333333 / math.pow(N, 3.0)) - (0.5 / (N * N)))
	else:
		tmp = math.log((1.0 + (1.0 / N)))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (Float64(log(Float64(N + 1.0)) - log(N)) <= 5e-7)
		tmp = Float64(Float64(1.0 / N) + Float64(Float64(0.3333333333333333 / (N ^ 3.0)) - Float64(0.5 / Float64(N * N))));
	else
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if ((log((N + 1.0)) - log(N)) <= 5e-7)
		tmp = (1.0 / N) + ((0.3333333333333333 / (N ^ 3.0)) - (0.5 / (N * N)));
	else
		tmp = log((1.0 + (1.0 / N)));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N[(N[Log[N[(N + 1.0), $MachinePrecision]], $MachinePrecision] - N[Log[N], $MachinePrecision]), $MachinePrecision], 5e-7], N[(N[(1.0 / N), $MachinePrecision] + N[(N[(0.3333333333333333 / N[Power[N, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N)) < 4.99999999999999977e-7

   1. Initial program 7.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{N} + 0.3333333333333333 \cdot \frac{1}{{N}^{3}}\right) - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{\frac{1}{N} + \left(0.3333333333333333 \cdot \frac{1}{{N}^{3}} - 0.5 \cdot \frac{1}{{N}^{2}}\right)} \]

      2. associate-*r/ 100.0%

         \[\leadsto \frac{1}{N} + \left(\color{blue}{\frac{0.3333333333333333 \cdot 1}{{N}^{3}}} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      3. metadata-eval 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{\color{blue}{0.3333333333333333}}{{N}^{3}} - 0.5 \cdot \frac{1}{{N}^{2}}\right) \]

      4. associate-*r/ 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}}\right) \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{\color{blue}{0.5}}{{N}^{2}}\right) \]

      6. unpow2 100.0%

         \[\leadsto \frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{\color{blue}{N \cdot N}}\right) \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)} \]

   if 4.99999999999999977e-7 < (-.f64 (log.f64 (+.f64 N 1)) (log.f64 N))

   1. Initial program 99.9%

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

      1. diff-log 99.9%

         \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]

   4. Taylor expanded in N around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(N + 1\right) - \log N \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{N} + \left(\frac{0.3333333333333333}{{N}^{3}} - \frac{0.5}{N \cdot N}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 195000.0) (log (+ 1.0 (/ 1.0 N))) (- (/ 1.0 N) (/ 0.5 (* N N)))))

C

double code(double N) {
	double tmp;
	if (N <= 195000.0) {
		tmp = log((1.0 + (1.0 / N)));
	} else {
		tmp = (1.0 / N) - (0.5 / (N * N));
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 195000.0d0) then
        tmp = log((1.0d0 + (1.0d0 / n)))
    else
        tmp = (1.0d0 / n) - (0.5d0 / (n * n))
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 195000.0) {
		tmp = Math.log((1.0 + (1.0 / N)));
	} else {
		tmp = (1.0 / N) - (0.5 / (N * N));
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 195000.0:
		tmp = math.log((1.0 + (1.0 / N)))
	else:
		tmp = (1.0 / N) - (0.5 / (N * N))
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 195000.0)
		tmp = log(Float64(1.0 + Float64(1.0 / N)));
	else
		tmp = Float64(Float64(1.0 / N) - Float64(0.5 / Float64(N * N)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 195000.0)
		tmp = log((1.0 + (1.0 / N)));
	else
		tmp = (1.0 / N) - (0.5 / (N * N));
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 195000.0], N[Log[N[(1.0 + N[(1.0 / N), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 / N), $MachinePrecision] - N[(0.5 / N[(N * N), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 195000

   1. Initial program 99.9%

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

      1. diff-log 99.9%

         \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\log \left(\frac{N + 1}{N}\right)} \]

   4. Taylor expanded in N around 0 99.9%

      \[\leadsto \log \color{blue}{\left(1 + \frac{1}{N}\right)} \]

   if 195000 < N

   1. Initial program 7.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 99.9%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 99.9%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 99.9%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 99.9%

         \[\leadsto \frac{1}{N} - \frac{0.5}{\color{blue}{N \cdot N}} \]

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{0.5}{N \cdot N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 195000:\\ \;\;\;\;\log \left(1 + \frac{1}{N}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{N} - \frac{0.5}{N \cdot N}\\ \end{array} \]

Alternative 4: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{N \cdot 2}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.9) (- N (log N)) (/ (- 1.0 (/ 1.0 (* N 2.0))) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - log(N);
	} else {
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.9d0) then
        tmp = n - log(n)
    else
        tmp = (1.0d0 - (1.0d0 / (n * 2.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.9) {
		tmp = N - Math.log(N);
	} else {
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.9:
		tmp = N - math.log(N)
	else:
		tmp = (1.0 - (1.0 / (N * 2.0))) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.9)
		tmp = Float64(N - log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(1.0 / Float64(N * 2.0))) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.9)
		tmp = N - log(N);
	else
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.9], N[(N - N[Log[N], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(1.0 / N[(N * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.900000000000000022

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 99.5%

      \[\leadsto \color{blue}{N + -1 \cdot \log N} \]
   3. Step-by-step derivation

      1. neg-mul-1 99.5%

         \[\leadsto N + \color{blue}{\left(-\log N\right)} \]

      2. unsub-neg 99.5%

         \[\leadsto \color{blue}{N - \log N} \]

   4. Simplified 99.5%

      \[\leadsto \color{blue}{N - \log N} \]

   if 0.900000000000000022 < N

   1. Initial program 8.3%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 98.9%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.9%

         \[\leadsto \frac{1}{N} - \frac{0.5}{\color{blue}{N \cdot N}} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{0.5}{N \cdot N}} \]
   5. Step-by-step derivation

      1. clear-num 98.9%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{1}{\frac{N \cdot N}{0.5}}} \]

      2. frac-sub 33.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{N \cdot N}{0.5} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}}} \]

      3. *-un-lft-identity 33.6%

         \[\leadsto \frac{\color{blue}{\frac{N \cdot N}{0.5}} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      4. div-inv 33.6%

         \[\leadsto \frac{\color{blue}{\left(N \cdot N\right) \cdot \frac{1}{0.5}} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      5. metadata-eval 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot \color{blue}{2} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      6. *-rgt-identity 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot 2 - \color{blue}{N}}{N \cdot \frac{N \cdot N}{0.5}} \]

      7. div-inv 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot 2 - N}{N \cdot \color{blue}{\left(\left(N \cdot N\right) \cdot \frac{1}{0.5}\right)}} \]

      8. metadata-eval 33.6%

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

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{\left(N \cdot N\right) \cdot 2 - N}{N \cdot \left(\left(N \cdot N\right) \cdot 2\right)}} \]
   7. Step-by-step derivation

      1. div-inv 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*r* 33.6%

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

      4. cube-mult 33.5%

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

   8. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{\left(N \cdot \left(N \cdot 2\right) - N\right) \cdot \frac{1}{{N}^{3} \cdot 2}} \]
   9. Step-by-step derivation

      1. associate-*r/ 33.5%

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

      2. *-rgt-identity 33.5%

         \[\leadsto \frac{\color{blue}{N \cdot \left(N \cdot 2\right) - N}}{{N}^{3} \cdot 2} \]

      3. *-commutative 33.5%

         \[\leadsto \frac{N \cdot \left(N \cdot 2\right) - N}{\color{blue}{2 \cdot {N}^{3}}} \]

      4. associate-/r* 33.5%

         \[\leadsto \color{blue}{\frac{\frac{N \cdot \left(N \cdot 2\right) - N}{2}}{{N}^{3}}} \]

      5. unpow3 33.6%

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

      6. associate-/r* 49.3%

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

      7. associate-/r* 49.3%

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

      8. *-commutative 49.3%

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

      9. associate-*r* 49.3%

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

      10. associate-/r* 50.1%

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

      11. div-sub 50.1%

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

      12. *-inverses 50.1%

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

      13. div-sub 50.1%

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

      14. associate-/r* 49.3%

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

      15. *-inverses 98.9%

          \[\leadsto \frac{\color{blue}{1} - \frac{1}{N \cdot 2}}{N} \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{1 - \frac{1}{N \cdot 2}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.9:\\ \;\;\;\;N - \log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{N \cdot 2}}{N}\\ \end{array} \]

Alternative 5: 98.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;N \leq 0.66:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{N \cdot 2}}{N}\\ \end{array} \end{array} \]

FPCore

(FPCore (N)
 :precision binary64
 (if (<= N 0.66) (- (log N)) (/ (- 1.0 (/ 1.0 (* N 2.0))) N)))

C

double code(double N) {
	double tmp;
	if (N <= 0.66) {
		tmp = -log(N);
	} else {
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	}
	return tmp;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= 0.66d0) then
        tmp = -log(n)
    else
        tmp = (1.0d0 - (1.0d0 / (n * 2.0d0))) / n
    end if
    code = tmp
end function

Java

public static double code(double N) {
	double tmp;
	if (N <= 0.66) {
		tmp = -Math.log(N);
	} else {
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	}
	return tmp;
}

Python

def code(N):
	tmp = 0
	if N <= 0.66:
		tmp = -math.log(N)
	else:
		tmp = (1.0 - (1.0 / (N * 2.0))) / N
	return tmp

Julia

function code(N)
	tmp = 0.0
	if (N <= 0.66)
		tmp = Float64(-log(N));
	else
		tmp = Float64(Float64(1.0 - Float64(1.0 / Float64(N * 2.0))) / N);
	end
	return tmp
end

MATLAB

function tmp_2 = code(N)
	tmp = 0.0;
	if (N <= 0.66)
		tmp = -log(N);
	else
		tmp = (1.0 - (1.0 / (N * 2.0))) / N;
	end
	tmp_2 = tmp;
end

Wolfram

code[N_] := If[LessEqual[N, 0.66], (-N[Log[N], $MachinePrecision]), N[(N[(1.0 - N[(1.0 / N[(N * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if N < 0.660000000000000031

   1. Initial program 100.0%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around 0 98.9%

      \[\leadsto \color{blue}{-1 \cdot \log N} \]
   3. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto \color{blue}{-\log N} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{-\log N} \]

   if 0.660000000000000031 < N

   1. Initial program 8.3%

      \[\log \left(N + 1\right) - \log N \]

   2. Taylor expanded in N around inf 98.9%

      \[\leadsto \color{blue}{\frac{1}{N} - 0.5 \cdot \frac{1}{{N}^{2}}} \]
   3. Step-by-step derivation

      1. associate-*r/ 98.9%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{0.5 \cdot 1}{{N}^{2}}} \]

      2. metadata-eval 98.9%

         \[\leadsto \frac{1}{N} - \frac{\color{blue}{0.5}}{{N}^{2}} \]

      3. unpow2 98.9%

         \[\leadsto \frac{1}{N} - \frac{0.5}{\color{blue}{N \cdot N}} \]

   4. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{1}{N} - \frac{0.5}{N \cdot N}} \]
   5. Step-by-step derivation

      1. clear-num 98.9%

         \[\leadsto \frac{1}{N} - \color{blue}{\frac{1}{\frac{N \cdot N}{0.5}}} \]

      2. frac-sub 33.6%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{N \cdot N}{0.5} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}}} \]

      3. *-un-lft-identity 33.6%

         \[\leadsto \frac{\color{blue}{\frac{N \cdot N}{0.5}} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      4. div-inv 33.6%

         \[\leadsto \frac{\color{blue}{\left(N \cdot N\right) \cdot \frac{1}{0.5}} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      5. metadata-eval 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot \color{blue}{2} - N \cdot 1}{N \cdot \frac{N \cdot N}{0.5}} \]

      6. *-rgt-identity 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot 2 - \color{blue}{N}}{N \cdot \frac{N \cdot N}{0.5}} \]

      7. div-inv 33.6%

         \[\leadsto \frac{\left(N \cdot N\right) \cdot 2 - N}{N \cdot \color{blue}{\left(\left(N \cdot N\right) \cdot \frac{1}{0.5}\right)}} \]

      8. metadata-eval 33.6%

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

   6. Applied egg-rr 33.6%

      \[\leadsto \color{blue}{\frac{\left(N \cdot N\right) \cdot 2 - N}{N \cdot \left(\left(N \cdot N\right) \cdot 2\right)}} \]
   7. Step-by-step derivation

      1. div-inv 33.6%

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

      2. associate-*l* 33.6%

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

      3. associate-*r* 33.6%

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

      4. cube-mult 33.5%

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

   8. Applied egg-rr 33.5%

      \[\leadsto \color{blue}{\left(N \cdot \left(N \cdot 2\right) - N\right) \cdot \frac{1}{{N}^{3} \cdot 2}} \]
   9. Step-by-step derivation

      1. associate-*r/ 33.5%

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

      2. *-rgt-identity 33.5%

         \[\leadsto \frac{\color{blue}{N \cdot \left(N \cdot 2\right) - N}}{{N}^{3} \cdot 2} \]

      3. *-commutative 33.5%

         \[\leadsto \frac{N \cdot \left(N \cdot 2\right) - N}{\color{blue}{2 \cdot {N}^{3}}} \]

      4. associate-/r* 33.5%

         \[\leadsto \color{blue}{\frac{\frac{N \cdot \left(N \cdot 2\right) - N}{2}}{{N}^{3}}} \]

      5. unpow3 33.6%

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

      6. associate-/r* 49.3%

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

      7. associate-/r* 49.3%

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

      8. *-commutative 49.3%

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

      9. associate-*r* 49.3%

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

      10. associate-/r* 50.1%

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

      11. div-sub 50.1%

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

      12. *-inverses 50.1%

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

      13. div-sub 50.1%

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

      14. associate-/r* 49.3%

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

      15. *-inverses 98.9%

          \[\leadsto \frac{\color{blue}{1} - \frac{1}{N \cdot 2}}{N} \]

   10. Simplified 98.9%

       \[\leadsto \color{blue}{\frac{1 - \frac{1}{N \cdot 2}}{N}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;N \leq 0.66:\\ \;\;\;\;-\log N\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{N \cdot 2}}{N}\\ \end{array} \]

Alternative 6: 51.9% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{N} \end{array} \]

FPCore

(FPCore (N) :precision binary64 (/ 1.0 N))

C

double code(double N) {
	return 1.0 / N;
}

Fortran

real(8) function code(n)
    real(8), intent (in) :: n
    code = 1.0d0 / n
end function

Java

public static double code(double N) {
	return 1.0 / N;
}

Python

def code(N):
	return 1.0 / N

Julia

function code(N)
	return Float64(1.0 / N)
end

MATLAB

function tmp = code(N)
	tmp = 1.0 / N;
end

Wolfram

code[N_] := N[(1.0 / N), $MachinePrecision]

Derivation

1. Initial program 53.8%

   \[\log \left(N + 1\right) - \log N \]

2. Taylor expanded in N around inf 52.0%

   \[\leadsto \color{blue}{\frac{1}{N}} \]

3. Final simplification 52.0%

   \[\leadsto \frac{1}{N} \]

Reproduce

herbie shell --seed 2023200 
(FPCore (N)
  :name "2log (problem 3.3.6)"
  :precision binary64
  (- (log (+ N 1.0)) (log N)))