Hyperbolic arc-cosine

Percentage Accurate: 51.8% → 99.6%

Time: 4.3s

Alternatives: 5

Speedup: 2.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end

MATLAB

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

Wolfram

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

Initial Program: 51.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (- (* x x) 1.0)))))

C

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

Fortran

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

Java

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

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) - 1.0))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) - 1.0))))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ x (- (+ x (/ -0.5 x)) (/ 0.125 (pow x 3.0))))))

C

double code(double x) {
	return log((x + ((x + (-0.5 / x)) - (0.125 / pow(x, 3.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + ((x + ((-0.5d0) / x)) - (0.125d0 / (x ** 3.0d0)))))
end function

Java

public static double code(double x) {
	return Math.log((x + ((x + (-0.5 / x)) - (0.125 / Math.pow(x, 3.0)))));
}

Python

def code(x):
	return math.log((x + ((x + (-0.5 / x)) - (0.125 / math.pow(x, 3.0)))))

Julia

function code(x)
	return log(Float64(x + Float64(Float64(x + Float64(-0.5 / x)) - Float64(0.125 / (x ^ 3.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + ((x + (-0.5 / x)) - (0.125 / (x ^ 3.0)))));
end

Wolfram

code[x_] := N[Log[N[(x + N[(N[(x + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] - N[(0.125 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.4%

   \[\leadsto \log \left(x + \color{blue}{\left(x - \left(0.125 \cdot \frac{1}{{x}^{3}} + 0.5 \cdot \frac{1}{x}\right)\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative 99.4%

      \[\leadsto \log \left(x + \left(x - \color{blue}{\left(0.5 \cdot \frac{1}{x} + 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right)\right) \]

   2. associate--r+ 99.4%

      \[\leadsto \log \left(x + \color{blue}{\left(\left(x - 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\right) \]

   3. sub-neg 99.4%

      \[\leadsto \log \left(x + \left(\color{blue}{\left(x + \left(-0.5 \cdot \frac{1}{x}\right)\right)} - 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   4. associate-*r/ 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   5. metadata-eval 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \left(-\frac{\color{blue}{0.5}}{x}\right)\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   6. distribute-neg-frac 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \color{blue}{\frac{-0.5}{x}}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \frac{\color{blue}{-0.5}}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\right) \]

   8. associate-*r/ 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) - \color{blue}{\frac{0.125 \cdot 1}{{x}^{3}}}\right)\right) \]

   9. metadata-eval 99.4%

      \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) - \frac{\color{blue}{0.125}}{{x}^{3}}\right)\right) \]

5. Simplified 99.4%

   \[\leadsto \log \left(x + \color{blue}{\left(\left(x + \frac{-0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)}\right) \]

6. Final simplification 99.4%

   \[\leadsto \log \left(x + \left(\left(x + \frac{-0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(x \cdot 2 + \left(-1 + 0.5 \cdot \frac{-1}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log1p (+ (* x 2.0) (+ -1.0 (* 0.5 (/ -1.0 x))))))

C

double code(double x) {
	return log1p(((x * 2.0) + (-1.0 + (0.5 * (-1.0 / x)))));
}

Java

public static double code(double x) {
	return Math.log1p(((x * 2.0) + (-1.0 + (0.5 * (-1.0 / x)))));
}

Python

def code(x):
	return math.log1p(((x * 2.0) + (-1.0 + (0.5 * (-1.0 / x)))))

Julia

function code(x)
	return log1p(Float64(Float64(x * 2.0) + Float64(-1.0 + Float64(0.5 * Float64(-1.0 / x)))))
end

Wolfram

code[x_] := N[Log[1 + N[(N[(x * 2.0), $MachinePrecision] + N[(-1.0 + N[(0.5 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. log1p-expm1-u 50.0%

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

   2. expm1-udef 50.0%

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

   3. add-exp-log 50.0%

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

   4. fma-neg 50.0%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 50.0%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 50.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around inf 99.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{2 \cdot x - \left(1 + 0.5 \cdot \frac{1}{x}\right)}\right) \]

6. Final simplification 99.2%

   \[\leadsto \mathsf{log1p}\left(x \cdot 2 + \left(-1 + 0.5 \cdot \frac{-1}{x}\right)\right) \]
7. Add Preprocessing

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x (- x (/ 0.5 x)))))

C

double code(double x) {
	return log((x + (x - (0.5 / x))));
}

Fortran

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

Java

public static double code(double x) {
	return Math.log((x + (x - (0.5 / x))));
}

Python

def code(x):
	return math.log((x + (x - (0.5 / x))))

Julia

function code(x)
	return log(Float64(x + Float64(x - Float64(0.5 / x))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (x - (0.5 / x))));
end

Wolfram

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

Derivation

1. Initial program 50.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. metadata-eval 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

   \[\leadsto \log \left(x + \left(x - \frac{0.5}{x}\right)\right) \]
7. Add Preprocessing

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x x)))

C

double code(double x) {
	return log((x + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + x))
end function

Java

public static double code(double x) {
	return Math.log((x + x));
}

Python

def code(x):
	return math.log((x + x))

Julia

function code(x)
	return log(Float64(x + x))
end

MATLAB

function tmp = code(x)
	tmp = log((x + x));
end

Wolfram

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

Derivation

1. Initial program 50.0%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 5: 0.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(-1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log1p -1.0))

C

double code(double x) {
	return log1p(-1.0);
}

Java

public static double code(double x) {
	return Math.log1p(-1.0);
}

Python

def code(x):
	return math.log1p(-1.0)

Julia

function code(x)
	return log1p(-1.0)
end

Wolfram

code[x_] := N[Log[1 + -1.0], $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. log1p-expm1-u 50.0%

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

   2. expm1-udef 50.0%

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

   3. add-exp-log 50.0%

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

   4. fma-neg 50.0%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

   5. metadata-eval 50.0%

      \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

4. Applied egg-rr 50.0%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

5. Taylor expanded in x around -inf 0.6%

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

6. Final simplification 0.6%

   \[\leadsto \mathsf{log1p}\left(-1\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))