Hyperbolic arc-cosine

Percentage Accurate: 51.6% → 98.9%

Time: 7.7s

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: 51.6% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (log 2.0) (log x)))

C

double code(double x) {
	return log(2.0) + log(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(2.0d0) + log(x)
end function

Java

public static double code(double x) {
	return Math.log(2.0) + Math.log(x);
}

Python

def code(x):
	return math.log(2.0) + math.log(x)

Julia

function code(x)
	return Float64(log(2.0) + log(x))
end

MATLAB

function tmp = code(x)
	tmp = log(2.0) + log(x);
end

Wolfram

code[x_] := N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.2%

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

3. Taylor expanded in x around inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. log-rec 98.8%

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

   3. remove-double-neg 98.8%

      \[\leadsto \log 2 + \color{blue}{\log x} \]

5. Simplified 98.8%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
6. Add Preprocessing

Alternative 2: 99.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.2%

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

3. Taylor expanded in x around inf 98.7%

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

   1. log1p-expm1-u 98.7%

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

   2. expm1-undefine 98.7%

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

   3. add-exp-log 98.7%

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

   4. count-2 98.7%

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

   5. *-commutative 98.7%

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

   6. fmm-def 98.7%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x, 2, -1\right)}\right) \]

   7. metadata-eval 98.7%

      \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, 2, \color{blue}{-1}\right)\right) \]

5. Applied egg-rr 98.7%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, 2, -1\right)\right)} \]

6. Taylor expanded in x around 0 98.7%

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

7. Final simplification 98.7%

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

Alternative 3: 99.1% 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 48.2%

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

3. Taylor expanded in x around inf 98.7%

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

Alternative 4: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.2%

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

3. Taylor expanded in x around inf 98.7%

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

4. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{\log 2 + \log x} \]

6. Simplified 31.5%

   \[\leadsto \color{blue}{\log x} \]
7. Add Preprocessing

Alternative 5: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -2.0d0
end function

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 48.2%

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

   1. pow1/2 48.2%

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

   2. metadata-eval 48.2%

      \[\leadsto \log \left(x + {\left(x \cdot x - 1\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}\right) \]

   3. metadata-eval 48.2%

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

   4. pow-pow 32.2%

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

   5. fmm-def 32.2%

      \[\leadsto \log \left(x + {\left({\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(0.5 \cdot 3\right)}\right)}^{0.3333333333333333}\right) \]

   6. metadata-eval 32.2%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(0.5 \cdot 3\right)}\right)}^{0.3333333333333333}\right) \]

   7. metadata-eval 32.2%

      \[\leadsto \log \left(x + {\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right) \]

4. Applied egg-rr 32.2%

   \[\leadsto \log \left(x + \color{blue}{{\left({\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}\right)}^{0.3333333333333333}}\right) \]
5. Step-by-step derivation

   1. unpow1/3 32.3%

      \[\leadsto \log \left(x + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}}\right) \]

6. Simplified 32.3%

   \[\leadsto \log \left(x + \color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}}\right) \]

7. Taylor expanded in x around 0 0.0%

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

9. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))