Hyperbolic arc-cosine

Percentage Accurate: 49.8% → 99.5%

Time: 8.4s

Alternatives: 3

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: 49.8% 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: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * 2.0d0) + ((-0.5d0) / x)))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = log(((x * 2.0) + (-0.5 / x)));
end

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. log1p-expm1-u 49.7%

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

   2. expm1-undefine 49.7%

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

   3. add-exp-log 49.7%

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

   4. fma-neg 49.7%

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

   5. metadata-eval 49.7%

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

4. Applied egg-rr 49.7%

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

5. Taylor expanded in x around inf 99.5%

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

6. Taylor expanded in x around 0 49.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{x \cdot \left(2 \cdot x - 1\right) - 0.5}{x}}\right) \]
7. Step-by-step derivation

   1. *-un-lft-identity 49.2%

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

   2. fma-neg 49.2%

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

   3. fma-neg 49.2%

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

   4. metadata-eval 49.2%

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

   5. metadata-eval 49.2%

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

8. Applied egg-rr 49.2%

   \[\leadsto \color{blue}{1 \cdot \mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(2, x, -1\right), -0.5\right)}{x}\right)} \]
9. Step-by-step derivation

   1. *-lft-identity 49.2%

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

   2. log1p-define 49.2%

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

   3. *-lft-identity 49.2%

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

   4. associate-*l/ 49.2%

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

   5. fma-undefine 49.2%

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

   6. +-commutative 49.2%

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

   7. distribute-lft-in 49.2%

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

   8. associate-*l/ 49.2%

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

   9. metadata-eval 49.2%

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

   10. metadata-eval 49.2%

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

   11. distribute-neg-frac 49.2%

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

   12. associate-*l/ 49.2%

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

   13. *-lft-identity 49.2%

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

   14. associate-*r/ 99.5%

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

   15. +-commutative 99.5%

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

   16. associate-+r+ 99.5%

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

10. Simplified 99.5%

    \[\leadsto \color{blue}{\log \left(x \cdot 2 + \frac{-0.5}{x}\right)} \]
11. Add Preprocessing

Alternative 2: 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 49.7%

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

3. Taylor expanded in x around inf 99.2%

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

Alternative 3: 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 49.7%

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

   1. log1p-expm1-u 49.7%

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

   2. expm1-undefine 49.7%

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

   3. add-exp-log 49.7%

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

   4. fma-neg 49.7%

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

   5. metadata-eval 49.7%

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

4. Applied egg-rr 49.7%

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

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 1.6%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))