Hyperbolic arc-cosine

Percentage Accurate: 51.5% → 99.7%

Time: 8.1s

Alternatives: 4

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: 51.5% 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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (+
   (/ -0.5 x)
   (* x (+ 2.0 (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x)))))))))

C

double code(double x) {
	return log(((-0.5 / x) + (x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x))))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((-0.5d0) / x) + (x * (2.0d0 + (((-0.125d0) + ((-0.0625d0) / (x * x))) / (x * (x * (x * x))))))))
end function

Java

public static double code(double x) {
	return Math.log(((-0.5 / x) + (x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x))))))));
}

Python

def code(x):
	return math.log(((-0.5 / x) + (x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x))))))))

Julia

function code(x)
	return log(Float64(Float64(-0.5 / x) + Float64(x * Float64(2.0 + Float64(Float64(-0.125 + Float64(-0.0625 / Float64(x * x))) / Float64(x * Float64(x * Float64(x * x))))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((-0.5 / x) + (x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x))))))));
end

Wolfram

code[x_] := N[Log[N[(N[(-0.5 / x), $MachinePrecision] + N[(x * N[(2.0 + N[(N[(-0.125 + N[(-0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right)\right) \]

5. Simplified 100.0%

   \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 + \frac{-0.5 - \frac{0.125}{x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (* x 2.0) (/ (- -0.5 (/ 0.125 (* x x))) x))))

C

double code(double x) {
	return log(((x * 2.0) + ((-0.5 - (0.125 / (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * 2.0d0) + (((-0.5d0) - (0.125d0 / (x * x))) / x)))
end function

Java

public static double code(double x) {
	return Math.log(((x * 2.0) + ((-0.5 - (0.125 / (x * x))) / x)));
}

Python

def code(x):
	return math.log(((x * 2.0) + ((-0.5 - (0.125 / (x * x))) / x)))

Julia

function code(x)
	return log(Float64(Float64(x * 2.0) + Float64(Float64(-0.5 - Float64(0.125 / Float64(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * 2.0) + ((-0.5 - (0.125 / (x * x))) / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(-0.5 - N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right) \cdot x\right)\right) \]

   3. distribute-neg-frac N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{{x}^{2}} \cdot x\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x}{{x}^{2}}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x}{x \cdot x}\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\frac{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot x}{x}}{x}\right)\right) \]

   7. associate-/l* N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot \frac{x}{x}}{x}\right)\right) \]

   8. *-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right) \cdot 1}{x}\right)\right) \]

   9. *-rgt-identity N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \frac{\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right), x\right)\right)\right) \]

5. Simplified 99.9%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{-0.5 - \frac{0.125}{x \cdot x}}{x}\right)} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot 2 + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{1}{{x}^{2}} \cdot \frac{1}{2}\right)\right)\right)\right)\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(\frac{1}{{x}^{2}} \cdot \frac{-1}{2}\right)\right)\right)\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(x \cdot \frac{1}{{x}^{2}}\right) \cdot \frac{-1}{2}\right)\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x \cdot 1}{{x}^{2}} \cdot \frac{-1}{2}\right)\right)\right) \]

   12. *-rgt-identity N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x}{{x}^{2}} \cdot \frac{-1}{2}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x}{x \cdot x} \cdot \frac{-1}{2}\right)\right)\right) \]

   14. associate-/r* N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{x}{x}}{x} \cdot \frac{-1}{2}\right)\right)\right) \]

   15. *-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{1}{x} \cdot \frac{-1}{2}\right)\right)\right) \]

   16. associate-*l/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{1 \cdot \frac{-1}{2}}{x}\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{-1}{2}}{x}\right)\right)\right) \]

   18. /-lowering-/.f64 99.6%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right)\right) \]

5. Simplified 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \log \left(\frac{-0.5}{x} + x \cdot 2\right) \]
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 51.2%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.0%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))