Hyperbolic arc-cosine

Percentage Accurate: 52.4% → 99.2%

Time: 10.8s

Alternatives: 2

Speedup: 1.2×

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: 52.4% 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: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 46.6

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval 46.6

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

4. Applied rewrites 46.6%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. flip3-+ N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-plus N/A

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

   9. pow1/2 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. cube-unmult N/A

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

   14. lift-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lift-fma.f64 N/A

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

6. Applied rewrites 32.0%

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

7. Taylor expanded in x around inf

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

   1. lower-/.f64 99.1

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

9. Applied rewrites 99.1%

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

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 99.1

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

5. Applied rewrites 99.1%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))