Hyperbolic arc-cosine

Percentage Accurate: 51.5% → 99.6%

Time: 7.8s

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: 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: 99.6% 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 50.8%

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

   1. log1p-expm1-u 50.8%

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

   2. expm1-undefine 50.8%

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

   3. add-exp-log 50.8%

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

   4. fma-neg 50.8%

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

   5. metadata-eval 50.8%

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

4. Applied egg-rr 50.8%

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

5. Taylor expanded in x around inf 99.4%

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

6. Taylor expanded in x around 0 49.8%

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

   1. *-un-lft-identity 49.8%

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

   2. div-inv 49.8%

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

   3. div-inv 49.8%

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

   4. fma-neg 49.8%

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

   5. *-commutative 49.8%

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

   6. fma-neg 49.8%

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

   7. metadata-eval 49.8%

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

   8. metadata-eval 49.8%

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

8. Applied egg-rr 49.8%

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

   1. *-lft-identity 49.8%

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

   2. log1p-define 49.8%

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

   3. +-commutative 49.8%

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

   4. *-lft-identity 49.8%

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

   5. associate-*l/ 49.8%

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

   6. fma-undefine 49.8%

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

   7. distribute-rgt-in 49.8%

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

   8. associate-*r/ 49.8%

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

   9. *-rgt-identity 49.8%

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

   10. *-commutative 49.8%

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

   11. associate-*r/ 99.4%

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

   12. metadata-eval 99.4%

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

   13. cancel-sign-sub-inv 99.4%

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

   14. associate-*r/ 99.4%

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

   15. metadata-eval 99.4%

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

   16. sub-neg 99.4%

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

   17. +-commutative 99.4%

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

   18. associate-+l+ 99.4%

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

10. Simplified 99.4%

    \[\leadsto \color{blue}{\log \left(\frac{-0.5}{x} + x \cdot 2\right)} \]
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 50.8%

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

3. Taylor expanded in x around inf 98.9%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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 50.8%

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

   1. log1p-expm1-u 50.8%

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

   2. expm1-undefine 50.8%

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

   3. add-exp-log 50.8%

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

   4. fma-neg 50.8%

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

   5. metadata-eval 50.8%

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

4. Applied egg-rr 50.8%

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

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 1.1%

   \[\leadsto \color{blue}{-2 + \frac{x}{-2}} \]

8. Taylor expanded in x around 0 1.6%

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

Reproduce

herbie shell --seed 2024117 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))