Hyperbolic arc-cosine

Percentage Accurate: 51.7% → 99.1%

Time: 6.2s

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: 51.7% 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: 99.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf 98.8%

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

   1. count-2 98.8%

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

   2. sum-log 98.9%

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

   3. +-commutative 98.9%

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

   4. add-sqr-sqrt 98.3%

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

   5. fma-define 98.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]

5. Applied egg-rr 98.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)} \]
6. Step-by-step derivation

   1. add-log-exp 98.0%

      \[\leadsto \color{blue}{\log \left(e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}\right)} \]

   2. add-sqr-sqrt 98.0%

      \[\leadsto \log \color{blue}{\left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}} \cdot \sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right)} \]

   3. log-prod 98.0%

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right)} \]

   4. fma-undefine 98.0%

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

   5. add-sqr-sqrt 98.3%

      \[\leadsto \log \left(\sqrt{e^{\color{blue}{\log x} + \log 2}}\right) + \log \left(\sqrt{e^{\mathsf{fma}\left(\sqrt{\log x}, \sqrt{\log x}, \log 2\right)}}\right) \]

   6. sum-log 98.4%

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

   7. add-exp-log 98.4%

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

   8. fma-undefine 98.4%

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

   9. add-sqr-sqrt 98.6%

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

7. Applied egg-rr 98.8%

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

   1. count-2 98.8%

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

   2. *-commutative 98.8%

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

9. Simplified 98.8%

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

10. Taylor expanded in x around 0 99.2%

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

11. Final simplification 99.2%

    \[\leadsto 2 \cdot \log \left(\sqrt{x} \cdot \sqrt{2}\right) \]
12. Add Preprocessing

Alternative 2: 98.8% 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 50.4%

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

3. Taylor expanded in x around inf 98.9%

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

   1. mul-1-neg 98.9%

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

   2. log-rec 98.9%

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

   3. remove-double-neg 98.9%

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

5. Simplified 98.9%

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

6. Final simplification 98.9%

   \[\leadsto \log 2 + \log x \]
7. 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 50.4%

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

3. Taylor expanded in x around inf 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \log \left(x + x\right) \]
5. Add Preprocessing

Alternative 4: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf 98.8%

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

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. +-inverses 0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}\right) \]

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}\right) \]

   6. +-inverses 0.0%

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

   7. associate-*r/ 0.0%

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

   8. +-inverses 0.0%

      \[\leadsto \log \left(\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right) \]

   9. +-inverses 0.0%

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

   10. flip-+ 10.8%

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

   11. sum-log 18.7%

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

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\log \left(\frac{0}{0}\right) + \log \left(\frac{0}{0}\right)} \]

7. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 3.1%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))