Rust f64::acosh

Percentage Accurate: 50.6% → 99.6%

Time: 4.9s

Alternatives: 12

Speedup: 2.0×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acosh x))

C

double code(double x) {
	return acosh(x);
}

Python

def code(x):
	return math.acosh(x)

Julia

function code(x)
	return acosh(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcCosh[x], $MachinePrecision]

Initial Program: 50.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.3%

   \[\leadsto \log \left(x + \color{blue}{\left(x - 0.5 \cdot \frac{1}{x}\right)}\right) \]
3. Step-by-step derivation

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

4. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 99.2% 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 59.1%

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

2. Taylor expanded in x around inf 99.0%

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

3. Final simplification 99.0%

   \[\leadsto \log \left(x + x\right) \]

Alternative 3: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

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

4. Applied egg-rr 0.0%

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

6. Simplified 1.6%

   \[\leadsto \color{blue}{-1} \]

7. Final simplification 1.6%

   \[\leadsto -1 \]

Alternative 4: 13.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.6666666666666666)

C

double code(double x) {
	return 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return 0.6666666666666666;
}

Python

def code(x):
	return 0.6666666666666666

Julia

function code(x)
	return 0.6666666666666666
end

MATLAB

function tmp = code(x)
	tmp = 0.6666666666666666;
end

Wolfram

code[x_] := 0.6666666666666666

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 14.0%

   \[\leadsto \color{blue}{0.6666666666666666} \]

9. Final simplification 14.0%

   \[\leadsto 0.6666666666666666 \]

Alternative 5: 15.0% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 6.0)

C

double code(double x) {
	return 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 6.0d0
end function

Java

public static double code(double x) {
	return 6.0;
}

Python

def code(x):
	return 6.0

Julia

function code(x)
	return 6.0
end

MATLAB

function tmp = code(x)
	tmp = 6.0;
end

Wolfram

code[x_] := 6.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 15.1%

   \[\leadsto \color{blue}{6} \]

9. Final simplification 15.1%

   \[\leadsto 6 \]

Alternative 6: 15.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 9 \end{array} \]

FPCore

(FPCore (x) :precision binary64 9.0)

C

double code(double x) {
	return 9.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 9.0d0
end function

Java

public static double code(double x) {
	return 9.0;
}

Python

def code(x):
	return 9.0

Julia

function code(x)
	return 9.0
end

MATLAB

function tmp = code(x)
	tmp = 9.0;
end

Wolfram

code[x_] := 9.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 15.4%

   \[\leadsto \color{blue}{9} \]

9. Final simplification 15.4%

   \[\leadsto 9 \]

Alternative 7: 15.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 16 \end{array} \]

FPCore

(FPCore (x) :precision binary64 16.0)

C

double code(double x) {
	return 16.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 16.0d0
end function

Java

public static double code(double x) {
	return 16.0;
}

Python

def code(x):
	return 16.0

Julia

function code(x)
	return 16.0
end

MATLAB

function tmp = code(x)
	tmp = 16.0;
end

Wolfram

code[x_] := 16.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 15.9%

   \[\leadsto \color{blue}{16} \]

9. Final simplification 15.9%

   \[\leadsto 16 \]

Alternative 8: 15.9% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 17 \end{array} \]

FPCore

(FPCore (x) :precision binary64 17.0)

C

double code(double x) {
	return 17.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 17.0d0
end function

Java

public static double code(double x) {
	return 17.0;
}

Python

def code(x):
	return 17.0

Julia

function code(x)
	return 17.0
end

MATLAB

function tmp = code(x)
	tmp = 17.0;
end

Wolfram

code[x_] := 17.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 15.9%

   \[\leadsto \color{blue}{17} \]

9. Final simplification 15.9%

   \[\leadsto 17 \]

Alternative 9: 16.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 27 \end{array} \]

FPCore

(FPCore (x) :precision binary64 27.0)

C

double code(double x) {
	return 27.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 27.0d0
end function

Java

public static double code(double x) {
	return 27.0;
}

Python

def code(x):
	return 27.0

Julia

function code(x)
	return 27.0
end

MATLAB

function tmp = code(x)
	tmp = 27.0;
end

Wolfram

code[x_] := 27.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 16.4%

   \[\leadsto \color{blue}{27} \]

9. Final simplification 16.4%

   \[\leadsto 27 \]

Alternative 10: 17.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 64 \end{array} \]

FPCore

(FPCore (x) :precision binary64 64.0)

C

double code(double x) {
	return 64.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 64.0d0
end function

Java

public static double code(double x) {
	return 64.0;
}

Python

def code(x):
	return 64.0

Julia

function code(x)
	return 64.0
end

MATLAB

function tmp = code(x)
	tmp = 64.0;
end

Wolfram

code[x_] := 64.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 17.5%

   \[\leadsto \color{blue}{64} \]

9. Final simplification 17.5%

   \[\leadsto 64 \]

Alternative 11: 17.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 65 \end{array} \]

FPCore

(FPCore (x) :precision binary64 65.0)

C

double code(double x) {
	return 65.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 65.0d0
end function

Java

public static double code(double x) {
	return 65.0;
}

Python

def code(x):
	return 65.0

Julia

function code(x)
	return 65.0
end

MATLAB

function tmp = code(x)
	tmp = 65.0;
end

Wolfram

code[x_] := 65.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 17.6%

   \[\leadsto \color{blue}{65} \]

9. Final simplification 17.6%

   \[\leadsto 65 \]

Alternative 12: 19.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 256 \end{array} \]

FPCore

(FPCore (x) :precision binary64 256.0)

C

double code(double x) {
	return 256.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 256.0d0
end function

Java

public static double code(double x) {
	return 256.0;
}

Python

def code(x):
	return 256.0

Julia

function code(x)
	return 256.0
end

MATLAB

function tmp = code(x)
	tmp = 256.0;
end

Wolfram

code[x_] := 256.0

Derivation

1. Initial program 59.1%

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

2. Taylor expanded in x around inf 99.0%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. 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. associate-*r/ 0.0%

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

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

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

   6. flip-+ 12.1%

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

   7. count-2 12.1%

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

   8. count-2 12.1%

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

   9. swap-sqr 12.1%

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

   10. log-prod 12.2%

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

   11. metadata-eval 12.2%

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

4. Applied egg-rr 12.2%

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

6. Simplified 13.8%

   \[\leadsto \color{blue}{-1 + \log 4} \]

8. Applied egg-rr 19.6%

   \[\leadsto \color{blue}{256} \]

9. Final simplification 19.6%

   \[\leadsto 256 \]

Developer target: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x - 1} \cdot \sqrt{x + 1}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0))))))

C

double code(double x) {
	return log((x + (sqrt((x - 1.0)) * sqrt((x + 1.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + (sqrt((x - 1.0d0)) * sqrt((x + 1.0d0)))))
end function

Java

public static double code(double x) {
	return Math.log((x + (Math.sqrt((x - 1.0)) * Math.sqrt((x + 1.0)))));
}

Python

def code(x):
	return math.log((x + (math.sqrt((x - 1.0)) * math.sqrt((x + 1.0)))))

Julia

function code(x)
	return log(Float64(x + Float64(sqrt(Float64(x - 1.0)) * sqrt(Float64(x + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + (sqrt((x - 1.0)) * sqrt((x + 1.0)))));
end

Wolfram

code[x_] := N[Log[N[(x + N[(N[Sqrt[N[(x - 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))