Hyperbolic arc-cosine

Percentage Accurate: 52.1% → 99.6%

Time: 4.8s

Alternatives: 10

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: 52.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in x around inf 99.1%

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

   1. *-commutative 99.1%

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

   2. associate-*r/ 99.1%

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

   3. metadata-eval 99.1%

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

4. Simplified 99.1%

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

5. Final simplification 99.1%

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

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 49.7%

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

2. Taylor expanded in x around inf 98.8%

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

3. Final simplification 98.8%

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

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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. sum-log 18.7%

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

   12. *-commutative 18.7%

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

   13. count-2 18.7%

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

   14. flip-+ 0.0%

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

   15. +-inverses 0.0%

       \[\leadsto \log \left(\frac{\color{blue}{0}}{x - x}\right) + \log \left(x \cdot 2\right) \]

   16. +-inverses 0.0%

       \[\leadsto \log \left(\frac{0}{\color{blue}{0}}\right) + \log \left(x \cdot 2\right) \]

   17. *-commutative 0.0%

       \[\leadsto \log \left(\frac{0}{0}\right) + \log \color{blue}{\left(2 \cdot x\right)} \]

   18. count-2 0.0%

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

   19. flip-+ 0.0%

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

   20. +-inverses 0.0%

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

   21. +-inverses 0.0%

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

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: 14.3% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.5)

C

double code(double x) {
	return 1.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.5d0
end function

Java

public static double code(double x) {
	return 1.5;
}

Python

def code(x):
	return 1.5

Julia

function code(x)
	return 1.5
end

MATLAB

function tmp = code(x)
	tmp = 1.5;
end

Wolfram

code[x_] := 1.5

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 14.3%

   \[\leadsto \color{blue}{1.5} \]

9. Final simplification 14.3%

   \[\leadsto 1.5 \]

Alternative 5: 14.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 3 \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.0)

C

double code(double x) {
	return 3.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.0d0
end function

Java

public static double code(double x) {
	return 3.0;
}

Python

def code(x):
	return 3.0

Julia

function code(x)
	return 3.0
end

MATLAB

function tmp = code(x)
	tmp = 3.0;
end

Wolfram

code[x_] := 3.0

Derivation

1. Initial program 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 14.6%

   \[\leadsto \color{blue}{3} \]

9. Final simplification 14.6%

   \[\leadsto 3 \]

Alternative 6: 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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 15.0%

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

9. Final simplification 15.0%

   \[\leadsto 6 \]

Alternative 7: 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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 15.3%

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

9. Final simplification 15.3%

   \[\leadsto 9 \]

Alternative 8: 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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 15.8%

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

9. Final simplification 15.8%

   \[\leadsto 16 \]

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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 16.3%

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

9. Final simplification 16.3%

   \[\leadsto 27 \]

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 49.7%

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

2. Taylor expanded in x around inf 98.8%

   \[\leadsto \log \left(x + \color{blue}{x}\right) \]
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. count-2 0.0%

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

   4. *-commutative 0.0%

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

   5. associate-*r/ 0.0%

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

   6. +-inverses 0.0%

      \[\leadsto \log \left(\left(x \cdot 2\right) \cdot \frac{\color{blue}{0}}{x - x}\right) \]

   7. +-inverses 0.0%

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

   8. flip-+ 10.6%

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

   9. count-2 10.6%

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

   10. *-commutative 10.6%

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

   11. swap-sqr 10.6%

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

   12. rem-exp-log 10.6%

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

   13. rem-exp-log 10.6%

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

   14. log-prod 10.6%

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

   15. pow2 10.6%

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

   16. rem-exp-log 10.6%

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

   17. rem-exp-log 10.6%

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

   18. metadata-eval 10.6%

       \[\leadsto \log \left({x}^{2}\right) + \log \color{blue}{4} \]

4. Applied egg-rr 10.6%

   \[\leadsto \color{blue}{\log \left({x}^{2}\right) + \log 4} \]

6. Simplified 13.7%

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

8. Applied egg-rr 17.4%

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

9. Final simplification 17.4%

   \[\leadsto 64 \]

Reproduce

herbie shell --seed 2023325 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))