Hyperbolic arc-cosine

Percentage Accurate: 51.8% → 99.7%

Time: 9.1s

Alternatives: 5

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: 51.8% 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: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - \frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (-
   (* x (+ 2.0 (/ (+ -0.125 (/ -0.0625 (* x x))) (* (* x x) (* x x)))))
   (/ 0.5 x))))

C

double code(double x) {
	return log(((x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))))) - (0.5 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * (2.0d0 + (((-0.125d0) + ((-0.0625d0) / (x * x))) / ((x * x) * (x * x))))) - (0.5d0 / x)))
end function

Java

public static double code(double x) {
	return Math.log(((x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))))) - (0.5 / x)));
}

Python

def code(x):
	return math.log(((x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))))) - (0.5 / x)))

Julia

function code(x)
	return log(Float64(Float64(x * Float64(2.0 + Float64(Float64(-0.125 + Float64(-0.0625 / Float64(x * x))) / Float64(Float64(x * x) * Float64(x * x))))) - Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * (2.0 + ((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))))) - (0.5 / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(x * N[(2.0 + N[(N[(-0.125 + N[(-0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) + 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right) \]

   2. fma-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, 1 \cdot \frac{\frac{-1}{2}}{x}\right)\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \frac{\frac{-1}{2}}{x}\right)\right)\right) \]

   4. frac-2neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   5. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{neg}\left(\frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{x}\right)\right)\right)\right) \]

   6. fmm-undef N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8} - \frac{\frac{1}{16}}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), \left(\frac{\mathsf{neg}\left(\frac{-1}{2}\right)}{x}\right)\right)\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - \frac{0.5}{x}\right)} \]
8. Add Preprocessing

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot \left(2 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.5}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (- (* x (+ 2.0 (/ -0.125 (* x (* x (* x x)))))) (/ 0.5 x))))

C

double code(double x) {
	return log(((x * (2.0 + (-0.125 / (x * (x * (x * x)))))) - (0.5 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * (2.0d0 + ((-0.125d0) / (x * (x * (x * x)))))) - (0.5d0 / x)))
end function

Java

public static double code(double x) {
	return Math.log(((x * (2.0 + (-0.125 / (x * (x * (x * x)))))) - (0.5 / x)));
}

Python

def code(x):
	return math.log(((x * (2.0 + (-0.125 / (x * (x * (x * x)))))) - (0.5 / x)))

Julia

function code(x)
	return log(Float64(Float64(x * Float64(2.0 + Float64(-0.125 / Float64(x * Float64(x * Float64(x * x)))))) - Float64(0.5 / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * (2.0 + (-0.125 / (x * (x * (x * x)))))) - (0.5 / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(x * N[(2.0 + N[(-0.125 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right)\right) \]

   3. distribute-lft-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + x \cdot \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right), \left(\left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) \cdot x\right)\right)\right) \]

5. Simplified 99.8%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 + \frac{-0.125 - \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \color{blue}{\left(x \cdot \left(2 - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)}\right)\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \left(x \cdot \left(2 + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \left(x \cdot \left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right) + 2\right)\right)\right)\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \left(x \cdot \left(\left(0 - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right) + 2\right)\right)\right)\right) \]

   4. associate-+l- N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \left(x \cdot \left(0 - \left(\frac{1}{8} \cdot \frac{1}{{x}^{4}} - 2\right)\right)\right)\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \left(x \cdot \left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}} - 2\right)\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}} - 2\right)\right)\right)\right)\right)\right) \]

   7. neg-sub0 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \left(0 - \left(\frac{1}{8} \cdot \frac{1}{{x}^{4}} - 2\right)\right)\right)\right)\right) \]

   8. associate-+l- N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(0 - \frac{1}{8} \cdot \frac{1}{{x}^{4}}\right) + 2\right)\right)\right)\right) \]

   9. neg-sub0 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right) + 2\right)\right)\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \left(2 + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

   12. associate-*r/ N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

   14. distribute-neg-frac N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{8}}{{x}^{4}}\right)\right)\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{8}}{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right)\right)\right) \]

   17. pow-sqr N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{8}}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   19. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

   20. associate-*l* N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   21. unpow2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

   22. cube-mult N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot {x}^{3}\right)\right)\right)\right)\right)\right) \]

   23. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left({x}^{3}\right)\right)\right)\right)\right)\right)\right) \]

   24. cube-mult N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(1, \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right) \]

8. Simplified 99.7%

   \[\leadsto \log \left(1 \cdot \frac{-0.5}{x} + \color{blue}{x \cdot \left(2 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)}\right) \]
9. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2}}{x} + x \cdot \left(2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) + \frac{\frac{-1}{2}}{x}\right)\right) \]

   3. fma-define N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \frac{\frac{-1}{2}}{x}\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}, \mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right)\right) \]

   6. fmm-undef N/A

      \[\leadsto \mathsf{log.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{\frac{1}{2}}{x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(2 + \frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \left(\frac{\frac{-1}{8}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

   14. /-lowering-/.f64 99.7%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(2, \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right) \]

10. Applied egg-rr 99.7%

    \[\leadsto \log \color{blue}{\left(x \cdot \left(2 + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) - \frac{0.5}{x}\right)} \]
11. Add Preprocessing

Alternative 3: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \log \left(x \cdot 2 + \frac{-0.5 + \frac{-0.125}{x \cdot x}}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log (+ (* x 2.0) (/ (+ -0.5 (/ -0.125 (* x x))) x))))

C

double code(double x) {
	return log(((x * 2.0) + ((-0.5 + (-0.125 / (x * x))) / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((x * 2.0d0) + (((-0.5d0) + ((-0.125d0) / (x * x))) / x)))
end function

Java

public static double code(double x) {
	return Math.log(((x * 2.0) + ((-0.5 + (-0.125 / (x * x))) / x)));
}

Python

def code(x):
	return math.log(((x * 2.0) + ((-0.5 + (-0.125 / (x * x))) / x)))

Julia

function code(x)
	return log(Float64(Float64(x * 2.0) + Float64(Float64(-0.5 + Float64(-0.125 / Float64(x * x))) / x)))
end

MATLAB

function tmp = code(x)
	tmp = log(((x * 2.0) + ((-0.5 + (-0.125 / (x * x))) / x)));
end

Wolfram

code[x_] := N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(-0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 51.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(-1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{2}}\right)\right)\right)\right)\right) \]

   7. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(x \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{\mathsf{neg}\left({x}^{2}\right)}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{\mathsf{neg}\left({x}^{2}\right)}\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{\mathsf{neg}\left(x \cdot x\right)}\right)\right)\right) \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x \cdot \left(\mathsf{neg}\left(x\right)\right)}\right)\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x \cdot \left(-1 \cdot x\right)}\right)\right)\right) \]

   12. times-frac N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{x}{x} \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{-1 \cdot x}\right)\right)\right) \]

   13. *-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{-1 \cdot x}\right)\right)\right) \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   15. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(1 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right)\right) \]

   16. distribute-rgt-neg-out N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\mathsf{neg}\left(1 \cdot \frac{\frac{1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)\right)\right)\right) \]

5. Simplified 99.7%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{-0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
6. Add Preprocessing

Alternative 4: 99.5% 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 51.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.0%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - x \cdot x}{x - x}\right)\right) \]

   2. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{0}{x - x}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{0 + 0}{x - x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot 0 + 0}{x - x}\right)\right) \]

   5. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + 0}{x - x}\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + \left(x \cdot x - x \cdot x\right)}{x - x}\right)\right) \]

   7. distribute-lft-out-- N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x - x\right)}{x - x}\right)\right) \]

   8. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot 0}{x - x}\right)\right) \]

   9. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{x - x}\right)\right) \]

   10. +-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{0}\right)\right) \]

   11. +-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot x - x \cdot x}\right)\right) \]

   12. distribute-lft-out-- N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{x \cdot \left(x - x\right)}\right)\right) \]

   13. frac-add N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2}}{x} + \frac{x \cdot x - x \cdot x}{x - x}\right)\right) \]

   14. *-lft-identity N/A

       \[\leadsto \mathsf{log.f64}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \frac{x \cdot x - x \cdot x}{x - x}\right)\right) \]

   15. flip-+ N/A

       \[\leadsto \mathsf{log.f64}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + \left(x + x\right)\right)\right) \]

   16. associate-+r+ N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x\right) + x\right)\right) \]

   17. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x} + x\right), x\right)\right) \]

   18. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot \frac{\frac{-1}{2}}{x}\right), x\right), x\right)\right) \]

   19. *-lft-identity N/A

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{-1}{2}}{x}\right), x\right), x\right)\right) \]

   20. /-lowering-/.f64 99.4%

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), x\right), x\right)\right) \]

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 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 51.6%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
4. Step-by-step derivation

5. Simplified 99.0%

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

Reproduce

herbie shell --seed 2024163 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))