Hyperbolic arc-(co)secant

Percentage Accurate: 100.0% → 100.0%

Time: 7.3s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. log-lowering-log.f64 N/A

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

   2. div-inv N/A

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

   3. distribute-rgt1-in N/A

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

   4. un-div-inv N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. pow1/2 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

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

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

   13. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

   1. +-commutative N/A

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

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

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

   3. unpow1/2 N/A

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

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

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

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(1, \left(x \cdot x\right)\right)\right), 1\right), x\right)\right) \]

   6. *-lowering-*.f64 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{1 + \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0625 + -0.125\right)\right)\right)\right)}{x}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (/
   (+
    1.0
    (+ 1.0 (* x (* x (+ -0.5 (* (* x x) (+ (* (* x x) -0.0625) -0.125)))))))
   x)))

C

double code(double x) {
	return log(((1.0 + (1.0 + (x * (x * (-0.5 + ((x * x) * (((x * x) * -0.0625) + -0.125))))))) / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(((1.0d0 + (1.0d0 + (x * (x * ((-0.5d0) + ((x * x) * (((x * x) * (-0.0625d0)) + (-0.125d0)))))))) / x))
end function

Java

public static double code(double x) {
	return Math.log(((1.0 + (1.0 + (x * (x * (-0.5 + ((x * x) * (((x * x) * -0.0625) + -0.125))))))) / x));
}

Python

def code(x):
	return math.log(((1.0 + (1.0 + (x * (x * (-0.5 + ((x * x) * (((x * x) * -0.0625) + -0.125))))))) / x))

Julia

function code(x)
	return log(Float64(Float64(1.0 + Float64(1.0 + Float64(x * Float64(x * Float64(-0.5 + Float64(Float64(x * x) * Float64(Float64(Float64(x * x) * -0.0625) + -0.125))))))) / x))
end

MATLAB

function tmp = code(x)
	tmp = log(((1.0 + (1.0 + (x * (x * (-0.5 + ((x * x) * (((x * x) * -0.0625) + -0.125))))))) / x));
end

Wolfram

code[x_] := N[Log[N[(N[(1.0 + N[(1.0 + N[(x * N[(x * N[(-0.5 + N[(N[(x * x), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * -0.0625), $MachinePrecision] + -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. log-lowering-log.f64 N/A

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

   2. div-inv N/A

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

   3. distribute-rgt1-in N/A

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

   4. un-div-inv N/A

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

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

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

   6. +-commutative N/A

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

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

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

   8. pow1/2 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. rem-square-sqrt N/A

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

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

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

   13. *-lowering-*.f64 99.6%

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

4. Applied egg-rr 99.6%

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

5. Taylor expanded in x around 0

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

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

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. unpow2 N/A

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

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

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

   \[\leadsto \log \left(\frac{1 + \left(1 + x \cdot \left(x \cdot \left(-0.5 + \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.0625 + -0.125\right)\right)\right)\right)}{x}\right) \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 N/A

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

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

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

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

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

   4. unpow2 N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. unpow2 N/A

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

   13. *-lowering-*.f64 99.5%

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

5. Simplified 99.5%

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

Alternative 5: 99.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0 - \log \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 0.0 (log (/ x 2.0))))

C

double code(double x) {
	return 0.0 - log((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - log((x / 2.0d0))
end function

Java

public static double code(double x) {
	return 0.0 - Math.log((x / 2.0));
}

Python

def code(x):
	return 0.0 - math.log((x / 2.0))

Julia

function code(x)
	return Float64(0.0 - log(Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - log((x / 2.0));
end

Wolfram

code[x_] := N[(0.0 - N[Log[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 99.0%

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

5. Simplified 99.0%

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

   1. clear-num N/A

      \[\leadsto \log \left(\frac{1}{\frac{x}{2}}\right) \]

   2. log-rec N/A

      \[\leadsto \mathsf{neg}\left(\log \left(\frac{x}{2}\right)\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{x}{2}\right)\right) \]

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

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{x}{2}\right)\right)\right) \]

   5. /-lowering-/.f64 99.3%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right) \]

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

   \[\leadsto 0 - \log \left(\frac{x}{2}\right) \]
9. Add Preprocessing

Alternative 6: 99.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (/ 2.0 x)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(Float64(2.0 / x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0

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

   1. /-lowering-/.f64 99.0%

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

5. Simplified 99.0%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))