Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.9%

Time: 3.1s

Alternatives: 4

Speedup: N/A×

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: 99.9% 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: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{\sqrt{\mathsf{fma}\left(-x, x, 1\right)} - -1}{x}\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return log(((sqrt(fma(-x, x, 1.0)) - -1.0) / x));
}

Julia

function code(x)
	return log(Float64(Float64(sqrt(fma(Float64(-x), x, 1.0)) - -1.0) / x))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-add-rev N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 100.0

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

   8. lift-sqrt.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. sin-acos-rev N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-acos.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-sin.f64 N/A

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

   2. lift-acos.f64 N/A

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

   3. sin-acos-rev N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-fma.f64 N/A

       \[\leadsto \log \left(\frac{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(x\right), x, 1\right)}} + 1}{x}\right) \]

   11. lower-neg.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 2\right)}}{x}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 3: 99.2% accurate, 1.3× 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 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 4: 0.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \log \left(\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{2}, 2\right)}}{x}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 0.0%

   \[\leadsto \log \left(-0.5 \cdot \color{blue}{x}\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))