Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.5%

Time: 5.7s

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: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-1}{x} \cdot \mathsf{fma}\left(0.5, x \cdot x, -2\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (* (/ -1.0 x) (fma 0.5 (* x x) -2.0))))

C

double code(double x) {
	return log(((-1.0 / x) * fma(0.5, (x * x), -2.0)));
}

Julia

function code(x)
	return log(Float64(Float64(-1.0 / x) * fma(0.5, Float64(x * x), -2.0)))
end

Wolfram

code[x_] := N[Log[N[(N[(-1.0 / x), $MachinePrecision] * N[(0.5 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $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. metadata-eval N/A

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

   4. distribute-neg-frac N/A

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

   5. div-inv N/A

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

   6. lift-/.f64 N/A

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

   7. neg-mul-1 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. distribute-frac-neg2 N/A

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

   12. lift-/.f64 N/A

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

   13. neg-mul-1 N/A

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

   14. distribute-rgt-out N/A

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

   15. lower-*.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. div-inv N/A

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

   18. lower-/.f64 N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 100.0

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

   21. lift--.f64 N/A

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

   22. lift-*.f64 N/A

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

   23. cancel-sign-sub-inv N/A

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

   24. +-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 100.0

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

7. Applied rewrites 100.0%

   \[\leadsto \log \left(\frac{-1}{x} \cdot \color{blue}{\mathsf{fma}\left(0.5, x \cdot x, -2\right)}\right) \]
8. Add Preprocessing

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Log[N[(-0.5 * x + N[(2.0 / x), $MachinePrecision]), $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 100.0

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

5. Applied rewrites 100.0%

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

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. Taylor expanded in x around inf

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

11. Applied rewrites 100.0%

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

Alternative 3: 99.1% 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 99.8

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

5. Applied rewrites 99.8%

   \[\leadsto \log \color{blue}{\left(\frac{2}{x}\right)} \]
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 100.0

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

5. Applied rewrites 100.0%

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

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 2024307 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))