Hyperbolic arc-(co)secant

Percentage Accurate: 99.9% → 99.7%

Time: 7.1s

Alternatives: 5

Speedup: 1.1×

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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (+
  (log (pow x -0.5))
  (log (* (pow x -0.5) (+ (sqrt (fma (- x) x 1.0)) 1.0)))))

C

double code(double x) {
	return log(pow(x, -0.5)) + log((pow(x, -0.5) * (sqrt(fma(-x, x, 1.0)) + 1.0)));
}

Julia

function code(x)
	return Float64(log((x ^ -0.5)) + log(Float64((x ^ -0.5) * Float64(sqrt(fma(Float64(-x), x, 1.0)) + 1.0))))
end

Wolfram

code[x_] := N[(N[Log[N[Power[x, -0.5], $MachinePrecision]], $MachinePrecision] + N[Log[N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[Sqrt[N[((-x) * x + 1.0), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. distribute-rgt1-in N/A

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

   7. *-commutative N/A

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

   8. lift-/.f64 N/A

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

   9. inv-pow N/A

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

   10. sqr-pow N/A

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

   11. associate-*l* N/A

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

   12. log-prod N/A

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

   13. lower-+.f64 N/A

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

   14. lower-log.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\log \left({x}^{-0.5}\right) + \log \left({x}^{-0.5} \cdot \left(\sqrt{\mathsf{fma}\left(-x, x, 1\right)} + 1\right)\right)} \]
5. Add Preprocessing

Alternative 2: 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 99.6%

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

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

   3. div-inv N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-rgt1-in N/A

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

   6. lift-/.f64 N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-+.f64 99.6

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-neg.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.5% 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 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

   \[\leadsto \log \color{blue}{\left(\frac{2 + \frac{-1}{2} \cdot {x}^{2}}{x}\right)} \]
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 99.5

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   \[\leadsto \color{blue}{\log 2 + -1 \cdot \log x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\log 2 - \log x} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{\log 2 - \log x} \]

   4. lower-log.f64 N/A

      \[\leadsto \color{blue}{\log 2} - \log x \]

   5. lower-log.f64 99.0

      \[\leadsto \log 2 - \color{blue}{\log x} \]

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\log 2 - \log x} \]
6. Step-by-step derivation

7. Applied rewrites 99.3%

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

9. Applied rewrites 99.3%

   \[\leadsto -\log \left(x \cdot 0.5\right) \]
10. Add Preprocessing

Alternative 5: 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 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

   \[\leadsto \log \color{blue}{\left(\frac{2 + \frac{-1}{2} \cdot {x}^{2}}{x}\right)} \]
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 99.5

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

5. Applied rewrites 99.5%

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

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 2024323 
(FPCore (x)
  :name "Hyperbolic arc-(co)secant"
  :precision binary64
  (log (+ (/ 1.0 x) (/ (sqrt (- 1.0 (* x x))) x))))