Rust f64::acosh

Percentage Accurate: 52.3% → 99.5%

Time: 6.0s

Alternatives: 5

Speedup: 1.2×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acosh x))

C

double code(double x) {
	return acosh(x);
}

Python

def code(x):
	return math.acosh(x)

Julia

function code(x)
	return acosh(x)
end

MATLAB

function tmp = code(x)
	tmp = acosh(x);
end

Wolfram

code[x_] := N[ArcCosh[x], $MachinePrecision]

Initial Program: 52.3% 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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

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

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

   8. distribute-lft-neg-out N/A

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

   9. mul-1-neg N/A

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

   10. *-commutative N/A

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

   11. lower--.f64 N/A

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

   12. *-commutative N/A

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

   13. mul-1-neg N/A

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

   14. distribute-lft-neg-out N/A

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

   15. *-commutative N/A

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

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

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

   17. metadata-eval N/A

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.2× speedup

Math

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

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 53.8%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 99.2

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

5. Applied rewrites 99.2%

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

Alternative 3: 2.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{0.25}{x}}{x}\\ \frac{1}{\frac{t\_0 \cdot 0 - \frac{0.25}{x}}{t\_0 \cdot \frac{0.25}{x}}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (/ 0.25 x) x)))
   (/ 1.0 (/ (- (* t_0 0.0) (/ 0.25 x)) (* t_0 (/ 0.25 x))))))

C

double code(double x) {
	double t_0 = (0.25 / x) / x;
	return 1.0 / (((t_0 * 0.0) - (0.25 / x)) / (t_0 * (0.25 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = (0.25d0 / x) / x
    code = 1.0d0 / (((t_0 * 0.0d0) - (0.25d0 / x)) / (t_0 * (0.25d0 / x)))
end function

Java

public static double code(double x) {
	double t_0 = (0.25 / x) / x;
	return 1.0 / (((t_0 * 0.0) - (0.25 / x)) / (t_0 * (0.25 / x)));
}

Python

def code(x):
	t_0 = (0.25 / x) / x
	return 1.0 / (((t_0 * 0.0) - (0.25 / x)) / (t_0 * (0.25 / x)))

Julia

function code(x)
	t_0 = Float64(Float64(0.25 / x) / x)
	return Float64(1.0 / Float64(Float64(Float64(t_0 * 0.0) - Float64(0.25 / x)) / Float64(t_0 * Float64(0.25 / x))))
end

MATLAB

function tmp = code(x)
	t_0 = (0.25 / x) / x;
	tmp = 1.0 / (((t_0 * 0.0) - (0.25 / x)) / (t_0 * (0.25 / x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.25 / x), $MachinePrecision] / x), $MachinePrecision]}, N[(1.0 / N[(N[(N[(t$95$0 * 0.0), $MachinePrecision] - N[(0.25 / x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(0.25 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\log x + \log 2\right) - \color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} \]

   12. unpow2 N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\frac{1}{4}}{\color{blue}{x \cdot x}} \]

   13. lower-*.f64 99.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 2.7%

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

12. Applied rewrites 2.7%

    \[\leadsto \frac{1}{\frac{0 \cdot \frac{\frac{0.25}{x}}{x} - \frac{0.25}{x}}{\frac{0.25}{x} \cdot \color{blue}{\frac{\frac{0.25}{x}}{x}}}} \]

13. Final simplification 2.7%

    \[\leadsto \frac{1}{\frac{\frac{\frac{0.25}{x}}{x} \cdot 0 - \frac{0.25}{x}}{\frac{\frac{0.25}{x}}{x} \cdot \frac{0.25}{x}}} \]
14. Add Preprocessing

Alternative 4: 2.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-4 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (* -4.0 (* x x))))

C

double code(double x) {
	return 1.0 / (-4.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((-4.0d0) * (x * x))
end function

Java

public static double code(double x) {
	return 1.0 / (-4.0 * (x * x));
}

Python

def code(x):
	return 1.0 / (-4.0 * (x * x))

Julia

function code(x)
	return Float64(1.0 / Float64(-4.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (-4.0 * (x * x));
end

Wolfram

code[x_] := N[(1.0 / N[(-4.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\log x + \log 2\right) - \color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} \]

   12. unpow2 N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\frac{1}{4}}{\color{blue}{x \cdot x}} \]

   13. lower-*.f64 99.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Step-by-step derivation

10. Applied rewrites 2.7%

    \[\leadsto \frac{1}{\left(x \cdot x\right) \cdot \color{blue}{-4}} \]

11. Final simplification 2.7%

    \[\leadsto \frac{1}{-4 \cdot \left(x \cdot x\right)} \]
12. Add Preprocessing

Alternative 5: 2.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.25}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -0.25 (* x x)))

C

double code(double x) {
	return -0.25 / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-0.25d0) / (x * x)
end function

Java

public static double code(double x) {
	return -0.25 / (x * x);
}

Python

def code(x):
	return -0.25 / (x * x)

Julia

function code(x)
	return Float64(-0.25 / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = -0.25 / (x * x);
end

Wolfram

code[x_] := N[(-0.25 / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.8%

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

3. Taylor expanded in x around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 N/A

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

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right)} + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   5. log-rec N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right)\right) + \log 2\right) - \frac{1}{4} \cdot \frac{1}{{x}^{2}} \]

   6. remove-double-neg N/A

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

   7. lower-log.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\color{blue}{\frac{1}{4}}}{{x}^{2}} \]

   11. lower-/.f64 N/A

       \[\leadsto \left(\log x + \log 2\right) - \color{blue}{\frac{\frac{1}{4}}{{x}^{2}}} \]

   12. unpow2 N/A

       \[\leadsto \left(\log x + \log 2\right) - \frac{\frac{1}{4}}{\color{blue}{x \cdot x}} \]

   13. lower-*.f64 99.5

       \[\leadsto \left(\log x + \log 2\right) - \frac{0.25}{\color{blue}{x \cdot x}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\left(\log x + \log 2\right) - \frac{0.25}{x \cdot x}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{-1}{4}}{\color{blue}{{x}^{2}}} \]
7. Step-by-step derivation

8. Applied rewrites 2.7%

   \[\leadsto \frac{-0.25}{\color{blue}{x \cdot x}} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024332 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :alt
  (! :herbie-platform default (log (+ x (* (sqrt (- x 1)) (sqrt (+ x 1))))))

  (log (+ x (sqrt (- (* x x) 1.0)))))