Rust f64::acosh

Percentage Accurate: 51.2% → 99.7%

Time: 6.4s

Alternatives: 4

Speedup: 2.0×

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: 51.2% 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: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

5. Simplified 100.0%

   \[\leadsto \log \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(2 - \frac{0.125 + \frac{0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)} \]

7. Applied egg-rr 100.0%

   \[\leadsto \log \color{blue}{\left(\frac{x}{0.5} - \left(\frac{0.5}{x} - \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot x\right)}\right)\right)} \]
8. Step-by-step derivation

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

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

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

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

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

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

   4. associate-/l/ N/A

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

   5. sub-div N/A

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

   6. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

   12. *-lowering-*.f64 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

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

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

   11. mul-1-neg N/A

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

   12. times-frac N/A

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

   13. *-inverses N/A

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

   14. mul-1-neg N/A

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

   15. distribute-neg-frac2 N/A

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

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

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

   17. metadata-eval N/A

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

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

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

5. Simplified 100.0%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

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

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

   5. *-commutative N/A

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. associate-*r/ N/A

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

   12. unpow2 N/A

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

   13. times-frac N/A

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

   14. *-inverses N/A

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

   15. metadata-eval N/A

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

   16. distribute-neg-frac N/A

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

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

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

   18. distribute-neg-frac N/A

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

   19. metadata-eval N/A

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

   20. /-lowering-/.f64 99.9%

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

5. Simplified 99.9%

   \[\leadsto \log \color{blue}{\left(x \cdot 2 + 1 \cdot \frac{-0.5}{x}\right)} \]
6. Step-by-step derivation

   1. fma-define N/A

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

   2. clear-num N/A

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

   3. div-inv N/A

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

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2, \frac{\mathsf{neg}\left(-1\right)}{\frac{x}{\frac{-1}{2}}}\right)\right)\right) \]

   5. distribute-neg-frac N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\mathsf{fma}\left(x, 2, \mathsf{neg}\left(\frac{-1}{\frac{x}{\frac{-1}{2}}}\right)\right)\right)\right) \]

   6. fmm-undef N/A

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

   7. div-inv N/A

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

   8. clear-num N/A

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

   9. *-lft-identity N/A

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

   10. neg-mul-1 N/A

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

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

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. /-lowering-/.f64 N/A

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

   16. metadata-eval N/A

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

   17. *-lft-identity N/A

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

   18. distribute-neg-frac N/A

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

   19. /-lowering-/.f64 N/A

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

   20. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

Alternative 4: 99.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (+ x x)))

C

double code(double x) {
	return log((x + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + x))
end function

Java

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

Python

def code(x):
	return math.log((x + x))

Julia

function code(x)
	return log(Float64(x + x))
end

MATLAB

function tmp = code(x)
	tmp = log((x + x));
end

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

5. Simplified 99.5%

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

Developer Target 1: 100.0% accurate, 0.7× 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 2024159 
(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)))))