Rust f64::acosh

Percentage Accurate: 51.0% → 99.7%

Time: 11.2s

Alternatives: 7

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: 51.0% 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, 1.6× speedup

Math

\[\begin{array}{l} \\ 0 - \log \left(\frac{-1}{\frac{0.5}{x} - x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 2\right)}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  0.0
  (log
   (/
    -1.0
    (-
     (/ 0.5 x)
     (* x (+ (/ (+ -0.125 (/ -0.0625 (* x x))) (* x (* x (* x x)))) 2.0)))))))

C

double code(double x) {
	return 0.0 - log((-1.0 / ((0.5 / x) - (x * (((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x)))) + 2.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - log(((-1.0d0) / ((0.5d0 / x) - (x * ((((-0.125d0) + ((-0.0625d0) / (x * x))) / (x * (x * (x * x)))) + 2.0d0)))))
end function

Java

public static double code(double x) {
	return 0.0 - Math.log((-1.0 / ((0.5 / x) - (x * (((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x)))) + 2.0)))));
}

Python

def code(x):
	return 0.0 - math.log((-1.0 / ((0.5 / x) - (x * (((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x)))) + 2.0)))))

Julia

function code(x)
	return Float64(0.0 - log(Float64(-1.0 / Float64(Float64(0.5 / x) - Float64(x * Float64(Float64(Float64(-0.125 + Float64(-0.0625 / Float64(x * x))) / Float64(x * Float64(x * Float64(x * x)))) + 2.0))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - log((-1.0 / ((0.5 / x) - (x * (((-0.125 + (-0.0625 / (x * x))) / (x * (x * (x * x)))) + 2.0)))));
end

Wolfram

code[x_] := N[(0.0 - N[Log[N[(-1.0 / N[(N[(0.5 / x), $MachinePrecision] - N[(x * N[(N[(N[(-0.125 + N[(-0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

   4. *-lft-identity N/A

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

   5. /-lowering-/.f64 N/A

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

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

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

7. Applied egg-rr 99.1%

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

9. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{-\log \left(\frac{1}{\frac{-0.5}{x} + \left(\left(1 + \frac{\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x}}{x \cdot \left(x \cdot x\right)}\right) + 1\right) \cdot x}\right)} \]
10. Step-by-step derivation

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. fma-define N/A

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

    4. frac-2neg N/A

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

    5. distribute-frac-neg2 N/A

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

    6. distribute-neg-frac N/A

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

    7. fmm-undef N/A

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

11. Applied egg-rr 99.1%

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

12. Final simplification 99.1%

    \[\leadsto 0 - \log \left(\frac{-1}{\frac{0.5}{x} - x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + 2\right)}\right) \]
13. Add Preprocessing

Alternative 2: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-0.5}{x} + \left(x + x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (+
   (/ -0.5 x)
   (+
    x
    (* x (+ (/ (+ -0.125 (/ -0.0625 (* x x))) (* (* x x) (* x x))) 1.0))))))

C

double code(double x) {
	return log(((-0.5 / x) + (x + (x * (((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))) + 1.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((-0.5d0) / x) + (x + (x * ((((-0.125d0) + ((-0.0625d0) / (x * x))) / ((x * x) * (x * x))) + 1.0d0)))))
end function

Java

public static double code(double x) {
	return Math.log(((-0.5 / x) + (x + (x * (((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))) + 1.0)))));
}

Python

def code(x):
	return math.log(((-0.5 / x) + (x + (x * (((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))) + 1.0)))))

Julia

function code(x)
	return log(Float64(Float64(-0.5 / x) + Float64(x + Float64(x * Float64(Float64(Float64(-0.125 + Float64(-0.0625 / Float64(x * x))) / Float64(Float64(x * x) * Float64(x * x))) + 1.0)))))
end

MATLAB

function tmp = code(x)
	tmp = log(((-0.5 / x) + (x + (x * (((-0.125 + (-0.0625 / (x * x))) / ((x * x) * (x * x))) + 1.0)))));
end

Wolfram

code[x_] := N[Log[N[(N[(-0.5 / x), $MachinePrecision] + N[(x + N[(x * N[(N[(N[(-0.125 + N[(-0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

   4. *-lft-identity N/A

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

   5. /-lowering-/.f64 N/A

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

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

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

   \[\leadsto \log \left(\frac{-0.5}{x} + \left(x + x \cdot \left(\frac{-0.125 + \frac{-0.0625}{x \cdot x}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{-0.5}{x} + x \cdot \left(2 - \frac{\frac{0.0625}{x \cdot x} + 0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (log
  (+
   (/ -0.5 x)
   (* x (- 2.0 (/ (+ (/ 0.0625 (* x x)) 0.125) (* (* x x) (* x x))))))))

C

double code(double x) {
	return log(((-0.5 / x) + (x * (2.0 - (((0.0625 / (x * x)) + 0.125) / ((x * x) * (x * x)))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((((-0.5d0) / x) + (x * (2.0d0 - (((0.0625d0 / (x * x)) + 0.125d0) / ((x * x) * (x * x)))))))
end function

Java

public static double code(double x) {
	return Math.log(((-0.5 / x) + (x * (2.0 - (((0.0625 / (x * x)) + 0.125) / ((x * x) * (x * x)))))));
}

Python

def code(x):
	return math.log(((-0.5 / x) + (x * (2.0 - (((0.0625 / (x * x)) + 0.125) / ((x * x) * (x * x)))))))

Julia

function code(x)
	return log(Float64(Float64(-0.5 / x) + Float64(x * Float64(2.0 - Float64(Float64(Float64(0.0625 / Float64(x * x)) + 0.125) / Float64(Float64(x * x) * Float64(x * x)))))))
end

MATLAB

function tmp = code(x)
	tmp = log(((-0.5 / x) + (x * (2.0 - (((0.0625 / (x * x)) + 0.125) / ((x * x) * (x * x)))))));
end

Wolfram

code[x_] := N[Log[N[(N[(-0.5 / x), $MachinePrecision] + N[(x * N[(2.0 - N[(N[(N[(0.0625 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.125), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

   4. *-lft-identity N/A

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

   5. /-lowering-/.f64 N/A

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

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

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \color{blue}{\left(x \cdot \left(2 + -1 \cdot \frac{\frac{1}{8} + \frac{1}{16} \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right)\right)}\right)\right) \]
9. Step-by-step derivation

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

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

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

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

   5. /-lowering-/.f64 N/A

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

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

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{\frac{1}{16}}{{x}^{2}}\right)\right), \left({x}^{4}\right)\right)\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{/.f64}\left(\frac{1}{16}, \left({x}^{2}\right)\right)\right), \left({x}^{4}\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

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

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

       \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(2, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{8}, \mathsf{/.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left({x}^{4}\right)\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

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

   13. pow-sqr N/A

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

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

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

   15. unpow2 N/A

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

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

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 99.1%

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

10. Simplified 99.1%

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

11. Final simplification 99.1%

    \[\leadsto \log \left(\frac{-0.5}{x} + x \cdot \left(2 - \frac{\frac{0.0625}{x \cdot x} + 0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)\right) \]
12. Add Preprocessing

Alternative 4: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around inf

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

   1. distribute-lft-in N/A

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

   2. *-rgt-identity N/A

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

   3. cancel-sign-sub N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

   7. remove-double-neg N/A

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

   8. associate-*r/ N/A

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

   9. unpow2 N/A

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

   10. times-frac N/A

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

   11. *-inverses N/A

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

   12. metadata-eval N/A

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

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

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

   14. neg-mul-1 N/A

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

   15. remove-double-neg N/A

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0 - \log \left(\frac{1}{x \cdot \left(2 + \frac{-0.5}{x \cdot x}\right)}\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.0 - log((1.0 / (x * (2.0 + (-0.5 / (x * x))))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0 - log((1.0d0 / (x * (2.0d0 + ((-0.5d0) / (x * x))))))
end function

Java

public static double code(double x) {
	return 0.0 - Math.log((1.0 / (x * (2.0 + (-0.5 / (x * x))))));
}

Python

def code(x):
	return 0.0 - math.log((1.0 / (x * (2.0 + (-0.5 / (x * x))))))

Julia

function code(x)
	return Float64(0.0 - log(Float64(1.0 / Float64(x * Float64(2.0 + Float64(-0.5 / Float64(x * x)))))))
end

MATLAB

function tmp = code(x)
	tmp = 0.0 - log((1.0 / (x * (2.0 + (-0.5 / (x * x))))));
end

Wolfram

code[x_] := N[(0.0 - N[Log[N[(1.0 / N[(x * N[(2.0 + N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around inf

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

5. Simplified 99.1%

   \[\leadsto \log \left(x + \color{blue}{\left(1 \cdot \frac{-0.5}{x} + x \cdot \left(1 + \frac{-0.125 + \frac{-0.0625}{x \cdot x}}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

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

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

   4. *-lft-identity N/A

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

   5. /-lowering-/.f64 N/A

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

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

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

7. Applied egg-rr 99.1%

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

9. Applied egg-rr 99.1%

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

10. Taylor expanded in x around inf

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

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

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

    2. sub-neg N/A

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

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

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

    4. associate-*r/ N/A

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

    5. metadata-eval N/A

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

    6. distribute-neg-frac N/A

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

    7. metadata-eval N/A

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

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

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

    9. unpow2 N/A

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

    10. *-lowering-*.f64 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

    \[\leadsto 0 - \log \left(\frac{1}{x \cdot \left(2 + \frac{-0.5}{x \cdot x}\right)}\right) \]
14. Add Preprocessing

Alternative 6: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 97.8%

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

   1. flip-+ N/A

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

   2. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{0}{x - x}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{0 + 0}{x - x}\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot 0 + 0}{x - x}\right)\right) \]

   5. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + 0}{x - x}\right)\right) \]

   6. +-inverses N/A

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

   7. distribute-lft-out-- N/A

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

   8. +-inverses N/A

      \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot 0}{x - x}\right)\right) \]

   9. +-inverses N/A

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

   10. +-inverses N/A

       \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{-1}{2} \cdot \left(x - x\right) + x \cdot \left(x \cdot x - x \cdot x\right)}{0}\right)\right) \]

   11. +-inverses N/A

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

   12. distribute-lft-out-- N/A

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

   13. frac-add N/A

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

   14. *-lft-identity N/A

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

   15. flip-+ N/A

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

   16. associate-+r+ N/A

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

   17. +-commutative N/A

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

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

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

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

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

   20. *-lft-identity N/A

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

   21. /-lowering-/.f64 98.6%

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

7. Applied egg-rr 98.6%

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

8. Final simplification 98.6%

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

Alternative 7: 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 48.5%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
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 97.8%

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

Developer Target 1: 99.9% 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 2024138 
(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)))))