Rust f64::acosh

Percentage Accurate: 51.6% → 99.6%

Time: 8.4s

Alternatives: 6

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: 51.6% 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: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. *-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \color{blue}{\frac{1}{{x}^{2}} \cdot 0.5}\right)\right)\right) \]

   3. cancel-sign-sub-inv 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{\left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)}\right)\right) \]

   4. associate-*r/ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \left(0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4}}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)\right)\right) \]

   5. metadata-eval 99.8%

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

   6. pow-sqr 99.8%

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

   7. unpow2 99.8%

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

   8. unpow2 99.8%

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

   9. times-frac 99.8%

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

   10. metadata-eval 99.8%

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

   11. distribute-neg-frac 99.8%

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

   12. unpow2 99.8%

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

5. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. clear-num 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{{x}^{2} \cdot \left(\left(\frac{0.125}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) - 2\right)}}\right)\right) \]
9. Step-by-step derivation

   1. unpow2 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\left(\frac{0.125}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) - 2\right)}\right)\right) \]

   2. *-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{\left(\left(\frac{0.125}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right) - 2\right) \cdot \left(x \cdot x\right)}}\right)\right) \]

   3. associate--l+ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{\left(\frac{0.125}{{x}^{4}} + \left(0.5 \cdot \frac{1}{{x}^{2}} - 2\right)\right)} \cdot \left(x \cdot x\right)}\right)\right) \]

   4. +-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{\left(\left(0.5 \cdot \frac{1}{{x}^{2}} - 2\right) + \frac{0.125}{{x}^{4}}\right)} \cdot \left(x \cdot x\right)}\right)\right) \]

   5. sub-neg 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\color{blue}{\left(0.5 \cdot \frac{1}{{x}^{2}} + \left(-2\right)\right)} + \frac{0.125}{{x}^{4}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   6. unpow2 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(0.5 \cdot \frac{1}{\color{blue}{x \cdot x}} + \left(-2\right)\right) + \frac{0.125}{{x}^{4}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   7. associate-*r/ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(\color{blue}{\frac{0.5 \cdot 1}{x \cdot x}} + \left(-2\right)\right) + \frac{0.125}{{x}^{4}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   8. metadata-eval 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(\frac{\color{blue}{0.5}}{x \cdot x} + \left(-2\right)\right) + \frac{0.125}{{x}^{4}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   9. metadata-eval 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(\frac{0.5}{x \cdot x} + \color{blue}{-2}\right) + \frac{0.125}{{x}^{4}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   10. metadata-eval 99.8%

       \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(\frac{0.5}{x \cdot x} + -2\right) + \frac{0.125}{{x}^{\color{blue}{\left(3 + 1\right)}}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   11. pow-plus 99.8%

       \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\left(\left(\frac{0.5}{x \cdot x} + -2\right) + \frac{0.125}{\color{blue}{{x}^{3} \cdot x}}\right) \cdot \left(x \cdot x\right)}\right)\right) \]

   12. cube-unmult 99.8%

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

   13. *-commutative 99.8%

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

10. Simplified 99.8%

    \[\leadsto \log \left(x + x \cdot \left(1 + \frac{1}{\color{blue}{\left(\left(\frac{0.5}{x \cdot x} + -2\right) + \frac{0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right) \cdot \left(x \cdot x\right)}}\right)\right) \]
11. Step-by-step derivation

    1. *-un-lft-identity 99.8%

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

    2. *-commutative 99.8%

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

    3. associate-/r* 99.8%

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

    4. associate-+l+ 99.8%

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

    5. associate-*r* 99.8%

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

12. Applied egg-rr 99.8%

    \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{1 \cdot \frac{\frac{1}{x \cdot x}}{\frac{0.5}{x \cdot x} + \left(-2 + \frac{0.125}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right)}}\right)\right) \]
13. Step-by-step derivation

    1. *-lft-identity 99.8%

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

    2. associate-/r* 99.8%

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

    3. associate-/r* 99.8%

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

    4. distribute-lft-in 99.8%

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

    5. associate-*r/ 99.8%

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

    6. times-frac 99.8%

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

    7. *-inverses 99.8%

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

    8. *-commutative 99.8%

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

    9. *-rgt-identity 99.8%

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

14. Simplified 99.8%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. *-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \color{blue}{\frac{1}{{x}^{2}} \cdot 0.5}\right)\right)\right) \]

   3. cancel-sign-sub-inv 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{\left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)}\right)\right) \]

   4. associate-*r/ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \left(0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4}}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)\right)\right) \]

   5. metadata-eval 99.8%

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

   6. pow-sqr 99.8%

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

   7. unpow2 99.8%

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

   8. unpow2 99.8%

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

   9. times-frac 99.8%

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

   10. metadata-eval 99.8%

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

   11. distribute-neg-frac 99.8%

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

   12. unpow2 99.8%

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

5. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. clear-num 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. *-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \color{blue}{\frac{1}{{x}^{2}} \cdot 0.5}\right)\right)\right) \]

   3. cancel-sign-sub-inv 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{\left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)}\right)\right) \]

   4. associate-*r/ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \left(0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4}}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)\right)\right) \]

   5. metadata-eval 99.8%

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

   6. pow-sqr 99.8%

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

   7. unpow2 99.8%

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

   8. unpow2 99.8%

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

   9. times-frac 99.8%

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

   10. metadata-eval 99.8%

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

   11. distribute-neg-frac 99.8%

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

   12. unpow2 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{\frac{-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{2}} - 0.5}{{x}^{2}}}\right)\right) \]
7. Step-by-step derivation

8. Simplified 99.8%

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

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in x around inf 99.8%

   \[\leadsto \log \left(x + \color{blue}{x \cdot \left(\left(1 + -1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}}\right) - 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right) \]
4. Step-by-step derivation

   1. associate--l+ 99.8%

      \[\leadsto \log \left(x + x \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]

   2. *-commutative 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \color{blue}{\frac{1}{{x}^{2}} \cdot 0.5}\right)\right)\right) \]

   3. cancel-sign-sub-inv 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \color{blue}{\left(-1 \cdot \frac{0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}}{{x}^{4}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)}\right)\right) \]

   4. associate-*r/ 99.8%

      \[\leadsto \log \left(x + x \cdot \left(1 + \left(\color{blue}{\frac{-1 \cdot \left(0.125 + 0.0625 \cdot \frac{1}{{x}^{2}}\right)}{{x}^{4}}} + \left(-\frac{1}{{x}^{2}}\right) \cdot 0.5\right)\right)\right) \]

   5. metadata-eval 99.8%

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

   6. pow-sqr 99.8%

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

   7. unpow2 99.8%

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

   8. unpow2 99.8%

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

   9. times-frac 99.8%

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

   10. metadata-eval 99.8%

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

   11. distribute-neg-frac 99.8%

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

   12. unpow2 99.8%

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

5. Simplified 99.8%

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

   1. associate-*l/ 99.8%

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

   2. clear-num 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-lft-in 99.8%

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

   5. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in x around inf 99.7%

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

   1. unpow2 99.7%

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

   2. associate-*l* 99.7%

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

   3. sub-neg 99.7%

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

   4. unpow2 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. metadata-eval 99.7%

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

10. Simplified 99.7%

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

    1. distribute-lft-in 99.7%

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

    2. *-rgt-identity 99.7%

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

    3. un-div-inv 99.7%

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

12. Applied egg-rr 99.7%

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

    1. associate-/r* 99.7%

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

    2. *-inverses 99.7%

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

    3. distribute-lft-in 99.7%

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

    4. +-commutative 99.7%

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

    5. associate-*r/ 99.7%

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

    6. times-frac 99.7%

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

    7. *-inverses 99.7%

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

    8. *-commutative 99.7%

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

    9. *-rgt-identity 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 5: 99.3% 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 53.1%

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

3. Taylor expanded in x around inf 99.6%

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

   1. sub-neg 99.6%

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

   2. distribute-rgt-in 99.6%

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

   3. *-lft-identity 99.6%

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

   4. distribute-lft-neg-in 99.6%

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

   5. associate-*l* 99.6%

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

   6. unpow2 99.6%

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

   7. associate-*l/ 99.6%

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

   8. *-lft-identity 99.6%

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

   9. associate-/r* 99.6%

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

   10. *-inverses 99.6%

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

   11. associate-*r/ 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

5. Simplified 99.6%

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

Alternative 6: 98.9% 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 53.1%

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

3. Taylor expanded in x around inf 99.3%

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

Developer target: 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 2024107 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

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

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