Hyperbolic arc-cosine

Percentage Accurate: 52.6% → 100.0%

Time: 8.5s

Alternatives: 5

Speedup: 1.2×

Specification

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]

Initial Program: 52.6% 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: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x - 1} \leq 1000000:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x, x, x\right) - \left(x - -1\right)} + x\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (- (* x x) 1.0))) 1000000.0)
   (log (+ (sqrt (- (fma x x x) (- x -1.0))) x))
   (- (log (/ 0.5 x)))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) - 1.0))) <= 1000000.0) {
		tmp = log((sqrt((fma(x, x, x) - (x - -1.0))) + x));
	} else {
		tmp = -log((0.5 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + sqrt(Float64(Float64(x * x) - 1.0))) <= 1000000.0)
		tmp = log(Float64(sqrt(Float64(fma(x, x, x) - Float64(x - -1.0))) + x));
	else
		tmp = Float64(-log(Float64(0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1000000.0], N[Log[N[(N[Sqrt[N[(N[(x * x + x), $MachinePrecision] - N[(x - -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision], (-N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 1e6

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.8

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. lift-*.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x, x, -1\right)} + x\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. difference-of-sqr--1 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-rgt-in N/A

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

      8. neg-mul-1 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. metadata-eval N/A

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

      14. sub-neg N/A

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

      15. lower--.f64 N/A

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

      16. lower-neg.f64 100.0

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. metadata-eval N/A

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

      20. sub-neg N/A

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

      21. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

      \[\leadsto \log \left(\sqrt{\color{blue}{x \cdot \left(x - -1\right) + \left(-\left(x - -1\right)\right)}} + x\right) \]
   7. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft1-in N/A

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

      11. lower-fma.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 1e6 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 48.6%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. lower-log.f64 N/A

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

      7. lower-log.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 100.0%

      \[\leadsto -\log \left(\frac{0.5}{x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. lift--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. difference-of-sqr-1 N/A

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

   8. *-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. lower-fma.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lower--.f64 N/A

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

   14. pow1/2 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× 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 50.4%

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

3. Taylor expanded in x around inf

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

   1. sub-neg N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

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

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

   5. unsub-neg N/A

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

   6. remove-double-neg N/A

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

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

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

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

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

   9. mul-1-neg N/A

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

   10. lower--.f64 N/A

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

   11. mul-1-neg N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. associate-*r* N/A

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

5. Applied rewrites 98.4%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. mul-1-neg N/A

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

   4. log-rec N/A

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

   5. remove-double-neg N/A

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

   6. lower-log.f64 N/A

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

   7. lower-log.f64 97.9

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

5. Applied rewrites 97.9%

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

7. Applied rewrites 98.2%

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

Alternative 5: 99.0% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (log (* 2.0 x)))

C

double code(double x) {
	return log((2.0 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((2.0d0 * x))
end function

Java

public static double code(double x) {
	return Math.log((2.0 * x));
}

Python

def code(x):
	return math.log((2.0 * x))

Julia

function code(x)
	return log(Float64(2.0 * x))
end

MATLAB

function tmp = code(x)
	tmp = log((2.0 * x));
end

Wolfram

code[x_] := N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around inf

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

   1. lower-*.f64 97.8

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

5. Applied rewrites 97.8%

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

Reproduce

herbie shell --seed 2024323 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))