Hyperbolic arc-cosine

Percentage Accurate: 50.8% → 99.9%

Time: 8.7s

Alternatives: 3

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: 50.8% 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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Log[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(x + -1.0), $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. unpow-prod-down N/A

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

   9. lower-fma.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-+.f64 N/A

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

   17. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 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. lower-+.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. metadata-eval N/A

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

   8. associate-*r* N/A

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

   9. associate-/l* N/A

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

   10. *-rgt-identity N/A

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

   11. unpow2 N/A

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

   12. associate-/r* N/A

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

   13. *-inverses N/A

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

   14. associate-*l/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.4%

   \[\log \left(x + \sqrt{x \cdot x - 1}\right) \]
2. Add Preprocessing
3. 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. unpow-prod-down N/A

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

   9. lower-fma.f64 N/A

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

   10. pow1/2 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. pow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. sub-neg N/A

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

   16. lower-+.f64 N/A

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

   17. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around inf

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. count-2 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. *-lft-identity N/A

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

   11. lower-+.f64 99.5

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

7. Applied rewrites 99.5%

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1.0)))))