Rust f64::acosh

Percentage Accurate: 51.8% → 99.0%

Time: 3.0s

Alternatives: 4

Speedup: 2.0×

Specification

\[x \geq 1\]

Math

\[\begin{array}{l} \\ \cosh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (acosh x))

C

double code(double x) {
	return acosh(x);
}

Python

def code(x):
	return math.acosh(x)

Julia

function code(x)
	return acosh(x)
end

MATLAB

function tmp = code(x)
	tmp = acosh(x);
end

Wolfram

code[x_] := N[ArcCosh[x], $MachinePrecision]

Initial Program: 51.8% 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: 99.0% 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 46.7%

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

3. Taylor expanded in x around inf 99.1%

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

Alternative 2: 31.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return log(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 99.1%

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

4. Taylor expanded in x around 0 98.8%

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

6. Simplified 31.4%

   \[\leadsto \color{blue}{\log x} \]
7. Add Preprocessing

Alternative 3: 14.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \log 4 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log 4.0))

C

double code(double x) {
	return log(4.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(4.0d0)
end function

Java

public static double code(double x) {
	return Math.log(4.0);
}

Python

def code(x):
	return math.log(4.0)

Julia

function code(x)
	return log(4.0)
end

MATLAB

function tmp = code(x)
	tmp = log(4.0);
end

Wolfram

code[x_] := N[Log[4.0], $MachinePrecision]

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 99.1%

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

   1. flip-+ 0.0%

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

   2. difference-of-squares 0.0%

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

   3. +-inverses 0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}\right) \]

   4. +-inverses 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto \log \left(\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}\right) \]

   6. +-inverses 0.0%

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

   7. associate-*r/ 0.0%

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

   8. +-inverses 0.0%

      \[\leadsto \log \left(\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right) \]

   9. +-inverses 0.0%

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

   10. flip-+ 10.2%

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

   11. pow2 10.2%

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

5. Applied egg-rr 0.0%

   \[\leadsto \log \color{blue}{\left(4 \cdot {\left(x \cdot \frac{0}{0}\right)}^{2}\right)} \]

7. Simplified 14.2%

   \[\leadsto \log \color{blue}{4} \]
8. Add Preprocessing

Alternative 4: 13.8% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 46.7%

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

3. Taylor expanded in x around inf 99.1%

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

   1. flip-+ 0.0%

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

   2. +-inverses 0.0%

      \[\leadsto \log \left(\frac{x \cdot x - x \cdot x}{\color{blue}{0}}\right) \]

   3. +-inverses 0.0%

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

   4. diff-log 0.0%

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

   5. +-inverses 0.0%

      \[\leadsto \log \color{blue}{0} - \log \left(x \cdot x - x \cdot x\right) \]

   6. metadata-eval 0.0%

      \[\leadsto \log \color{blue}{\left({0}^{0.5}\right)} - \log \left(x \cdot x - x \cdot x\right) \]

   7. +-inverses 0.0%

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

   8. pow1/2 0.0%

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

   9. +-inverses 0.0%

      \[\leadsto \log \left(\sqrt{x \cdot x - x \cdot x}\right) - \log \color{blue}{0} \]

   10. metadata-eval 0.0%

       \[\leadsto \log \left(\sqrt{x \cdot x - x \cdot x}\right) - \log \color{blue}{\left({0}^{0.5}\right)} \]

   11. +-inverses 0.0%

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

   12. pow1/2 0.0%

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

   13. log-div 0.0%

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

   14. +-inverses 0.0%

       \[\leadsto \log \left(\frac{\sqrt{x \cdot x - x \cdot x}}{\sqrt{\color{blue}{0}}}\right) \]

   15. +-inverses 0.0%

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

   16. sqrt-div 0.0%

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

   17. flip-+ 18.8%

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

   18. pow1/2 18.8%

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

   19. log-pow 18.8%

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

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{0.5 \cdot \log \left(\frac{0}{0}\right)} \]

7. Simplified 13.8%

   \[\leadsto \color{blue}{0.5} \]
8. 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 2024131 
(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)))))