Rust f64::acosh

Percentage Accurate: 52.0% → 98.9%

Time: 5.2s

Alternatives: 5

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: 52.0% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log 2 + \log x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. log-rec 98.8%

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

   3. remove-double-neg 98.8%

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

5. Simplified 98.8%

   \[\leadsto \color{blue}{\log 2 + \log x} \]
6. Add Preprocessing

Alternative 2: 99.1% 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 47.4%

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

3. Taylor expanded in x around inf 98.7%

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

Alternative 3: 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 47.4%

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

3. Taylor expanded in x around inf 98.7%

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

4. Taylor expanded in x around 0 98.8%

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

6. Simplified 31.5%

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

Alternative 4: 3.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 47.4%

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

   1. add-cbrt-cube 33.4%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)}\right)} \]

   2. pow1/3 33.2%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 33.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 33.2%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 47.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fmm-def 47.2%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 47.2%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

5. Taylor expanded in x around inf 98.3%

   \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \color{blue}{x}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* 98.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \log \left(x + x\right)} \]

   2. metadata-eval 98.7%

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

   3. *-un-lft-identity 98.7%

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

   4. flip-+ 0.0%

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

   5. difference-of-squares 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. associate-*r/ 0.0%

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

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

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

   13. flip-+ 10.3%

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

   14. sum-log 18.7%

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

   15. *-un-lft-identity 18.7%

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

7. Applied egg-rr 0.0%

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

9. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Alternative 5: 1.6% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

double code(double x) {
	return -2.0;
}

Fortran

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

Java

public static double code(double x) {
	return -2.0;
}

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

function tmp = code(x)
	tmp = -2.0;
end

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 47.4%

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

   1. add-cbrt-cube 33.4%

      \[\leadsto \log \color{blue}{\left(\sqrt[3]{\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)}\right)} \]

   2. pow1/3 33.2%

      \[\leadsto \log \color{blue}{\left({\left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)}^{0.3333333333333333}\right)} \]

   3. log-pow 33.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(x + \sqrt{x \cdot x - 1}\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right) \cdot \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   4. pow3 33.2%

      \[\leadsto 0.3333333333333333 \cdot \log \color{blue}{\left({\left(x + \sqrt{x \cdot x - 1}\right)}^{3}\right)} \]

   5. log-pow 47.2%

      \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(x + \sqrt{x \cdot x - 1}\right)\right)} \]

   6. fmm-def 47.2%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right)\right) \]

   7. metadata-eval 47.2%

      \[\leadsto 0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right)\right) \]

4. Applied egg-rr 47.2%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\right)} \]

5. Taylor expanded in x around 0 0.0%

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

7. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]
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 2024149 
(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)))))