Rust f64::acosh

Percentage Accurate: 51.3% → 99.9%

Time: 4.8s

Alternatives: 7

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: 51.3% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (+ (* x x) -1.0))) 20000000.0)
   (log (+ x (pow (cbrt (fma x x -1.0)) 1.5)))
   (log (+ x x))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) + -1.0))) <= 20000000.0) {
		tmp = log((x + pow(cbrt(fma(x, x, -1.0)), 1.5)));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + sqrt(Float64(Float64(x * x) + -1.0))) <= 20000000.0)
		tmp = log(Float64(x + (cbrt(fma(x, x, -1.0)) ^ 1.5)));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 20000000.0], N[Log[N[(x + N[Power[N[Power[N[(x * x + -1.0), $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 2e7

   1. Initial program 99.6%

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

      1. pow1/2 99.6%

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

      2. fma-neg 99.8%

         \[\leadsto \log \left(x + {\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{0.5}\right) \]

      3. metadata-eval 99.8%

         \[\leadsto \log \left(x + {\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{0.5}\right) \]

      4. add-cube-cbrt 100.0%

         \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5}\right) \]

      5. pow3 100.0%

         \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{3}\right)}}^{0.5}\right) \]

      6. pow-pow 100.0%

         \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

      7. metadata-eval 100.0%

         \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

   if 2e7 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (+ (* x x) -1.0))) 20000000.0)
   (log (+ x (sqrt (fma x x -1.0))))
   (log (+ x x))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) + -1.0))) <= 20000000.0) {
		tmp = log((x + sqrt(fma(x, x, -1.0))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(x + sqrt(Float64(Float64(x * x) + -1.0))) <= 20000000.0)
		tmp = log(Float64(x + sqrt(fma(x, x, -1.0))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 20000000.0], N[Log[N[(x + N[Sqrt[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 2e7

   1. Initial program 99.6%

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

      1. fma-neg 99.8%

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

      2. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   if 2e7 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t\_0 \leq 20000000:\\ \;\;\;\;\log t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 20000000.0) (log t_0) (log (+ x x)))))

C

double code(double x) {
	double t_0 = x + sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 20000000.0) {
		tmp = log(t_0);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + sqrt(((x * x) + (-1.0d0)))
    if (t_0 <= 20000000.0d0) then
        tmp = log(t_0)
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x + Math.sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 20000000.0) {
		tmp = Math.log(t_0);
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + math.sqrt(((x * x) + -1.0))
	tmp = 0
	if t_0 <= 20000000.0:
		tmp = math.log(t_0)
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x + sqrt(Float64(Float64(x * x) + -1.0)))
	tmp = 0.0
	if (t_0 <= 20000000.0)
		tmp = log(t_0);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x + sqrt(((x * x) + -1.0));
	tmp = 0.0;
	if (t_0 <= 20000000.0)
		tmp = log(t_0);
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 20000000.0], N[Log[t$95$0], $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64)))) < 2e7

   1. Initial program 99.6%

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

   if 2e7 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) #s(literal 1 binary64))))

   1. Initial program 51.6%

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

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \log \left(x + \color{blue}{x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 20000000:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% 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 98.4%

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

Alternative 5: 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 53.1%

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

3. Taylor expanded in x around inf 98.4%

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

4. Taylor expanded in x around 0 98.1%

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

6. Simplified 31.4%

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

Alternative 6: 14.1% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

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 98.4%

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

   1. count-2 98.4%

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

   2. sum-log 98.1%

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

   3. flip-+ 97.8%

      \[\leadsto \color{blue}{\frac{\log 2 \cdot \log 2 - \log x \cdot \log x}{\log 2 - \log x}} \]

   4. div-sub 97.8%

      \[\leadsto \color{blue}{\frac{\log 2 \cdot \log 2}{\log 2 - \log x} - \frac{\log x \cdot \log x}{\log 2 - \log x}} \]

   5. pow2 97.8%

      \[\leadsto \frac{\color{blue}{{\log 2}^{2}}}{\log 2 - \log x} - \frac{\log x \cdot \log x}{\log 2 - \log x} \]

   6. diff-log 97.8%

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

   7. pow2 97.8%

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

   8. diff-log 97.6%

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

5. Applied egg-rr 97.6%

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

7. Simplified 14.2%

   \[\leadsto \color{blue}{\frac{\log x}{\log x}} \]

8. Taylor expanded in x around 0 14.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Alternative 7: 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 53.1%

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

   1. pow1/2 53.1%

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

   2. fma-neg 53.1%

      \[\leadsto \log \left(x + {\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{0.5}\right) \]

   3. metadata-eval 53.1%

      \[\leadsto \log \left(x + {\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{0.5}\right) \]

   4. add-cube-cbrt 53.1%

      \[\leadsto \log \left(x + {\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}}^{0.5}\right) \]

   5. pow3 53.1%

      \[\leadsto \log \left(x + {\color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{3}\right)}}^{0.5}\right) \]

   6. pow-pow 53.1%

      \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\left(3 \cdot 0.5\right)}}\right) \]

   7. metadata-eval 53.1%

      \[\leadsto \log \left(x + {\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{\color{blue}{1.5}}\right) \]

4. Applied egg-rr 53.1%

   \[\leadsto \log \left(x + \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(x, x, -1\right)}\right)}^{1.5}}\right) \]

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 2024141 
(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)))))