Rust f64::acosh

Percentage Accurate: 51.4% → 99.8%

Time: 4.5s

Alternatives: 6

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: 51.4% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 5000:\\ \;\;\;\;\log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (+ (* x x) -1.0))) 5000.0)
   (log (+ x (cbrt (pow (fma x x -1.0) 1.5))))
   (- (log (/ 0.5 x)))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) + -1.0))) <= 5000.0) {
		tmp = log((x + cbrt(pow(fma(x, x, -1.0), 1.5))));
	} 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))) <= 5000.0)
		tmp = log(Float64(x + cbrt((fma(x, x, -1.0) ^ 1.5))));
	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], 5000.0], N[Log[N[(x + N[Power[N[Power[N[(x * x + -1.0), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]), $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) 1))) < 5e3

   1. Initial program 99.2%

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

      1. add-cbrt-cube 99.5%

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

      2. pow3 99.2%

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

      3. sqrt-pow2 99.7%

         \[\leadsto \log \left(x + \sqrt[3]{\color{blue}{{\left(x \cdot x - 1\right)}^{\left(\frac{3}{2}\right)}}}\right) \]

      4. fma-neg 99.7%

         \[\leadsto \log \left(x + \sqrt[3]{{\color{blue}{\left(\mathsf{fma}\left(x, x, -1\right)\right)}}^{\left(\frac{3}{2}\right)}}\right) \]

      5. metadata-eval 99.7%

         \[\leadsto \log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, \color{blue}{-1}\right)\right)}^{\left(\frac{3}{2}\right)}}\right) \]

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   if 5e3 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1)))

   1. Initial program 49.2%

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

      1. flip-+ 0.0%

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

      2. div-inv 0.0%

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

      3. log-prod 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fma-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-neg 0.0%

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

      8. metadata-eval 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. fma-udef 0.0%

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

      2. associate--r+ 1.5%

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

      3. sub-neg 1.5%

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

      4. +-inverses 1.8%

         \[\leadsto \log \left(\color{blue}{0} + \left(--1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      5. metadata-eval 1.8%

         \[\leadsto \log \left(0 + \color{blue}{1}\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      6. metadata-eval 1.8%

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

      7. metadata-eval 1.8%

         \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      8. +-lft-identity 1.8%

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

      9. log-rec 1.5%

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

   5. Simplified 1.5%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 5000:\\ \;\;\;\;\log \left(x + \sqrt[3]{{\left(\mathsf{fma}\left(x, x, -1\right)\right)}^{1.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ x (sqrt (+ (* x x) -1.0))) 2e+21)
   (log1p (+ (+ x (sqrt (fma x x -1.0))) -1.0))
   (- (log (/ 0.5 x)))))

C

double code(double x) {
	double tmp;
	if ((x + sqrt(((x * x) + -1.0))) <= 2e+21) {
		tmp = log1p(((x + sqrt(fma(x, x, -1.0))) + -1.0));
	} 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))) <= 2e+21)
		tmp = log1p(Float64(Float64(x + sqrt(fma(x, x, -1.0))) + -1.0));
	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], 2e+21], N[Log[1 + N[(N[(x + N[Sqrt[N[(x * x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $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) 1))) < 2e21

   1. Initial program 99.7%

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

      1. log1p-expm1-u 99.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + \sqrt{x \cdot x - 1}\right)\right)\right)} \]

      2. expm1-udef 99.7%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(x + \sqrt{x \cdot x - 1}\right)} - 1}\right) \]

      3. add-exp-log 99.8%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x + \sqrt{x \cdot x - 1}\right)} - 1\right) \]

      4. fma-neg 99.8%

         \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{\mathsf{fma}\left(x, x, -1\right)}}\right) - 1\right) \]

      5. metadata-eval 99.8%

         \[\leadsto \mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, \color{blue}{-1}\right)}\right) - 1\right) \]

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) - 1\right)} \]

   if 2e21 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1)))

   1. Initial program 46.9%

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

      1. flip-+ 0.0%

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

      2. div-inv 0.0%

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

      3. log-prod 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fma-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-neg 0.0%

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

      8. metadata-eval 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. fma-udef 0.0%

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

      2. associate--r+ 1.4%

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

      3. sub-neg 1.4%

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

      4. +-inverses 1.8%

         \[\leadsto \log \left(\color{blue}{0} + \left(--1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      5. metadata-eval 1.8%

         \[\leadsto \log \left(0 + \color{blue}{1}\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      6. metadata-eval 1.8%

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

      7. metadata-eval 1.8%

         \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      8. +-lft-identity 1.8%

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

      9. log-rec 1.4%

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

   5. Simplified 1.4%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{log1p}\left(\left(x + \sqrt{\mathsf{fma}\left(x, x, -1\right)}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \sqrt{x \cdot x + -1}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\log t_0\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ x (sqrt (+ (* x x) -1.0)))))
   (if (<= t_0 2e+21) (log t_0) (- (log (/ 0.5 x))))))

C

double code(double x) {
	double t_0 = x + sqrt(((x * x) + -1.0));
	double tmp;
	if (t_0 <= 2e+21) {
		tmp = log(t_0);
	} else {
		tmp = -log((0.5 / 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 <= 2d+21) then
        tmp = log(t_0)
    else
        tmp = -log((0.5d0 / 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 <= 2e+21) {
		tmp = Math.log(t_0);
	} else {
		tmp = -Math.log((0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x + math.sqrt(((x * x) + -1.0))
	tmp = 0
	if t_0 <= 2e+21:
		tmp = math.log(t_0)
	else:
		tmp = -math.log((0.5 / x))
	return tmp

Julia

function code(x)
	t_0 = Float64(x + sqrt(Float64(Float64(x * x) + -1.0)))
	tmp = 0.0
	if (t_0 <= 2e+21)
		tmp = log(t_0);
	else
		tmp = Float64(-log(Float64(0.5 / 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 <= 2e+21)
		tmp = log(t_0);
	else
		tmp = -log((0.5 / 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, 2e+21], N[Log[t$95$0], $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) 1))) < 2e21

   1. Initial program 99.7%

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

   if 2e21 < (+.f64 x (sqrt.f64 (-.f64 (*.f64 x x) 1)))

   1. Initial program 46.9%

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

      1. flip-+ 0.0%

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

      2. div-inv 0.0%

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

      3. log-prod 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. fma-neg 0.0%

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

      6. metadata-eval 0.0%

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

      7. fma-neg 0.0%

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

      8. metadata-eval 0.0%

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

   3. Applied egg-rr 0.0%

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

      1. fma-udef 0.0%

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

      2. associate--r+ 1.4%

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

      3. sub-neg 1.4%

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

      4. +-inverses 1.8%

         \[\leadsto \log \left(\color{blue}{0} + \left(--1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      5. metadata-eval 1.8%

         \[\leadsto \log \left(0 + \color{blue}{1}\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      6. metadata-eval 1.8%

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

      7. metadata-eval 1.8%

         \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

      8. +-lft-identity 1.8%

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

      9. log-rec 1.4%

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

   5. Simplified 1.4%

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

   6. Taylor expanded in x around inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \sqrt{x \cdot x + -1} \leq 2 \cdot 10^{+21}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{0.5}{x}\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 2.0× 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. Step-by-step derivation

   1. flip-+ 2.2%

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

   2. div-inv 2.2%

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

   3. log-prod 2.2%

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

   4. add-sqr-sqrt 2.2%

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

   5. fma-neg 2.2%

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

   6. metadata-eval 2.2%

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

   7. fma-neg 2.2%

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

   8. metadata-eval 2.2%

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

3. Applied egg-rr 2.2%

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

   1. fma-udef 2.2%

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

   2. associate--r+ 3.6%

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

   3. sub-neg 3.6%

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

   4. +-inverses 3.9%

      \[\leadsto \log \left(\color{blue}{0} + \left(--1\right)\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   5. metadata-eval 3.9%

      \[\leadsto \log \left(0 + \color{blue}{1}\right) + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   6. metadata-eval 3.9%

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

   7. metadata-eval 3.9%

      \[\leadsto \color{blue}{0} + \log \left(\frac{1}{x - \sqrt{\mathsf{fma}\left(x, x, -1\right)}}\right) \]

   8. +-lft-identity 3.9%

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

   9. log-rec 3.6%

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

5. Simplified 3.6%

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

6. Taylor expanded in x around inf 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 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 50.4%

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

2. Taylor expanded in x around inf 98.0%

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

3. Final simplification 98.0%

   \[\leadsto \log \left(x + x\right) \]

Alternative 6: 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 50.4%

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

2. Taylor expanded in x around inf 98.0%

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

   1. flip-+ 0.0%

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

   2. log-div 0.0%

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

   3. +-inverses 0.0%

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

   4. +-inverses 0.0%

      \[\leadsto \log 0 - \log \color{blue}{0} \]

4. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\log 0 - \log 0} \]
5. Step-by-step derivation

   1. +-inverses 3.1%

      \[\leadsto \color{blue}{0} \]

6. Simplified 3.1%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 3.1%

   \[\leadsto 0 \]

Developer target: 99.8% 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 2023201 
(FPCore (x)
  :name "Rust f64::acosh"
  :precision binary64
  :pre (>= x 1.0)

  :herbie-target
  (log (+ x (* (sqrt (- x 1.0)) (sqrt (+ x 1.0)))))

  (log (+ x (sqrt (- (* x x) 1.0)))))