Hyperbolic arcsine

Percentage Accurate: 17.8% → 99.6%

Time: 9.3s

Alternatives: 6

Speedup: 207.0×

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: 17.8% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00115:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.00115)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00115) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00115:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00115)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00115)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00115], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.6%

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

      1. sqr-neg 2.6%

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

      2. +-commutative 2.6%

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

      3. sqr-neg 2.6%

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

      4. hypot-1-def 3.6%

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

   3. Simplified 3.6%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.25 < x < 0.00115

   1. Initial program 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.6%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 99.6%

         \[\leadsto \color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 99.6%

         \[\leadsto x + \color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 99.6%

         \[\leadsto x + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      5. unpow3 99.6%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 0.00115 < x

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. +-commutative 53.9%

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

      3. sqr-neg 53.9%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.6%

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

      1. sqr-neg 2.6%

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

      2. +-commutative 2.6%

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

      3. sqr-neg 2.6%

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

      4. hypot-1-def 3.6%

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

   3. Simplified 3.6%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.25 < x < 1.25

   1. Initial program 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in x around 0 99.6%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.6%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 99.6%

         \[\leadsto \color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 99.6%

         \[\leadsto x + \color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 99.6%

         \[\leadsto x + -0.16666666666666666 \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      5. unpow3 99.6%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 1.25 < x

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. +-commutative 53.9%

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

      3. sqr-neg 53.9%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

Alternative 3: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ -0.5 x))
   (if (<= x 1.05) (* (* x (+ x 2.0)) (+ 0.5 (* x -0.25))) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.05d0) then
        tmp = (x * (x + 2.0d0)) * (0.5d0 + (x * (-0.25d0)))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((-0.5 / x))
	elif x <= 1.05:
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = Float64(Float64(x * Float64(x + 2.0)) * Float64(0.5 + Float64(x * -0.25)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((-0.5 / x));
	elseif (x <= 1.05)
		tmp = (x * (x + 2.0)) * (0.5 + (x * -0.25));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(x * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 2.6%

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

      1. sqr-neg 2.6%

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

      2. +-commutative 2.6%

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

      3. sqr-neg 2.6%

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

      4. hypot-1-def 3.6%

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

   3. Simplified 3.6%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.1000000000000001 < x < 1.05000000000000004

   1. Initial program 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.5%

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

   3. Simplified 6.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 6.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 6.5%

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

      3. log1p-undefine 6.5%

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

      4. rem-exp-log 6.5%

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

   6. Applied egg-rr 6.5%

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

   7. Taylor expanded in x around 0 6.0%

      \[\leadsto \color{blue}{\left(1 + x\right)} - 1 \]
   8. Step-by-step derivation

      1. +-commutative 6.0%

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

   9. Simplified 6.0%

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
   10. Step-by-step derivation

       1. flip-- 6.0%

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

       2. div-inv 6.0%

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

       3. metadata-eval 6.0%

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

       4. difference-of-sqr-1 6.0%

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

       5. associate-+l+ 6.0%

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

       6. metadata-eval 6.0%

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

       7. associate--l+ 99.5%

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

       8. metadata-eval 99.5%

          \[\leadsto \left(\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)\right) \cdot \frac{1}{\left(x + 1\right) + 1} \]

       9. +-rgt-identity 99.5%

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

       10. associate-+l+ 99.5%

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

       11. metadata-eval 99.5%

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

   11. Applied egg-rr 99.5%

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

   12. Taylor expanded in x around 0 99.5%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot x\right)} \]
   13. Step-by-step derivation

       1. *-commutative 99.5%

          \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \left(0.5 + \color{blue}{x \cdot -0.25}\right) \]

   14. Simplified 99.5%

       \[\leadsto \left(\left(x + 2\right) \cdot x\right) \cdot \color{blue}{\left(0.5 + x \cdot -0.25\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. +-commutative 53.9%

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

      3. sqr-neg 53.9%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \left(0.5 + x \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) x (log (* x 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 5.1%

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

      1. sqr-neg 5.1%

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

      2. +-commutative 5.1%

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

      3. sqr-neg 5.1%

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

      4. hypot-1-def 5.5%

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

   3. Simplified 5.5%

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

   5. Taylor expanded in x around 0 66.4%

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

   if 1.25 < x

   1. Initial program 53.9%

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

      1. sqr-neg 53.9%

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

      2. +-commutative 53.9%

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

      3. sqr-neg 53.9%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

Alternative 5: 53.1% accurate, 29.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (* x 2.0) (+ x 2.0)))

C

double code(double x) {
	return (x * 2.0) / (x + 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return (x * 2.0) / (x + 2.0);
}

Python

def code(x):
	return (x * 2.0) / (x + 2.0)

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(x + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / (x + 2.0);
end

Wolfram

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

Derivation

1. Initial program 17.1%

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

   1. sqr-neg 17.1%

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

   2. +-commutative 17.1%

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

   3. sqr-neg 17.1%

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

   4. hypot-1-def 28.7%

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

3. Simplified 28.7%

   \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. expm1-log1p-u 27.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

   2. expm1-undefine 27.4%

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

   3. log1p-undefine 27.4%

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

   4. rem-exp-log 28.7%

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

6. Applied egg-rr 28.7%

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

7. Taylor expanded in x around 0 5.9%

   \[\leadsto \color{blue}{\left(1 + x\right)} - 1 \]
8. Step-by-step derivation

   1. +-commutative 5.9%

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

9. Simplified 5.9%

   \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]
10. Step-by-step derivation

    1. flip-- 5.7%

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

    2. metadata-eval 5.7%

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

    3. difference-of-sqr-1 5.7%

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

    4. associate-+l+ 5.7%

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

    5. metadata-eval 5.7%

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

    6. associate--l+ 51.4%

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

    7. metadata-eval 51.4%

       \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

    8. +-rgt-identity 51.4%

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

    9. associate-+l+ 51.4%

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

    10. metadata-eval 51.4%

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

11. Applied egg-rr 51.4%

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

12. Taylor expanded in x around 0 52.5%

    \[\leadsto \frac{\color{blue}{2 \cdot x}}{x + 2} \]
13. Step-by-step derivation

    1. *-commutative 52.5%

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

14. Simplified 52.5%

    \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]
15. Add Preprocessing

Alternative 6: 52.4% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 17.1%

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

   1. sqr-neg 17.1%

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

   2. +-commutative 17.1%

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

   3. sqr-neg 17.1%

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

   4. hypot-1-def 28.7%

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

3. Simplified 28.7%

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

5. Taylor expanded in x around 0 51.5%

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

Developer Target 1: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot x + 1}\\ \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ (* x x) 1.0))))
   (if (< x 0.0) (log (/ -1.0 (- x t_0))) (log (+ x t_0)))))

C

double code(double x) {
	double t_0 = sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = log((-1.0 / (x - t_0)));
	} else {
		tmp = log((x + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((x * x) + 1.0d0))
    if (x < 0.0d0) then
        tmp = log(((-1.0d0) / (x - t_0)))
    else
        tmp = log((x + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt(((x * x) + 1.0));
	double tmp;
	if (x < 0.0) {
		tmp = Math.log((-1.0 / (x - t_0)));
	} else {
		tmp = Math.log((x + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt(((x * x) + 1.0))
	tmp = 0
	if x < 0.0:
		tmp = math.log((-1.0 / (x - t_0)))
	else:
		tmp = math.log((x + t_0))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, If[Less[x, 0.0], N[Log[N[(-1.0 / N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[N[(x + t$95$0), $MachinePrecision]], $MachinePrecision]]]

Reproduce

herbie shell --seed 2024120 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

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