Hyperbolic arcsine

Percentage Accurate: 17.4% → 96.9%

Time: 8.6s

Alternatives: 9

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: 17.4% 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: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.028:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33)
   (log (/ -0.5 x))
   (if (<= x 0.028)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.028) {
		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 <= (-4d+33)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.028d0) 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 <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.028) {
		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 <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 0.028:
		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 <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.028)
		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 <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 0.028)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.028], 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 < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 0.0280000000000000006

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 0 98.2%

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

      1. distribute-rgt-in 98.2%

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

      2. *-lft-identity 98.2%

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

      3. associate-*l* 98.2%

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

      4. unpow2 98.2%

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

      5. unpow3 98.2%

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

   5. Simplified 98.2%

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

   if 0.0280000000000000006 < x

   1. Initial program 63.0%

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

   3. Taylor expanded in x around inf 96.9%

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

      1. *-commutative 96.9%

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

   5. Simplified 96.9%

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

Alternative 2: 96.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33)
   (log (/ -0.5 x))
   (if (<= x 1.2e-23)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* (pow x 2.0) -0.044642857142857144)))
         0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.2e-23) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.2e-23) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (Math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 1.2e-23:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (math.pow(x, 2.0) * -0.044642857142857144))) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.2e-23)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 1.2e-23)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + ((x ^ 2.0) * -0.044642857142857144))) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.2e-23], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $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 < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 1.19999999999999998e-23

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)} \]

   if 1.19999999999999998e-23 < x

   1. Initial program 60.7%

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

      1. sqr-neg 60.7%

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

      2. +-commutative 60.7%

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

      3. sqr-neg 60.7%

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

      4. hypot-1-def 94.0%

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

   3. Simplified 94.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + {x}^{2} \cdot -0.044642857142857144\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33)
   (log (/ -0.5 x))
   (if (<= x 1.2e-23)
     (*
      x
      (+ 1.0 (* (pow x 2.0) (- (* (pow x 2.0) 0.075) 0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.2e-23) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * 0.075) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.2e-23) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * 0.075) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 1.2e-23:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * 0.075) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.2e-23)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * 0.075) - 0.16666666666666666))));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 1.2e-23)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * 0.075) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.2e-23], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * 0.075), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $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 < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 1.19999999999999998e-23

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]

   if 1.19999999999999998e-23 < x

   1. Initial program 60.7%

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

      1. sqr-neg 60.7%

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

      2. +-commutative 60.7%

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

      3. sqr-neg 60.7%

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

      4. hypot-1-def 94.0%

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

   3. Simplified 94.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot 0.075 - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-17}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33)
   (log (/ -0.5 x))
   (if (<= x 9.2e-17)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else if (x <= 9.2e-17) {
		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 <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 9.2e-17) {
		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 <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 9.2e-17:
		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 <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 9.2e-17)
		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 <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 9.2e-17)
		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, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9.2e-17], 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 < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 9.20000000000000035e-17

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0 98.1%

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

      1. distribute-rgt-in 98.1%

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

      2. *-lft-identity 98.1%

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

      3. associate-*l* 98.1%

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

      4. unpow2 98.1%

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

      5. unpow3 98.1%

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

   5. Simplified 98.1%

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

   if 9.20000000000000035e-17 < x

   1. Initial program 61.6%

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

      1. sqr-neg 61.6%

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

      2. +-commutative 61.6%

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

      3. sqr-neg 61.6%

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

      4. hypot-1-def 95.3%

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

   3. Simplified 95.3%

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

Alternative 5: 96.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-11}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33)
   (log (/ -0.5 x))
   (if (<= x 4e-11)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (+ (log 2.0) (log x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else if (x <= 4e-11) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(2.0) + log(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d+33)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 4d-11) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(2.0d0) + log(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 4e-11) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(2.0) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 4e-11:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(2.0) + math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 4e-11)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(log(2.0) + log(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 4e-11)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(2.0) + log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 4e-11], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 3.99999999999999976e-11

   1. Initial program 9.4%

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

   3. Taylor expanded in x around 0 98.1%

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

      1. distribute-rgt-in 98.1%

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

      2. *-lft-identity 98.1%

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

      3. associate-*l* 98.1%

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

      4. unpow2 98.1%

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

      5. unpow3 98.1%

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

   5. Simplified 98.1%

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

   if 3.99999999999999976e-11 < x

   1. Initial program 62.8%

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

   3. Taylor expanded in x around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. log-rec 96.6%

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

      3. remove-double-neg 96.6%

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

   5. Simplified 96.6%

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e+33) (log (/ -0.5 x)) (if (<= x 1.25) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -4e+33) {
		tmp = log((-0.5 / x));
	} else 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 <= (-4d+33)) then
        tmp = log(((-0.5d0) / x))
    else 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 <= -4e+33) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e+33:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e+33)
		tmp = log(Float64(-0.5 / x));
	elseif (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 <= -4e+33)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e+33], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.9999999999999998e33

   1. Initial program 1.7%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -3.9999999999999998e33 < x < 1.25

   1. Initial program 10.3%

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

   3. Taylor expanded in x around 0 97.3%

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

   if 1.25 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. *-commutative 98.4%

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

   5. Simplified 98.4%

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

Alternative 7: 75.1% 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 7.9%

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

   3. Taylor expanded in x around 0 71.1%

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

   if 1.25 < x

   1. Initial program 62.5%

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

   3. Taylor expanded in x around inf 98.4%

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

      1. *-commutative 98.4%

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

   5. Simplified 98.4%

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

Alternative 8: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -1.7e-9) x (log1p x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.7e-9) {
		tmp = x;
	} else {
		tmp = log1p(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.7e-9) {
		tmp = x;
	} else {
		tmp = Math.log1p(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.7e-9:
		tmp = x
	else:
		tmp = math.log1p(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.7e-9)
		tmp = x;
	else
		tmp = log1p(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.7e-9], x, N[Log[1 + x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-9

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0 10.4%

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

   if -1.6999999999999999e-9 < x

   1. Initial program 25.0%

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

   3. Taylor expanded in x around 0 14.7%

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

      1. +-commutative 14.7%

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

   5. Simplified 14.7%

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

      1. *-un-lft-identity 14.7%

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

      2. log-prod 14.7%

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

      3. metadata-eval 14.7%

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

      4. +-commutative 14.7%

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

      5. log1p-define 77.1%

         \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(x\right)} \]

   7. Applied egg-rr 77.1%

      \[\leadsto \color{blue}{0 + \mathsf{log1p}\left(x\right)} \]
   8. Step-by-step derivation

      1. +-lft-identity 77.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]

   9. Simplified 77.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.7% 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 21.1%

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

3. Taylor expanded in x around 0 55.2%

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

Developer target: 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 2024107 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

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