Hyperbolic arcsine

Percentage Accurate: 17.3% → 99.5%

Time: 6.8s

Alternatives: 8

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: 17.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + \sqrt{x \cdot x + 1}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (log (+ x (sqrt (+ (* x x) 1.0)))))

C

double code(double x) {
	return log((x + sqrt(((x * x) + 1.0))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

Java

public static double code(double x) {
	return Math.log((x + Math.sqrt(((x * x) + 1.0))));
}

Python

def code(x):
	return math.log((x + math.sqrt(((x * x) + 1.0))))

Julia

function code(x)
	return log(Float64(x + sqrt(Float64(Float64(x * x) + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = log((x + sqrt(((x * x) + 1.0))));
end

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -325:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -325.0)
   (log (- (* 0.125 (/ 1.0 (pow x 3.0))) (* 0.5 (/ 1.0 x))))
   (log1p (+ x (+ (hypot 1.0 x) -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -325.0) {
		tmp = log(((0.125 * (1.0 / pow(x, 3.0))) - (0.5 * (1.0 / x))));
	} else {
		tmp = log1p((x + (hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -325.0) {
		tmp = Math.log(((0.125 * (1.0 / Math.pow(x, 3.0))) - (0.5 * (1.0 / x))));
	} else {
		tmp = Math.log1p((x + (Math.hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -325.0:
		tmp = math.log(((0.125 * (1.0 / math.pow(x, 3.0))) - (0.5 * (1.0 / x))))
	else:
		tmp = math.log1p((x + (math.hypot(1.0, x) + -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -325.0)
		tmp = log(Float64(Float64(0.125 * Float64(1.0 / (x ^ 3.0))) - Float64(0.5 * Float64(1.0 / x))));
	else
		tmp = log1p(Float64(x + Float64(hypot(1.0, x) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -325.0], N[Log[N[(N[(0.125 * N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -325

   1. Initial program 3.0%

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

      1. +-commutative 3.0%

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

      2. hypot-1-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)} \]

   if -325 < x

   1. Initial program 27.8%

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

      1. +-commutative 27.8%

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

      2. hypot-1-def 41.1%

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

   3. Simplified 41.1%

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

      1. log1p-expm1-u 41.1%

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

      2. expm1-udef 41.1%

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

      3. add-exp-log 41.1%

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

      4. associate--l+ 99.7%

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

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -325:\\ \;\;\;\;\log \left(0.125 \cdot \frac{1}{{x}^{3}} - 0.5 \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00077:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.00077)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00077) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00077) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00077:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00077)
		tmp = Float64(x + Float64((x ^ 3.0) * -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 <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00077)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00077], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $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 < -1.26000000000000001

   1. Initial program 3.0%

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

      1. +-commutative 3.0%

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

      2. hypot-1-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around -inf 99.4%

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

   if -1.26000000000000001 < x < 7.6999999999999996e-4

   1. Initial program 7.1%

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

      1. +-commutative 7.1%

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

      2. hypot-1-def 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 99.6%

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

   if 7.6999999999999996e-4 < x

   1. Initial program 63.5%

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

      1. +-commutative 63.5%

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

      2. hypot-1-def 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00077:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9600:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -9600.0) (log (/ -0.5 x)) (log1p (+ x (+ (hypot 1.0 x) -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -9600.0) {
		tmp = log((-0.5 / x));
	} else {
		tmp = log1p((x + (hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -9600.0) {
		tmp = Math.log((-0.5 / x));
	} else {
		tmp = Math.log1p((x + (Math.hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -9600.0:
		tmp = math.log((-0.5 / x))
	else:
		tmp = math.log1p((x + (math.hypot(1.0, x) + -1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -9600.0)
		tmp = log(Float64(-0.5 / x));
	else
		tmp = log1p(Float64(x + Float64(hypot(1.0, x) + -1.0)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -9600.0], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9600

   1. Initial program 1.9%

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

      1. +-commutative 1.9%

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

      2. hypot-1-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around -inf 100.0%

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

   if -9600 < x

   1. Initial program 28.0%

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

      1. +-commutative 28.0%

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

      2. hypot-1-def 41.3%

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

   3. Simplified 41.3%

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

      1. log1p-expm1-u 41.3%

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

      2. expm1-udef 41.3%

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

      3. add-exp-log 41.3%

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

      4. associate--l+ 99.5%

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

   5. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9600:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(0.5 \cdot \frac{1}{x} + x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (+ (* 0.5 (/ 1.0 x)) (* x 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log(((0.5 * (1.0 / x)) + (x * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.95d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log(((0.5d0 * (1.0d0 / x)) + (x * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log(((0.5 * (1.0 / x)) + (x * 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log(((0.5 * (1.0 / x)) + (x * 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(Float64(0.5 * Float64(1.0 / x)) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.95)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log(((0.5 * (1.0 / x)) + (x * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.0%

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

      1. +-commutative 3.0%

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

      2. hypot-1-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around -inf 99.4%

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

   if -1.26000000000000001 < x < 0.94999999999999996

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 98.9%

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

   if 0.94999999999999996 < x

   1. Initial program 62.6%

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

      1. +-commutative 62.6%

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

      2. 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. Taylor expanded in x around inf 99.8%

      \[\leadsto \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(0.5 \cdot \frac{1}{x} + x \cdot 2\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (+ x (* (pow x 3.0) -0.16666666666666666))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x + ((x ** 3.0d0) * (-0.16666666666666666d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (Math.pow(x, 3.0) * -0.16666666666666666);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = x + (math.pow(x, 3.0) * -0.16666666666666666)
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64((x ^ 3.0) * -0.16666666666666666));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x + ((x ^ 3.0) * -0.16666666666666666);
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x + N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.0%

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

      1. +-commutative 3.0%

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

      2. hypot-1-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around -inf 99.4%

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

   if -1.26000000000000001 < x < 1.25

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 98.9%

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

   if 1.25 < x

   1. Initial program 62.6%

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

      1. +-commutative 62.6%

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

      2. 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + {x}^{3} \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 6: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26) (log (/ -0.5 x)) (if (<= x 1.25) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		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 <= (-1.26d0)) 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 <= -1.26) {
		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 <= -1.26:
		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 <= -1.26)
		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 <= -1.26)
		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, -1.26], 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 < -1.26000000000000001

   1. Initial program 3.0%

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

      1. +-commutative 3.0%

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

      2. hypot-1-def 4.2%

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

   3. Simplified 4.2%

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

   4. Taylor expanded in x around -inf 99.4%

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

   if -1.26000000000000001 < x < 1.25

   1. Initial program 8.4%

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

      1. +-commutative 8.4%

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

      2. hypot-1-def 8.4%

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

   3. Simplified 8.4%

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

   4. Taylor expanded in x around 0 98.5%

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

   if 1.25 < x

   1. Initial program 62.6%

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

      1. +-commutative 62.6%

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

      2. 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 7: 75.6% accurate, 2.0× 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 6.7%

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

      1. +-commutative 6.7%

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

      2. hypot-1-def 7.1%

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

   3. Simplified 7.1%

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

   4. Taylor expanded in x around 0 68.5%

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

   if 1.25 < x

   1. Initial program 62.6%

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

      1. +-commutative 62.6%

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

      2. 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. Taylor expanded in x around inf 99.0%

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

4. Final simplification 76.8%

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

Alternative 8: 52.6% 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 22.0%

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

   1. +-commutative 22.0%

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

   2. hypot-1-def 32.5%

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

3. Simplified 32.5%

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

4. Taylor expanded in x around 0 51.3%

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

5. Final simplification 51.3%

   \[\leadsto x \]

Developer target: 29.1% 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 2023305 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (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)))))