Hyperbolic arcsine

Percentage Accurate: 18.4% → 99.6%

Time: 10.6s

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: 18.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0073:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 0.0073)
     (* x (+ 1.0 (* (pow x 2.0) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.0073) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.0073) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 0.0073:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.0073)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.075 * Float64(x * x)) - 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 <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 0.0073)
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0073], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.075 * N[(x * x), $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 < -1.30000000000000004

   1. Initial program 3.4%

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

      1. sqr-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. sqr-neg 3.4%

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

      4. hypot-1-def 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in x around -inf 98.8%

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

   if -1.30000000000000004 < x < 0.00730000000000000007

   1. Initial program 7.6%

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

      1. sqr-neg 7.6%

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

      2. +-commutative 7.6%

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

      3. sqr-neg 7.6%

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

      4. hypot-1-def 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 0.00730000000000000007 < x

   1. Initial program 51.5%

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

      1. sqr-neg 51.5%

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

      2. +-commutative 51.5%

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

      3. sqr-neg 51.5%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7600:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -7600.0) (log (/ -0.5 x)) (log1p (+ x (+ (hypot 1.0 x) -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -7600.0) {
		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 <= -7600.0) {
		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 <= -7600.0:
		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 <= -7600.0)
		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, -7600.0], 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 < -7600

   1. Initial program 1.9%

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

      1. sqr-neg 1.9%

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

      2. +-commutative 1.9%

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

      3. sqr-neg 1.9%

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

      4. 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. Add Preprocessing

   5. Taylor expanded in x around -inf 100.0%

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

   if -7600 < x

   1. Initial program 22.1%

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

      1. sqr-neg 22.1%

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

      2. +-commutative 22.1%

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

      3. sqr-neg 22.1%

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

      4. hypot-1-def 37.5%

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

   3. Simplified 37.5%

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

      1. expm1-log1p-u 36.5%

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

      2. expm1-undefine 36.5%

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

      3. log1p-undefine 36.5%

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

      4. rem-exp-log 37.6%

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

   6. Applied egg-rr 37.6%

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

      1. add-exp-log 36.5%

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

      2. expm1-define 36.5%

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

      3. log1p-define 36.5%

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

      4. log1p-expm1-u 36.5%

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

      5. expm1-log1p-u 37.5%

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

      6. log1p-undefine 37.5%

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

      7. expm1-undefine 37.5%

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

      8. add-exp-log 37.5%

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

   8. Applied egg-rr 37.5%

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

      1. log1p-define 37.5%

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

      2. associate--l+ 99.5%

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

   10. Simplified 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 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7600:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.35:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.35)
     (* x (+ 1.0 (* (pow x 2.0) (- (* 0.075 (* x x)) 0.16666666666666666))))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.35) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((0.075 * (x * x)) - 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 <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.35d0) then
        tmp = x * (1.0d0 + ((x ** 2.0d0) * ((0.075d0 * (x * x)) - 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 <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.35) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.35:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.35)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64(0.075 * Float64(x * x)) - 0.16666666666666666))));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.35)
		tmp = x * (1.0 + ((x ^ 2.0) * ((0.075 * (x * x)) - 0.16666666666666666)));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.35], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.075 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 3.4%

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

      1. sqr-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. sqr-neg 3.4%

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

      4. hypot-1-def 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in x around -inf 98.8%

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

   if -1.30000000000000004 < x < 1.3500000000000001

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.7%

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

   7. Applied egg-rr 99.7%

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

   if 1.3500000000000001 < x

   1. Initial program 50.8%

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

      1. sqr-neg 50.8%

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

      2. +-commutative 50.8%

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

      3. sqr-neg 50.8%

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

      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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

Alternative 4: 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 3.4%

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

      1. sqr-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. sqr-neg 3.4%

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

      4. hypot-1-def 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in x around -inf 98.8%

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

   if -1.25 < x < 1.25

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. distribute-rgt-in 99.3%

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

      2. *-lft-identity 99.3%

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

      3. associate-*l* 99.3%

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

      4. unpow2 99.3%

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

      5. unpow3 99.3%

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

   7. Simplified 99.3%

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

   if 1.25 < x

   1. Initial program 50.8%

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

      1. sqr-neg 50.8%

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

      2. +-commutative 50.8%

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

      3. sqr-neg 50.8%

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

      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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

Alternative 5: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (* x (+ 1.0 (* (* x x) -0.16666666666666666)))
     (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 * (1.0 + ((x * x) * -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 <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x * (1.0d0 + ((x * x) * (-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 <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} 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 * (1.0 + ((x * x) * -0.16666666666666666))
	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(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	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[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $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 3.4%

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

      1. sqr-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. sqr-neg 3.4%

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

      4. hypot-1-def 4.6%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in x around -inf 98.8%

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

   if -1.25 < x < 1.25

   1. Initial program 8.2%

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

      1. sqr-neg 8.2%

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

      2. +-commutative 8.2%

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

      3. sqr-neg 8.2%

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

      4. hypot-1-def 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x around 0 99.3%

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

      1. unpow2 99.7%

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

   7. Applied egg-rr 99.3%

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

   if 1.25 < x

   1. Initial program 50.8%

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

      1. sqr-neg 50.8%

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

      2. +-commutative 50.8%

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

      3. sqr-neg 50.8%

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

      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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.2% 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 6.7%

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

      1. sqr-neg 6.7%

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

      2. +-commutative 6.7%

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

      3. sqr-neg 6.7%

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

      4. 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. Add Preprocessing

   5. Taylor expanded in x around 0 68.6%

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

   if 1.25 < x

   1. Initial program 50.8%

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

      1. sqr-neg 50.8%

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

      2. +-commutative 50.8%

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

      3. sqr-neg 50.8%

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

      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 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

Alternative 7: 52.1% 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 17.2%

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

   1. sqr-neg 17.2%

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

   2. +-commutative 17.2%

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

   3. sqr-neg 17.2%

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

   4. hypot-1-def 29.2%

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

3. Simplified 29.2%

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

   1. expm1-log1p-u 27.6%

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

   2. expm1-undefine 27.6%

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

   3. log1p-undefine 27.6%

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

   4. rem-exp-log 29.2%

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

6. Applied egg-rr 29.2%

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

7. Taylor expanded in x around 0 6.4%

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

   1. +-commutative 6.4%

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

9. Simplified 6.4%

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

    1. flip-- 6.2%

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

    2. metadata-eval 6.2%

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

    3. difference-of-sqr-1 6.2%

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

    4. associate-+l+ 6.2%

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

    5. metadata-eval 6.2%

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

    6. associate--l+ 53.4%

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

    7. metadata-eval 53.4%

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

    8. +-rgt-identity 53.4%

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

    9. associate-+l+ 53.4%

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

    10. metadata-eval 53.4%

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

11. Applied egg-rr 53.4%

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

12. Taylor expanded in x around 0 54.4%

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

13. Final simplification 54.4%

    \[\leadsto \frac{x \cdot 2}{x + 2} \]
14. Add Preprocessing

Alternative 8: 51.3% 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 17.2%

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

   1. sqr-neg 17.2%

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

   2. +-commutative 17.2%

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

   3. sqr-neg 17.2%

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

   4. hypot-1-def 29.2%

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

3. Simplified 29.2%

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

5. Taylor expanded in x around 0 53.6%

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

Developer Target 1: 30.7% 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 2024181 
(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)))))