Hyperbolic arcsine

Percentage Accurate: 18.2% → 99.8%

Time: 9.5s

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: 18.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.024:\\ \;\;\;\;x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(0.075 + -0.044642857142857144 \cdot {x}^{2}\right) - 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (* -1.0 (/ (- 0.5 (* 0.125 (/ (/ 1.0 x) x))) x)))
   (if (<= x 0.024)
     (*
      x
      (+
       1.0
       (*
        (pow x 2.0)
        (-
         (* (pow x 2.0) (+ 0.075 (* -0.044642857142857144 (pow x 2.0))))
         0.16666666666666666))))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.024) {
		tmp = x * (1.0 + (pow(x, 2.0) * ((pow(x, 2.0) * (0.075 + (-0.044642857142857144 * pow(x, 2.0)))) - 0.16666666666666666)));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.024) {
		tmp = x * (1.0 + (Math.pow(x, 2.0) * ((Math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * Math.pow(x, 2.0)))) - 0.16666666666666666)));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)))
	elif x <= 0.024:
		tmp = x * (1.0 + (math.pow(x, 2.0) * ((math.pow(x, 2.0) * (0.075 + (-0.044642857142857144 * math.pow(x, 2.0)))) - 0.16666666666666666)))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.125 * Float64(Float64(1.0 / x) / x))) / x)));
	elseif (x <= 0.024)
		tmp = Float64(x * Float64(1.0 + Float64((x ^ 2.0) * Float64(Float64((x ^ 2.0) * Float64(0.075 + Float64(-0.044642857142857144 * (x ^ 2.0)))) - 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 <= -1.15)
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	elseif (x <= 0.024)
		tmp = x * (1.0 + ((x ^ 2.0) * (((x ^ 2.0) * (0.075 + (-0.044642857142857144 * (x ^ 2.0)))) - 0.16666666666666666)));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(-1.0 * N[(N[(0.5 - N[(0.125 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.024], N[(x * N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(N[Power[x, 2.0], $MachinePrecision] * N[(0.075 + N[(-0.044642857142857144 * N[Power[x, 2.0], $MachinePrecision]), $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 < -1.1499999999999999

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   6. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{{\left({x}^{2}\right)}^{-1}}}{x}\right) \]

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right)}}{x}\right) \]
   8. Step-by-step derivation

      1. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.1499999999999999 < x < 0.024

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. +-commutative 10.9%

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

      3. sqr-neg 10.9%

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

      4. hypot-1-def 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 0.024 < x

   1. Initial program 56.0%

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

      1. sqr-neg 56.0%

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

      2. +-commutative 56.0%

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

      3. sqr-neg 56.0%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

Alternative 2: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.006:\\ \;\;\;\;{x}^{3} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right) + x\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (* -1.0 (/ (- 0.5 (* 0.125 (/ (/ 1.0 x) x))) x)))
   (if (<= x 0.006)
     (+ (* (pow x 3.0) (- (* 0.075 (pow x 2.0)) 0.16666666666666666)) x)
     (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.006) {
		tmp = (pow(x, 3.0) * ((0.075 * pow(x, 2.0)) - 0.16666666666666666)) + x;
	} else {
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.006) {
		tmp = (Math.pow(x, 3.0) * ((0.075 * Math.pow(x, 2.0)) - 0.16666666666666666)) + x;
	} else {
		tmp = 2.0 * Math.log(Math.sqrt((x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)))
	elif x <= 0.006:
		tmp = (math.pow(x, 3.0) * ((0.075 * math.pow(x, 2.0)) - 0.16666666666666666)) + x
	else:
		tmp = 2.0 * math.log(math.sqrt((x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.125 * Float64(Float64(1.0 / x) / x))) / x)));
	elseif (x <= 0.006)
		tmp = Float64(Float64((x ^ 3.0) * Float64(Float64(0.075 * (x ^ 2.0)) - 0.16666666666666666)) + x);
	else
		tmp = Float64(2.0 * log(sqrt(Float64(x + hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	elseif (x <= 0.006)
		tmp = ((x ^ 3.0) * ((0.075 * (x ^ 2.0)) - 0.16666666666666666)) + x;
	else
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(-1.0 * N[(N[(0.5 - N[(0.125 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.006], N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(0.075 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   6. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{{\left({x}^{2}\right)}^{-1}}}{x}\right) \]

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right)}}{x}\right) \]
   8. Step-by-step derivation

      1. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.1499999999999999 < x < 0.0060000000000000001

   1. Initial program 10.2%

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

      1. sqr-neg 10.2%

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

      2. +-commutative 10.2%

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

      3. sqr-neg 10.2%

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

      4. hypot-1-def 10.2%

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

   3. Simplified 10.2%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow-plus 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{\color{blue}{3}} + 1 \cdot x \]

      9. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + \color{blue}{x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)} + x \]

   if 0.0060000000000000001 < x

   1. Initial program 56.4%

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

      1. sqr-neg 56.4%

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

      2. +-commutative 56.4%

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

      3. sqr-neg 56.4%

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

      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
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. log-pow 99.9%

         \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]

   6. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.0072:\\ \;\;\;\;{x}^{3} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (* -1.0 (/ (- 0.5 (* 0.125 (/ (/ 1.0 x) x))) x)))
   (if (<= x 0.0072)
     (+ (* (pow x 3.0) (- (* 0.075 (pow x 2.0)) 0.16666666666666666)) x)
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.0072) {
		tmp = (pow(x, 3.0) * ((0.075 * pow(x, 2.0)) - 0.16666666666666666)) + x;
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = Math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.0072) {
		tmp = (Math.pow(x, 3.0) * ((0.075 * Math.pow(x, 2.0)) - 0.16666666666666666)) + x;
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.15:
		tmp = math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)))
	elif x <= 0.0072:
		tmp = (math.pow(x, 3.0) * ((0.075 * math.pow(x, 2.0)) - 0.16666666666666666)) + x
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.125 * Float64(Float64(1.0 / x) / x))) / x)));
	elseif (x <= 0.0072)
		tmp = Float64(Float64((x ^ 3.0) * Float64(Float64(0.075 * (x ^ 2.0)) - 0.16666666666666666)) + x);
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.15)
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	elseif (x <= 0.0072)
		tmp = ((x ^ 3.0) * ((0.075 * (x ^ 2.0)) - 0.16666666666666666)) + x;
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(-1.0 * N[(N[(0.5 - N[(0.125 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0072], N[(N[(N[Power[x, 3.0], $MachinePrecision] * N[(N[(0.075 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $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 < -1.1499999999999999

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   6. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{{\left({x}^{2}\right)}^{-1}}}{x}\right) \]

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right)}}{x}\right) \]
   8. Step-by-step derivation

      1. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.1499999999999999 < x < 0.0071999999999999998

   1. Initial program 10.2%

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

      1. sqr-neg 10.2%

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

      2. +-commutative 10.2%

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

      3. sqr-neg 10.2%

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

      4. hypot-1-def 10.2%

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

   3. Simplified 10.2%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow-plus 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{\color{blue}{3}} + 1 \cdot x \]

      9. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + \color{blue}{x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{{x}^{3} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)} + x \]

   if 0.0071999999999999998 < x

   1. Initial program 56.4%

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

      1. sqr-neg 56.4%

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

      2. +-commutative 56.4%

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

      3. sqr-neg 56.4%

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

      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 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 0.00105:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (* -1.0 (/ (- 0.5 (* 0.125 (/ (/ 1.0 x) x))) x)))
   (if (<= x 0.00105)
     (+ (* -0.16666666666666666 (pow x 3.0)) x)
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.00105) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + x;
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 0.00105) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + x;
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)))
	elif x <= 0.00105:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + x
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.125 * Float64(Float64(1.0 / x) / x))) / x)));
	elseif (x <= 0.00105)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + x);
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	elseif (x <= 0.00105)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + x;
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(-1.0 * N[(N[(0.5 - N[(0.125 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00105], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $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 < -1.1000000000000001

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   6. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{{\left({x}^{2}\right)}^{-1}}}{x}\right) \]

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right)}}{x}\right) \]
   8. Step-by-step derivation

      1. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.1000000000000001 < x < 0.00104999999999999994

   1. Initial program 10.2%

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

      1. sqr-neg 10.2%

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

      2. +-commutative 10.2%

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

      3. sqr-neg 10.2%

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

      4. hypot-1-def 10.2%

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

   3. Simplified 10.2%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   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. +-commutative 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. fma-neg 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow-plus 100.0%

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

      8. metadata-eval 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{\color{blue}{3}} + 1 \cdot x \]

      9. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + \color{blue}{x} \]

   7. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around 0 99.8%

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

   if 0.00104999999999999994 < x

   1. Initial program 56.4%

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

      1. sqr-neg 56.4%

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

      2. +-commutative 56.4%

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

      3. sqr-neg 56.4%

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

      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 5: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{\frac{1}{x}}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (* -1.0 (/ (- 0.5 (* 0.125 (/ (/ 1.0 x) x))) x)))
   (if (<= x 1.3) (+ (* -0.16666666666666666 (pow x 3.0)) x) (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 1.3) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + x;
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log(((-1.0d0) * ((0.5d0 - (0.125d0 * ((1.0d0 / x) / x))) / x)))
    else if (x <= 1.3d0) then
        tmp = ((-0.16666666666666666d0) * (x ** 3.0d0)) + x
    else
        tmp = log((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	} else if (x <= 1.3) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)))
	elif x <= 1.3:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(-1.0 * Float64(Float64(0.5 - Float64(0.125 * Float64(Float64(1.0 / x) / x))) / x)));
	elseif (x <= 1.3)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((-1.0 * ((0.5 - (0.125 * ((1.0 / x) / x))) / x)));
	elseif (x <= 1.3)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(-1.0 * N[(N[(0.5 - N[(0.125 * N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 100.0%

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{0.5 - 0.125 \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   6. Step-by-step derivation

      1. inv-pow 100.0%

         \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{{\left({x}^{2}\right)}^{-1}}}{x}\right) \]

      2. unpow2 100.0%

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

      3. unpow-prod-down 100.0%

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

      4. inv-pow 100.0%

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

      5. inv-pow 100.0%

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

   7. Applied egg-rr 100.0%

      \[\leadsto \log \left(-1 \cdot \frac{0.5 - 0.125 \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{1}{x}\right)}}{x}\right) \]
   8. Step-by-step derivation

      1. un-div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.1000000000000001 < x < 1.30000000000000004

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. +-commutative 10.9%

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

      3. sqr-neg 10.9%

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

      4. hypot-1-def 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow-plus 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{\color{blue}{3}} + 1 \cdot x \]

      9. *-un-lft-identity 99.9%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + \color{blue}{x} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around 0 99.5%

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

   if 1.30000000000000004 < x

   1. Initial program 56.0%

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

      1. sqr-neg 56.0%

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

      2. +-commutative 56.0%

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

      3. sqr-neg 56.0%

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

      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 97.4%

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

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.3) (+ (* -0.16666666666666666 (pow x 3.0)) x) (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + x;
	} else {
		tmp = log((2.0 * x));
	}
	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 <= 1.3d0) then
        tmp = ((-0.16666666666666666d0) * (x ** 3.0d0)) + x
    else
        tmp = log((2.0d0 * x))
    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 <= 1.3) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 99.1%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. +-commutative 10.9%

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

      3. sqr-neg 10.9%

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

      4. hypot-1-def 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in x around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-in 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. fma-neg 99.9%

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

      6. metadata-eval 99.9%

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

      7. pow-plus 99.9%

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

      8. metadata-eval 99.9%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{\color{blue}{3}} + 1 \cdot x \]

      9. *-un-lft-identity 99.9%

         \[\leadsto \mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + \color{blue}{x} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.075, {x}^{2}, -0.16666666666666666\right) \cdot {x}^{3} + x} \]

   8. Taylor expanded in x around 0 99.5%

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

   if 1.30000000000000004 < x

   1. Initial program 56.0%

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

      1. sqr-neg 56.0%

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

      2. +-commutative 56.0%

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

      3. sqr-neg 56.0%

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

      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 97.4%

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

Alternative 7: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25) (log (/ -0.5 x)) (if (<= x 1.3) x (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = log((2.0 * x));
	}
	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 <= 1.3d0) then
        tmp = x
    else
        tmp = log((2.0d0 * x))
    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 <= 1.3) {
		tmp = x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.3:
		tmp = x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = x;
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.3)
		tmp = x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], x, N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 4.0%

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

      1. sqr-neg 4.0%

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

      2. +-commutative 4.0%

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

      3. sqr-neg 4.0%

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

      4. hypot-1-def 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in x around -inf 99.1%

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

   if -1.25 < x < 1.30000000000000004

   1. Initial program 10.9%

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

      1. sqr-neg 10.9%

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

      2. +-commutative 10.9%

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

      3. sqr-neg 10.9%

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

      4. hypot-1-def 10.9%

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

   3. Simplified 10.9%

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

   5. Taylor expanded in x around 0 98.0%

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

   if 1.30000000000000004 < x

   1. Initial program 56.0%

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

      1. sqr-neg 56.0%

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

      2. +-commutative 56.0%

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

      3. sqr-neg 56.0%

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

      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 97.4%

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

Alternative 8: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.3) x (log (* 2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d0) then
        tmp = x
    else
        tmp = log((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3) {
		tmp = x;
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3:
		tmp = x
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3)
		tmp = x;
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3)
		tmp = x;
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3], x, N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 8.7%

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

      1. sqr-neg 8.7%

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

      2. +-commutative 8.7%

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

      3. sqr-neg 8.7%

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

      4. hypot-1-def 9.0%

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

   3. Simplified 9.0%

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

   5. Taylor expanded in x around 0 68.5%

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

   if 1.30000000000000004 < x

   1. Initial program 56.0%

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

      1. sqr-neg 56.0%

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

      2. +-commutative 56.0%

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

      3. sqr-neg 56.0%

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

      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 97.4%

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

Alternative 9: 51.8% 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 22.9%

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

   1. sqr-neg 22.9%

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

   2. +-commutative 22.9%

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

   3. sqr-neg 22.9%

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

   4. hypot-1-def 36.4%

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

3. Simplified 36.4%

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

5. Taylor expanded in x around 0 49.6%

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

Developer target: 30.5% 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 2024077 -o generate:simplify
(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)))))