Hyperbolic arcsine

Percentage Accurate: 17.5% → 99.9%

Time: 15.9s

Alternatives: 11

Speedup: 207.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.5% 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: 99.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right)\\ \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_0 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (pow (+ x (hypot 1.0 x)) 0.25))))
   (if (<= x -0.0011)
     (- (log (- (hypot 1.0 x) x)))
     (if (<= x 0.0011)
       (+ x (* -0.16666666666666666 (pow x 3.0)))
       (* 2.0 (+ t_0 t_0))))))

C

double code(double x) {
	double t_0 = log(pow((x + hypot(1.0, x)), 0.25));
	double tmp;
	if (x <= -0.0011) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0011) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = 2.0 * (t_0 + t_0);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.log(Math.pow((x + Math.hypot(1.0, x)), 0.25));
	double tmp;
	if (x <= -0.0011) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0011) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = 2.0 * (t_0 + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.log(math.pow((x + math.hypot(1.0, x)), 0.25))
	tmp = 0
	if x <= -0.0011:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0011:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = 2.0 * (t_0 + t_0)
	return tmp

Julia

function code(x)
	t_0 = log((Float64(x + hypot(1.0, x)) ^ 0.25))
	tmp = 0.0
	if (x <= -0.0011)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0011)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(2.0 * Float64(t_0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = log(((x + hypot(1.0, x)) ^ 0.25));
	tmp = 0.0;
	if (x <= -0.0011)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0011)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = 2.0 * (t_0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Log[N[Power[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -0.0011], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0011], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00110000000000000007

   1. Initial program 3.3%

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

      1. expm1-log1p-u 3.3%

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

      2. expm1-udef 3.3%

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

      3. +-commutative 3.3%

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

      4. hypot-1-def 4.6%

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

   4. Applied egg-rr 4.6%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. log1p-udef 4.6%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

   6. Applied egg-rr 4.6%

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

      1. associate--l+ 4.6%

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

      2. metadata-eval 4.6%

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

      3. +-rgt-identity 4.6%

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

      4. flip-+ 3.0%

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

      5. div-inv 3.0%

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

      6. pow2 3.0%

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

      7. hypot-udef 3.0%

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

      8. hypot-udef 2.9%

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

      9. metadata-eval 2.9%

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

      10. metadata-eval 2.9%

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

      11. add-sqr-sqrt 3.0%

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

      12. pow2 3.0%

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

   8. Applied egg-rr 3.0%

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

      1. associate-*r/ 3.0%

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

      2. *-rgt-identity 3.0%

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

      3. +-commutative 3.0%

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

      4. associate--r+ 47.7%

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

      5. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      6. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

       1. frac-2neg 100.0%

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

       2. metadata-eval 100.0%

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

       3. log-div 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0 - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-neg-in 100.0%

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

       4. remove-double-neg 100.0%

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

       5. +-commutative 100.0%

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

       6. sub-neg 100.0%

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

   14. Simplified 100.0%

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

   if -0.00110000000000000007 < x < 0.00110000000000000007

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 0.00110000000000000007 < x

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

      3. log-pow 53.8%

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

      4. +-commutative 53.8%

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

      5. hypot-1-def 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 99.9%

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

      2. log-prod 100.0%

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

      3. pow1/2 100.0%

         \[\leadsto 2 \cdot \left(\log \left(\sqrt{\color{blue}{{\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.5}}}\right) + \log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(1, x\right)}}\right)\right) \]

      4. sqrt-pow1 100.0%

         \[\leadsto 2 \cdot \left(\log \color{blue}{\left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)} + \log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(1, x\right)}}\right)\right) \]

      5. metadata-eval 100.0%

         \[\leadsto 2 \cdot \left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{0.25}}\right) + \log \left(\sqrt{\sqrt{x + \mathsf{hypot}\left(1, x\right)}}\right)\right) \]

      6. pow1/2 100.0%

         \[\leadsto 2 \cdot \left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right) + \log \left(\sqrt{\color{blue}{{\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.5}}}\right)\right) \]

      7. sqrt-pow1 100.0%

         \[\leadsto 2 \cdot \left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right) + \log \color{blue}{\left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}\right) \]

      8. metadata-eval 100.0%

         \[\leadsto 2 \cdot \left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right) + \log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{0.25}}\right)\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto 2 \cdot \color{blue}{\left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right) + \log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right) + \log \left({\left(x + \mathsf{hypot}\left(1, x\right)\right)}^{0.25}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0013:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + {\left(\sqrt{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0011)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0013)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (pow (sqrt (hypot 1.0 x)) 2.0))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0013) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + pow(sqrt(hypot(1.0, x)), 2.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0013) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.pow(Math.sqrt(Math.hypot(1.0, x)), 2.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0011:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0013:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.pow(math.sqrt(math.hypot(1.0, x)), 2.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0011)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0013)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + (sqrt(hypot(1.0, x)) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0011)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0013)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + (sqrt(hypot(1.0, x)) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0011], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0013], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Power[N[Sqrt[N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00110000000000000007

   1. Initial program 3.3%

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

      1. expm1-log1p-u 3.3%

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

      2. expm1-udef 3.3%

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

      3. +-commutative 3.3%

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

      4. hypot-1-def 4.6%

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

   4. Applied egg-rr 4.6%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. log1p-udef 4.6%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

   6. Applied egg-rr 4.6%

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

      1. associate--l+ 4.6%

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

      2. metadata-eval 4.6%

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

      3. +-rgt-identity 4.6%

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

      4. flip-+ 3.0%

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

      5. div-inv 3.0%

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

      6. pow2 3.0%

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

      7. hypot-udef 3.0%

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

      8. hypot-udef 2.9%

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

      9. metadata-eval 2.9%

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

      10. metadata-eval 2.9%

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

      11. add-sqr-sqrt 3.0%

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

      12. pow2 3.0%

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

   8. Applied egg-rr 3.0%

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

      1. associate-*r/ 3.0%

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

      2. *-rgt-identity 3.0%

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

      3. +-commutative 3.0%

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

      4. associate--r+ 47.7%

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

      5. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      6. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

       1. frac-2neg 100.0%

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

       2. metadata-eval 100.0%

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

       3. log-div 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0 - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-neg-in 100.0%

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

       4. remove-double-neg 100.0%

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

       5. +-commutative 100.0%

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

       6. sub-neg 100.0%

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

   14. Simplified 100.0%

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

   if -0.00110000000000000007 < x < 0.0012999999999999999

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 0.0012999999999999999 < x

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.8%

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

      2. pow2 53.8%

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

      3. +-commutative 53.8%

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

      4. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0013:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + {\left(\sqrt{\mathsf{hypot}\left(1, x\right)}\right)}^{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (log (/ -1.0 (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))
   (if (<= x 0.0012)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = Math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))))
	elif x <= 0.0012:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = log(Float64(-1.0 / Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x)))));
	elseif (x <= 0.0012)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	elseif (x <= 0.0012)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], N[Log[N[(-1.0 / N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.0012], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   1. Initial program 3.3%

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

      1. expm1-log1p-u 3.3%

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

      2. expm1-udef 3.3%

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

      3. +-commutative 3.3%

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

      4. hypot-1-def 4.6%

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

   4. Applied egg-rr 4.6%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. log1p-udef 4.6%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

   6. Applied egg-rr 4.6%

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

      1. associate--l+ 4.6%

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

      2. metadata-eval 4.6%

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

      3. +-rgt-identity 4.6%

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

      4. flip-+ 3.0%

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

      5. div-inv 3.0%

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

      6. pow2 3.0%

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

      7. hypot-udef 3.0%

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

      8. hypot-udef 2.9%

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

      9. metadata-eval 2.9%

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

      10. metadata-eval 2.9%

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

      11. add-sqr-sqrt 3.0%

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

      12. pow2 3.0%

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

   8. Applied egg-rr 3.0%

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

      1. associate-*r/ 3.0%

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

      2. *-rgt-identity 3.0%

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

      3. +-commutative 3.0%

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

      4. associate--r+ 47.7%

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

      5. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      6. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around -inf 98.8%

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

   if -0.95999999999999996 < x < 0.00119999999999999989

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 0.00119999999999999989 < x

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. +-commutative 53.8%

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

      3. sqr-neg 53.8%

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

      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. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0011)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0012)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.0011) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0012) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.0011:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0012:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0011)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0012)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0011)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0012)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.0011], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0012], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00110000000000000007

   1. Initial program 3.3%

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

      1. expm1-log1p-u 3.3%

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

      2. expm1-udef 3.3%

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

      3. +-commutative 3.3%

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

      4. hypot-1-def 4.6%

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

   4. Applied egg-rr 4.6%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. log1p-udef 4.6%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

   6. Applied egg-rr 4.6%

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

      1. associate--l+ 4.6%

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

      2. metadata-eval 4.6%

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

      3. +-rgt-identity 4.6%

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

      4. flip-+ 3.0%

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

      5. div-inv 3.0%

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

      6. pow2 3.0%

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

      7. hypot-udef 3.0%

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

      8. hypot-udef 2.9%

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

      9. metadata-eval 2.9%

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

      10. metadata-eval 2.9%

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

      11. add-sqr-sqrt 3.0%

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

      12. pow2 3.0%

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

   8. Applied egg-rr 3.0%

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

      1. associate-*r/ 3.0%

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

      2. *-rgt-identity 3.0%

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

      3. +-commutative 3.0%

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

      4. associate--r+ 47.7%

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

      5. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      6. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

       1. frac-2neg 100.0%

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

       2. metadata-eval 100.0%

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

       3. log-div 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto \color{blue}{0} - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right) \]

   12. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{0 - \log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
   13. Step-by-step derivation

       1. neg-sub0 100.0%

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

       2. sub-neg 100.0%

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

       3. distribute-neg-in 100.0%

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

       4. remove-double-neg 100.0%

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

       5. +-commutative 100.0%

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

       6. sub-neg 100.0%

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

   14. Simplified 100.0%

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

   if -0.00110000000000000007 < x < 0.00119999999999999989

   1. Initial program 7.5%

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

   3. Taylor expanded in x around 0 100.0%

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

   if 0.00119999999999999989 < x

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. +-commutative 53.8%

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

      3. sqr-neg 53.8%

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

      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. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0011:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0012:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	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 <= 0.95d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    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 <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / 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 <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + (x + (0.5 / 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, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -1.30000000000000004 < x < 0.94999999999999996

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 0.94999999999999996 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (log (/ -1.0 (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (+ x (/ 0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = log(((-1.0d0) / ((x * 2.0d0) + (0.5d0 * (1.0d0 / x)))))
    else if (x <= 0.95d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x + (x + (0.5d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = Math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + (x + (0.5 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + (x + (0.5 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = log(Float64(-1.0 / Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x)))));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + Float64(x + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	elseif (x <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + (x + (0.5 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], N[Log[N[(-1.0 / N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   1. Initial program 3.3%

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

      1. expm1-log1p-u 3.3%

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

      2. expm1-udef 3.3%

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

      3. +-commutative 3.3%

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

      4. hypot-1-def 4.6%

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

   4. Applied egg-rr 4.6%

      \[\leadsto \log \color{blue}{\left(e^{\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. log1p-udef 4.6%

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

      2. rem-exp-log 4.6%

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

      3. +-commutative 4.6%

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

   6. Applied egg-rr 4.6%

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

      1. associate--l+ 4.6%

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

      2. metadata-eval 4.6%

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

      3. +-rgt-identity 4.6%

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

      4. flip-+ 3.0%

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

      5. div-inv 3.0%

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

      6. pow2 3.0%

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

      7. hypot-udef 3.0%

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

      8. hypot-udef 2.9%

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

      9. metadata-eval 2.9%

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

      10. metadata-eval 2.9%

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

      11. add-sqr-sqrt 3.0%

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

      12. pow2 3.0%

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

   8. Applied egg-rr 3.0%

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

      1. associate-*r/ 3.0%

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

      2. *-rgt-identity 3.0%

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

      3. +-commutative 3.0%

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

      4. associate--r+ 47.7%

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

      5. +-inverses 100.0%

         \[\leadsto \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

      6. metadata-eval 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around -inf 98.8%

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

   if -0.95999999999999996 < x < 0.94999999999999996

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0 99.6%

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

   if 0.94999999999999996 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. associate-*r/ 99.8%

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

      2. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (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 <= -1.3) {
		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 <= (-1.3d0)) 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 <= -1.3) {
		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 <= -1.3:
		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 <= -1.3)
		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 <= -1.3)
		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, -1.3], 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 < -1.30000000000000004

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 1.25 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.3%

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

Alternative 8: 99.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.8%

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

   3. Taylor expanded in x around -inf 100.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0 98.4%

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

   if 1.25 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.0%

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

Alternative 9: 75.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0 67.6%

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

   if 1.25 < x

   1. Initial program 53.1%

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

   3. Taylor expanded in x around inf 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 75.2%

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

Alternative 10: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.1) x (log x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x;
	} else {
		tmp = log(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = x
    else
        tmp = log(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x;
	} else {
		tmp = Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = x
	else:
		tmp = math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = x;
	else
		tmp = log(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = x;
	else
		tmp = log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], x, N[Log[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 6.5%

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

   3. Taylor expanded in x around 0 67.6%

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

   if 3.10000000000000009 < x

   1. Initial program 53.1%

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

      1. add-sqr-sqrt 53.1%

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

      2. pow2 53.1%

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

      3. +-commutative 53.1%

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

      4. hypot-1-def 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 31.3%

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

      1. mul-1-neg 31.3%

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

      2. log-rec 31.3%

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

      3. remove-double-neg 31.3%

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

   7. Simplified 31.3%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.5% 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 17.8%

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

3. Taylor expanded in x around 0 52.6%

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

4. Final simplification 52.6%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 30.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

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

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