Hyperbolic arcsine

Percentage Accurate: 18.1% → 99.4%

Time: 13.2s

Alternatives: 10

Speedup: 122.0×

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: 18.1% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;x \leq -11000:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot t\_0\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - t\_1 \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x x 1.0))) (t_1 (sqrt t_0)))
   (if (<= x -11000.0)
     (log (/ -0.5 x))
     (if (<= x 200000.0)
       (-
        (log (fma x (* x x) (* (fma x x 1.0) t_0)))
        (log1p (fma x x (* x (- x (* t_1 t_1))))))
       (log (+ x x))))))

C

double code(double x) {
	double t_0 = sqrt(fma(x, x, 1.0));
	double t_1 = sqrt(t_0);
	double tmp;
	if (x <= -11000.0) {
		tmp = log((-0.5 / x));
	} else if (x <= 200000.0) {
		tmp = log(fma(x, (x * x), (fma(x, x, 1.0) * t_0))) - log1p(fma(x, x, (x * (x - (t_1 * t_1)))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(x, x, 1.0))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (x <= -11000.0)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 200000.0)
		tmp = Float64(log(fma(x, Float64(x * x), Float64(fma(x, x, 1.0) * t_0))) - log1p(fma(x, x, Float64(x * Float64(x - Float64(t_1 * t_1))))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[x, -11000.0], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 200000.0], N[(N[Log[N[(x * N[(x * x), $MachinePrecision] + N[(N[(x * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(x * x + N[(x * N[(x - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -11000

   1. Initial program 1.9%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -11000 < x < 2e5

   1. Initial program 9.9%

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

      1. flip3-+ N/A

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

      2. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. accelerator-lowering-fma.f64 99.2

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

   6. Applied egg-rr 99.2%

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

   if 2e5 < x

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{if}\;x \leq -11000:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 200000:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot t\_0\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, x \cdot \left(x - t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (fma x x 1.0))))
   (if (<= x -11000.0)
     (log (/ -0.5 x))
     (if (<= x 200000.0)
       (-
        (log (fma x (* x x) (* (fma x x 1.0) t_0)))
        (log1p (fma x x (* x (- x t_0)))))
       (log (+ x x))))))

C

double code(double x) {
	double t_0 = sqrt(fma(x, x, 1.0));
	double tmp;
	if (x <= -11000.0) {
		tmp = log((-0.5 / x));
	} else if (x <= 200000.0) {
		tmp = log(fma(x, (x * x), (fma(x, x, 1.0) * t_0))) - log1p(fma(x, x, (x * (x - t_0))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(fma(x, x, 1.0))
	tmp = 0.0
	if (x <= -11000.0)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 200000.0)
		tmp = Float64(log(fma(x, Float64(x * x), Float64(fma(x, x, 1.0) * t_0))) - log1p(fma(x, x, Float64(x * Float64(x - t_0)))));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, -11000.0], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 200000.0], N[(N[Log[N[(x * N[(x * x), $MachinePrecision] + N[(N[(x * x + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(x * x + N[(x * N[(x - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -11000

   1. Initial program 1.9%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

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

   if -11000 < x < 2e5

   1. Initial program 9.9%

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

      1. flip3-+ N/A

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

      2. log-div N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{\log \left({x}^{3} + {\left(\sqrt{x \cdot x + 1}\right)}^{3}\right) - \log \left(x \cdot x + \left(\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1} - x \cdot \sqrt{x \cdot x + 1}\right)\right)} \]

   4. Applied egg-rr 99.2%

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

   if 2e5 < x

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

Alternative 3: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, -x, \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.044642857142857144, -0.075\right), 0.16666666666666666\right), -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.1)
     (-
      (log (fma x (- x) (fma x x 1.0)))
      (*
       x
       (fma
        (* x x)
        (fma
         (* x x)
         (fma x (* x 0.044642857142857144) -0.075)
         0.16666666666666666)
        -1.0)))
     (log (fma x 2.0 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.1) {
		tmp = log(fma(x, -x, fma(x, x, 1.0))) - (x * fma((x * x), fma((x * x), fma(x, (x * 0.044642857142857144), -0.075), 0.16666666666666666), -1.0));
	} else {
		tmp = log(fma(x, 2.0, (0.5 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 1.1)
		tmp = Float64(log(fma(x, Float64(-x), fma(x, x, 1.0))) - Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(x, Float64(x * 0.044642857142857144), -0.075), 0.16666666666666666), -1.0)));
	else
		tmp = log(fma(x, 2.0, Float64(0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1], N[(N[Log[N[(x * (-x) + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.044642857142857144), $MachinePrecision] + -0.075), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 4.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \log \color{blue}{\left(\frac{-1 \cdot \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}}{x}\right) \]

      3. neg-sub0 N/A

         \[\leadsto \log \left(\frac{\color{blue}{0 - \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}}{x}\right) \]

      4. associate--r- N/A

         \[\leadsto \log \left(\frac{\color{blue}{\left(0 - \frac{1}{2}\right) + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      5. metadata-eval N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right) \]

      6. +-commutative N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \frac{-1}{2}}}{x}\right) \]

      7. metadata-eval N/A

         \[\leadsto \log \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x}\right) \]

      8. sub-neg N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}}{x}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}{x}\right)} \]

      10. sub-neg N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x}\right) \]

      11. metadata-eval N/A

          \[\leadsto \log \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-1}{2}}}{x}\right) \]

      12. +-commutative N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}}{x}\right) \]

      15. metadata-eval N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \frac{\color{blue}{\frac{1}{8}}}{{x}^{2}}}{x}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}}{x}\right) \]

      17. unpow2 N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}}{x}\right) \]

      18. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

   if -1.1499999999999999 < x < 1.1000000000000001

   1. Initial program 8.1%

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

      1. flip-+ N/A

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

      2. frac-2neg N/A

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

      3. log-div N/A

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

      4. --lowering--.f64 N/A

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

   4. Applied egg-rr 8.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right)\right) - 1\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right)\right) - 1\right)} \]

      2. sub-neg N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right)\right) + \left(\mathsf{neg}\left(1\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right)\right) + \color{blue}{-1}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right), -1\right)} \]

      5. unpow2 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right), -1\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{6} + {x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right), -1\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{5}{112} \cdot {x}^{2} - \frac{3}{40}\right) + \frac{1}{6}}, -1\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{5}{112} \cdot {x}^{2} - \frac{3}{40}, \frac{1}{6}\right)}, -1\right) \]

      9. unpow2 N/A

         \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{112} \cdot {x}^{2} - \frac{3}{40}, \frac{1}{6}\right), -1\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{5}{112} \cdot {x}^{2} - \frac{3}{40}, \frac{1}{6}\right), -1\right) \]

      11. sub-neg N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{5}{112} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{3}{40}\right)\right)}, \frac{1}{6}\right), -1\right) \]

      12. *-commutative N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{5}{112}} + \left(\mathsf{neg}\left(\frac{3}{40}\right)\right), \frac{1}{6}\right), -1\right) \]

      13. metadata-eval N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{5}{112} + \color{blue}{\frac{-3}{40}}, \frac{1}{6}\right), -1\right) \]

      14. unpow2 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{5}{112} + \frac{-3}{40}, \frac{1}{6}\right), -1\right) \]

      15. associate-*l* N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \frac{5}{112}\right)} + \frac{-3}{40}, \frac{1}{6}\right), -1\right) \]

      16. accelerator-lowering-fma.f64 N/A

          \[\leadsto \log \left(\mathsf{fma}\left(x, \mathsf{neg}\left(x\right), \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{5}{112}, \frac{-3}{40}\right)}, \frac{1}{6}\right), -1\right) \]

      17. *-lowering-*.f64 99.9

          \[\leadsto \log \left(\mathsf{fma}\left(x, -x, \mathsf{fma}\left(x, x, 1\right)\right)\right) - x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 0.044642857142857144}, -0.075\right), 0.16666666666666666\right), -1\right) \]

   7. Simplified 99.9%

      \[\leadsto \log \left(\mathsf{fma}\left(x, -x, \mathsf{fma}\left(x, x, 1\right)\right)\right) - \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.044642857142857144, -0.075\right), 0.16666666666666666\right), -1\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 98.9

          \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{x}}\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.1)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (fma x 2.0 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.1) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log(fma(x, 2.0, (0.5 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 1.1)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(fma(x, 2.0, Float64(0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x * 2.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1499999999999999

   1. Initial program 4.2%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \log \color{blue}{\left(\frac{-1 \cdot \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \log \left(\frac{\color{blue}{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}}{x}\right) \]

      3. neg-sub0 N/A

         \[\leadsto \log \left(\frac{\color{blue}{0 - \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}}{x}\right) \]

      4. associate--r- N/A

         \[\leadsto \log \left(\frac{\color{blue}{\left(0 - \frac{1}{2}\right) + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      5. metadata-eval N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right) \]

      6. +-commutative N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \frac{-1}{2}}}{x}\right) \]

      7. metadata-eval N/A

         \[\leadsto \log \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x}\right) \]

      8. sub-neg N/A

         \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}}{x}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}{x}\right)} \]

      10. sub-neg N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{x}\right) \]

      11. metadata-eval N/A

          \[\leadsto \log \left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{-1}{2}}}{x}\right) \]

      12. +-commutative N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \log \left(\frac{\color{blue}{\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}}}{x}\right) \]

      14. associate-*r/ N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{8} \cdot 1}{{x}^{2}}}}{x}\right) \]

      15. metadata-eval N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \frac{\color{blue}{\frac{1}{8}}}{{x}^{2}}}{x}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}}{x}\right) \]

      17. unpow2 N/A

          \[\leadsto \log \left(\frac{\frac{-1}{2} + \frac{\frac{1}{8}}{\color{blue}{x \cdot x}}}{x}\right) \]

      18. *-lowering-*.f64 98.8

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

   5. Simplified 98.8%

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

   if -1.1499999999999999 < x < 1.1000000000000001

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 98.9

          \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{x}}\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.1)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (fma x 2.0 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.1) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log(fma(x, 2.0, (0.5 / x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.1)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(fma(x, 2.0, Float64(0.5 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x * 2.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 4.2%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   if -1.30000000000000004 < x < 1.1000000000000001

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 1.1000000000000001 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 98.9

          \[\leadsto \log \left(\mathsf{fma}\left(x, 2, \color{blue}{\frac{0.5}{x}}\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 4.2%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 98.3

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

   5. Simplified 98.3%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 1.25 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

Alternative 7: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(\left(-x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (log (- (- x) x)))
   (if (<= x 1.25)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -log((-x - x));
	} else if (x <= 1.25) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(-log(Float64(Float64(-x) - x)));
	elseif (x <= 1.25)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], (-N[Log[N[((-x) - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 4.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. rem-square-sqrt N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. accelerator-lowering-fma.f64 3.9

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

   4. Applied egg-rr 3.9%

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. log-rec N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. accelerator-lowering-fma.f64 51.6

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 97.0

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

   9. Simplified 97.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 1.25 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

Alternative 8: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;-\log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (log (- 1.0 x)))
   (if (<= x 1.25)
     (fma
      (fma
       (* x x)
       (fma x (* x -0.044642857142857144) 0.075)
       -0.16666666666666666)
      (* x (* x x))
      x)
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = -log((1.0 - x));
	} else if (x <= 1.25) {
		tmp = fma(fma((x * x), fma(x, (x * -0.044642857142857144), 0.075), -0.16666666666666666), (x * (x * x)), x);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(-log(Float64(1.0 - x)));
	elseif (x <= 1.25)
		tmp = fma(fma(Float64(x * x), fma(x, Float64(x * -0.044642857142857144), 0.075), -0.16666666666666666), Float64(x * Float64(x * x)), x);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], (-N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * -0.044642857142857144), $MachinePrecision] + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 4.2%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. rem-square-sqrt N/A

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

      4. div-sub N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. --lowering--.f64 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. accelerator-lowering-fma.f64 3.9

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

   4. Applied egg-rr 3.9%

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

      4. +-inverses N/A

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

      5. metadata-eval N/A

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

      6. log-rec N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. accelerator-lowering-fma.f64 51.6

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

   6. Applied egg-rr 51.6%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 31.2

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

   9. Simplified 31.2%

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

   if -1.3999999999999999 < x < 1.25

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)} \cdot x \]

      3. distribute-lft1-in N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} + x \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x \cdot {x}^{2}, x\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]

   if 1.25 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

Alternative 9: 76.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 6.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 65.5%

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

   if 1.25 < x

   1. Initial program 56.5%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 98.6%

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

Alternative 10: 52.9% accurate, 122.0× 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 16.2%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 54.1%

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

Developer Target 1: 29.6% accurate, 0.9× 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 2024205 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

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