Hyperbolic arcsine

Percentage Accurate: 18.2% → 99.6%

Time: 12.5s

Alternatives: 7

Speedup: 24.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right)\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.05)
     (fma (* x (* x (fma x (* x 0.075) -0.16666666666666666))) x x)
     (log (+ x (- x (/ -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 <= 1.05) {
		tmp = fma((x * (x * fma(x, (x * 0.075), -0.16666666666666666))), x, x);
	} else {
		tmp = log((x + (x - (-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 <= 1.05)
		tmp = fma(Float64(x * Float64(x * fma(x, Float64(x * 0.075), -0.16666666666666666))), x, x);
	else
		tmp = log(Float64(x + Float64(x - 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, 1.05], N[(N[(x * N[(x * N[(x * N[(x * 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $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 < -1.1499999999999999

   1. Initial program 6.0%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. lower-/.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. lower-+.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. lower-/.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. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -1.1499999999999999 < x < 1.05000000000000004

   1. Initial program 8.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

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

   if 1.05000000000000004 < x

   1. Initial program 46.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower--.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right)\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right)\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (log (/ -0.5 x))
   (if (<= x 1.05)
     (fma (* x (* x (fma x (* x 0.075) -0.16666666666666666))) x x)
     (log (+ x (- x (/ -0.5 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = fma((x * (x * fma(x, (x * 0.075), -0.16666666666666666))), x, x);
	} else {
		tmp = log((x + (x - (-0.5 / x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = fma(Float64(x * Float64(x * fma(x, Float64(x * 0.075), -0.16666666666666666))), x, x);
	else
		tmp = log(Float64(x + Float64(x - Float64(-0.5 / x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(x * N[(x * N[(x * N[(x * 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + x), $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 < -1.32000000000000006

   1. Initial program 3.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. lower-/.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.32000000000000006 < x < 1.05000000000000004

   1. Initial program 8.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

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

   if 1.05000000000000004 < x

   1. Initial program 51.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower--.f64 N/A

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

      6. associate-*l* N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right)\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x - \frac{-0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 30.1% 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 2024219 
(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)))))