Hyperbolic arcsine

Percentage Accurate: 18.0% → 99.7%

Time: 9.7s

Alternatives: 8

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: 18.0% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144 \cdot x, x, 0.075\right) \cdot x, x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ (- (- (/ 0.125 (* x x)) 0.5) (/ 0.0625 (* (* (* x x) x) x))) x))
   (if (<= x 1.05)
     (*
      (fma
       (*
        (fma
         (* (fma (* -0.044642857142857144 x) x 0.075) x)
         x
         -0.16666666666666666)
        x)
       x
       1.0)
      x)
     (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log(((((0.125 / (x * x)) - 0.5) - (0.0625 / (((x * x) * x) * x))) / x));
	} else if (x <= 1.05) {
		tmp = fma((fma((fma((-0.044642857142857144 * x), x, 0.075) * x), x, -0.16666666666666666) * x), x, 1.0) * x;
	} else {
		tmp = log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(Float64(Float64(0.125 / Float64(x * x)) - 0.5) - Float64(0.0625 / Float64(Float64(Float64(x * x) * x) * x))) / x));
	elseif (x <= 1.05)
		tmp = Float64(fma(Float64(fma(Float64(fma(Float64(-0.044642857142857144 * x), x, 0.075) * x), x, -0.16666666666666666) * x), x, 1.0) * x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] - N[(0.0625 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(N[(N[(N[(N[(N[(N[(-0.044642857142857144 * x), $MachinePrecision] * x + 0.075), $MachinePrecision] * x), $MachinePrecision] * x + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 5.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \log \color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \log \color{blue}{\left(\frac{-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto \log \color{blue}{\left(\frac{-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \log \color{blue}{\left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}{x}\right)} \]

   if -1.1000000000000001 < x < 1.05000000000000004

   1. Initial program 9.0%

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

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

   5. Applied rewrites 99.6%

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

   if 1.05000000000000004 < x

   1. Initial program 55.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{\left(\frac{0.125}{x \cdot x} - 0.5\right) - \frac{0.0625}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144 \cdot x, x, 0.075\right) \cdot x, x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144 \cdot x, x, 0.075\right) \cdot x, x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (log (- (- (/ -0.5 x) x) x)))
   (if (<= x 1.05)
     (*
      (fma
       (*
        (fma
         (* (fma (* -0.044642857142857144 x) x 0.075) x)
         x
         -0.16666666666666666)
        x)
       x
       1.0)
      x)
     (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -log((((-0.5 / x) - x) - x));
	} else if (x <= 1.05) {
		tmp = fma((fma((fma((-0.044642857142857144 * x), x, 0.075) * x), x, -0.16666666666666666) * x), x, 1.0) * x;
	} else {
		tmp = log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(-log(Float64(Float64(Float64(-0.5 / x) - x) - x)));
	elseif (x <= 1.05)
		tmp = Float64(fma(Float64(fma(Float64(fma(Float64(-0.044642857142857144 * x), x, 0.075) * x), x, -0.16666666666666666) * x), x, 1.0) * x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], (-N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.05], N[(N[(N[(N[(N[(N[(N[(-0.044642857142857144 * x), $MachinePrecision] * x + 0.075), $MachinePrecision] * x), $MachinePrecision] * x + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision], N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 3.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate--l+ N/A

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

      11. lift-*.f64 N/A

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. *-rgt-identity N/A

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

      17. lower--.f64 N/A

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

      18. lower--.f64 2.8

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

   4. Applied rewrites 2.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-+r- N/A

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

      7. +-commutative N/A

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

      8. associate-+r- N/A

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

      9. +-inverses N/A

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

      10. metadata-eval N/A

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

      11. neg-log N/A

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

      12. lift-log.f64 N/A

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

      13. lower-neg.f64 52.4

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

   6. Applied rewrites 52.4%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. associate-*l/ N/A

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

      14. lft-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.4

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

   9. Applied rewrites 99.4%

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

   if -1.05000000000000004 < x < 1.05000000000000004

   1. Initial program 8.7%

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

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

   5. Applied rewrites 99.8%

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

   if 1.05000000000000004 < x

   1. Initial program 51.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144 \cdot x, x, 0.075\right) \cdot x, x, -0.16666666666666666\right) \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 30.4% 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 2024234 
(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)))))