Hyperbolic arcsine

Percentage Accurate: 17.5% → 99.5%

Time: 10.2s

Alternatives: 5

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: 17.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(\mathsf{fma}\left(-2, x, \frac{-0.5}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- 0.0 (log (fma -2.0 x (/ -0.5 x))))
   (if (<= x 1.32)
     (* x (fma (* x x) (fma x (* x 0.075) -0.16666666666666666) 1.0))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - log(fma(-2.0, x, (-0.5 / x)));
	} else if (x <= 1.32) {
		tmp = x * fma((x * x), fma(x, (x * 0.075), -0.16666666666666666), 1.0);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.0 - log(fma(-2.0, x, Float64(-0.5 / x))));
	elseif (x <= 1.32)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.075), -0.16666666666666666), 1.0));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.0 - N[Log[N[(-2.0 * x + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 3.8%

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

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

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

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

      4. accelerator-lowering-fma.f64 3.8

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

   4. Applied egg-rr 3.8%

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

      1. flip-+ N/A

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

      2. flip3-- N/A

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

      3. associate-/r/ N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-+r- N/A

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

   6. Applied egg-rr 50.1%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

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

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

      7. associate-*l* N/A

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

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

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. *-inverses N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 N/A

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

      19. metadata-eval 98.9

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

   9. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1.32000000000000006

   1. Initial program 8.4%

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.32000000000000006 < x

   1. Initial program 38.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(\mathsf{fma}\left(-2, x, \frac{-0.5}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- 0.0 (log (* x -2.0)))
   (if (<= x 1.32)
     (* x (fma (* x x) (fma x (* x 0.075) -0.16666666666666666) 1.0))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = 0.0 - log((x * -2.0));
	} else if (x <= 1.32) {
		tmp = x * fma((x * x), fma(x, (x * 0.075), -0.16666666666666666), 1.0);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(0.0 - log(Float64(x * -2.0)));
	elseif (x <= 1.32)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.075), -0.16666666666666666), 1.0));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(0.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 3.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 98.8

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

   5. Simplified 98.8%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval 98.8

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

   7. Applied egg-rr 98.8%

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

   if -1.30000000000000004 < x < 1.32000000000000006

   1. Initial program 8.4%

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.32000000000000006 < x

   1. Initial program 38.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;0 - \log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (- 0.0 (log (- 1.0 x)))
   (if (<= x 1.32)
     (* x (fma (* x x) (fma x (* x 0.075) -0.16666666666666666) 1.0))
     (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = 0.0 - log((1.0 - x));
	} else if (x <= 1.32) {
		tmp = x * fma((x * x), fma(x, (x * 0.075), -0.16666666666666666), 1.0);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(0.0 - log(Float64(1.0 - x)));
	elseif (x <= 1.32)
		tmp = Float64(x * fma(Float64(x * x), fma(x, Float64(x * 0.075), -0.16666666666666666), 1.0));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(0.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32], N[(x * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   1. Initial program 3.8%

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

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

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

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

      4. accelerator-lowering-fma.f64 3.8

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

   4. Applied egg-rr 3.8%

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

      1. flip-+ N/A

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

      2. flip3-- N/A

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

      3. associate-/r/ N/A

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

      4. rem-square-sqrt N/A

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

      5. associate-+r- N/A

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

   6. Applied egg-rr 50.1%

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

   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 31.5

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

   9. Simplified 31.5%

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

   if -1.55000000000000004 < x < 1.32000000000000006

   1. Initial program 8.4%

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

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

      2. +-commutative N/A

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

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

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

      4. unpow2 N/A

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

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

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

      6. sub-neg N/A

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

      7. unpow2 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.32000000000000006 < x

   1. Initial program 38.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;0 - \log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.32:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.075, -0.16666666666666666\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.9% 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 6.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.3%

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

   if 1.25 < x

   1. Initial program 38.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.5% 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 14.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 54.2%

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

Developer Target 1: 29.1% 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 2024197 
(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)))))