Hyperbolic arcsine

Percentage Accurate: 18.1% → 99.6%

Time: 9.4s

Alternatives: 8

Speedup: 7.2×

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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-1}{x - \left(\frac{-0.5}{x} - x\right)}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -1.0 (- x (- (/ -0.5 x) x))))
   (if (<= x 1.3)
     (fma
      (/
       (* (* x x) -0.027777777777777776)
       (- (* (* (* x x) 0.075) x) (* -0.16666666666666666 x)))
      (* x x)
      x)
     (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-1.0 / (x - ((-0.5 / x) - x))));
	} else if (x <= 1.3) {
		tmp = fma((((x * x) * -0.027777777777777776) / ((((x * x) * 0.075) * x) - (-0.16666666666666666 * x))), (x * x), x);
	} else {
		tmp = log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-1.0 / Float64(x - Float64(Float64(-0.5 / x) - x))));
	elseif (x <= 1.3)
		tmp = fma(Float64(Float64(Float64(x * x) * -0.027777777777777776) / Float64(Float64(Float64(Float64(x * x) * 0.075) * x) - Float64(-0.16666666666666666 * x))), Float64(x * x), x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-1.0 / N[(x - N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision] / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] * x), $MachinePrecision] - N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $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 < -1.30000000000000004

   1. Initial program 4.1%

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

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. lower--.f64 N/A

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

      12. sub-neg N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval N/A

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

      16. lower--.f64 5.2

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 5.2

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

   4. Applied rewrites 5.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 41.5%

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

   8. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*l* N/A

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

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

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. associate-*l/ N/A

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

      14. lft-mult-inverse N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 8.9%

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

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36} \cdot {x}^{2}}{\left(\frac{3}{40} \cdot \left(x \cdot x\right)\right) \cdot x - \frac{-1}{6} \cdot x}, x \cdot x, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot \left(x \cdot x\right)}{\left(0.075 \cdot \left(x \cdot x\right)\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right) \]

   if 1.30000000000000004 < x

   1. Initial program 49.6%

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

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. *-inverses N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-1}{x - \left(\frac{-0.5}{x} - x\right)}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (log (/ -0.5 x))
   (if (<= x 1.3)
     (fma
      (/
       (* (* x x) -0.027777777777777776)
       (- (* (* (* x x) 0.075) x) (* -0.16666666666666666 x)))
      (* x x)
      x)
     (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = fma((((x * x) * -0.027777777777777776) / ((((x * x) * 0.075) * x) - (-0.16666666666666666 * x))), (x * x), x);
	} else {
		tmp = log(((x - (-0.5 / x)) + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = fma(Float64(Float64(Float64(x * x) * -0.027777777777777776) / Float64(Float64(Float64(Float64(x * x) * 0.075) * x) - Float64(-0.16666666666666666 * x))), Float64(x * x), x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision] / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] * x), $MachinePrecision] - N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $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 < -1.8999999999999999

   1. Initial program 4.1%

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

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

   5. Applied rewrites 98.8%

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

   if -1.8999999999999999 < x < 1.30000000000000004

   1. Initial program 8.9%

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

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.7

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36} \cdot {x}^{2}}{\left(\frac{3}{40} \cdot \left(x \cdot x\right)\right) \cdot x - \frac{-1}{6} \cdot x}, x \cdot x, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot \left(x \cdot x\right)}{\left(0.075 \cdot \left(x \cdot x\right)\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right) \]

   if 1.30000000000000004 < x

   1. Initial program 49.6%

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

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

      10. *-lft-identity N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. *-inverses N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.88:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (log (/ -0.5 x))
   (if (<= x 1.88)
     (fma
      (/
       (* (* x x) -0.027777777777777776)
       (- (* (* (* x x) 0.075) x) (* -0.16666666666666666 x)))
      (* x x)
      x)
     (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.88) {
		tmp = fma((((x * x) * -0.027777777777777776) / ((((x * x) * 0.075) * x) - (-0.16666666666666666 * x))), (x * x), x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.88)
		tmp = fma(Float64(Float64(Float64(x * x) * -0.027777777777777776) / Float64(Float64(Float64(Float64(x * x) * 0.075) * x) - Float64(-0.16666666666666666 * x))), Float64(x * x), x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.88], N[(N[(N[(N[(x * x), $MachinePrecision] * -0.027777777777777776), $MachinePrecision] / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] * x), $MachinePrecision] - N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999

   1. Initial program 4.1%

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

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

   5. Applied rewrites 98.8%

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

   if -1.8999999999999999 < x < 1.8799999999999999

   1. Initial program 9.6%

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

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 99.1

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 99.1%

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

   10. Taylor expanded in x around 0

       \[\leadsto \mathsf{fma}\left(\frac{\frac{-1}{36} \cdot {x}^{2}}{\left(\frac{3}{40} \cdot \left(x \cdot x\right)\right) \cdot x - \frac{-1}{6} \cdot x}, x \cdot x, x\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 99.2%

       \[\leadsto \mathsf{fma}\left(\frac{-0.027777777777777776 \cdot \left(x \cdot x\right)}{\left(0.075 \cdot \left(x \cdot x\right)\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right) \]

   if 1.8799999999999999 < x

   1. Initial program 48.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.88:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(x \cdot x\right) \cdot -0.027777777777777776}{\left(\left(x \cdot x\right) \cdot 0.075\right) \cdot x - -0.16666666666666666 \cdot x}, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000004

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

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

   if 1.30000000000000004 < x

   1. Initial program 49.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.0

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

   5. Applied rewrites 98.0%

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

Alternative 5: 58.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.52:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.52)
   (fma (* (* x x) x) (fma 0.075 (* x x) -0.16666666666666666) x)
   (log (+ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.52) {
		tmp = fma(((x * x) * x), fma(0.075, (x * x), -0.16666666666666666), x);
	} else {
		tmp = log((1.0 + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.52)
		tmp = fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x);
	else
		tmp = log(Float64(1.0 + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.52], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.52

   1. Initial program 7.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 74.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

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

   if 1.52 < x

   1. Initial program 49.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 31.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.52:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (* x x) x) (fma 0.075 (* x x) -0.16666666666666666) x))

C

double code(double x) {
	return fma(((x * x) * x), fma(0.075, (x * x), -0.16666666666666666), x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * x), fma(0.075, Float64(x * x), -0.16666666666666666), x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(0.075 * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 54.3

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

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

Alternative 7: 50.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \left(\left(\left(0.075 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (* (* (* (* 0.075 x) x) x) x) x))

C

double code(double x) {
	return fma(x, ((((0.075 * x) * x) * x) * x), x);
}

Julia

function code(x)
	return fma(x, Float64(Float64(Float64(Float64(0.075 * x) * x) * x) * x), x)
end

Wolfram

code[x_] := N[(x * N[(N[(N[(N[(0.075 * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 54.3

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{3}{40} \cdot \color{blue}{{x}^{2}}, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 54.2%

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

12. Applied rewrites 54.2%

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

13. Final simplification 54.2%

    \[\leadsto \mathsf{fma}\left(x, \left(\left(\left(0.075 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x, x\right) \]
14. Add Preprocessing

Alternative 8: 49.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.16666666666666666 \cdot x, x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* -0.16666666666666666 x) (* x x) x))

C

double code(double x) {
	return fma((-0.16666666666666666 * x), (x * x), x);
}

Julia

function code(x)
	return fma(Float64(-0.16666666666666666 * x), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(-0.16666666666666666 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 19.6%

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

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 54.3

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

5. Applied rewrites 54.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075, x \cdot x, -0.16666666666666666\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 54.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{6} \cdot x, x \cdot x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 53.0%

    \[\leadsto \mathsf{fma}\left(-0.16666666666666666 \cdot x, x \cdot x, x\right) \]
11. 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 2024268 
(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)))))