Hyperbolic arcsine

Percentage Accurate: 17.5% → 99.7%

Time: 7.7s

Alternatives: 7

Speedup: 24.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-\log \left(\mathsf{fma}\left(-2, x, \frac{-0.5}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (log (fma -2.0 x (/ -0.5 x))))
   (if (<= x 1.05)
     (fma
      (*
       (fma
        (fma -0.044642857142857144 (* x x) 0.075)
        (* x x)
        -0.16666666666666666)
       (* x x))
      x
      x)
     (log (+ (- x (/ -0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = -log(fma(-2.0, x, (-0.5 / x)));
	} else if (x <= 1.05) {
		tmp = fma((fma(fma(-0.044642857142857144, (x * x), 0.075), (x * 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 <= -1.05)
		tmp = Float64(-log(fma(-2.0, x, Float64(-0.5 / x))));
	elseif (x <= 1.05)
		tmp = fma(Float64(fma(fma(-0.044642857142857144, Float64(x * x), 0.075), Float64(x * x), -0.16666666666666666) * Float64(x * x)), x, x);
	else
		tmp = log(Float64(Float64(x - Float64(-0.5 / x)) + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], (-N[Log[N[(-2.0 * x + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.05], N[(N[(N[(N[(-0.044642857142857144 * N[(x * x), $MachinePrecision] + 0.075), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 2.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.8%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-in N/A

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

      3. distribute-neg-in N/A

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

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

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. unpow2 N/A

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

      11. associate-/r* N/A

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

      12. distribute-neg-frac N/A

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

      13. *-lft-identity N/A

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

      14. times-frac N/A

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

      15. metadata-eval N/A

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

      16. *-inverses N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. metadata-eval 100.0

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

   7. Applied rewrites 100.0%

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

   if -1.05000000000000004 < x < 1.05000000000000004

   1. Initial program 9.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   if 1.05000000000000004 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \log \left(x + \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;-\log \left(\mathsf{fma}\left(-2, x, \frac{-0.5}{x}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(-2 \cdot x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (log (* -2.0 x)))
   (if (<= x 1.05)
     (fma
      (*
       (fma
        (fma -0.044642857142857144 (* x x) 0.075)
        (* x 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((-2.0 * x));
	} else if (x <= 1.05) {
		tmp = fma((fma(fma(-0.044642857142857144, (x * x), 0.075), (x * 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 = Float64(-log(Float64(-2.0 * x)));
	elseif (x <= 1.05)
		tmp = fma(Float64(fma(fma(-0.044642857142857144, Float64(x * x), 0.075), Float64(x * x), -0.16666666666666666) * Float64(x * 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[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.05], N[(N[(N[(N[(-0.044642857142857144 * N[(x * x), $MachinePrecision] + 0.075), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + 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 2.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.8%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1.30000000000000004 < x < 1.05000000000000004

   1. Initial program 9.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   if 1.05000000000000004 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \log \left(x + \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}\right) \]
   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 99.0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(-2 \cdot x\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x - \frac{-0.5}{x}\right) + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 2.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.8%

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

   5. Taylor expanded in x around -inf

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

      1. lower-*.f64 99.7

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

   7. Applied rewrites 99.7%

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

   if -1.30000000000000004 < x < 1.30000000000000004

   1. Initial program 9.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   if 1.30000000000000004 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 4: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;-\log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (log (- 1.0 x)))
   (if (<= x 1.3)
     (fma
      (*
       (fma
        (fma -0.044642857142857144 (* x x) 0.075)
        (* x x)
        -0.16666666666666666)
       (* x x))
      x
      x)
     (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = -log((1.0 - x));
	} else if (x <= 1.3) {
		tmp = fma((fma(fma(-0.044642857142857144, (x * x), 0.075), (x * x), -0.16666666666666666) * (x * x)), x, x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(-log(Float64(1.0 - x)));
	elseif (x <= 1.3)
		tmp = fma(Float64(fma(fma(-0.044642857142857144, Float64(x * x), 0.075), Float64(x * x), -0.16666666666666666) * Float64(x * x)), x, x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], (-N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.3], N[(N[(N[(N[(-0.044642857142857144 * N[(x * x), $MachinePrecision] + 0.075), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 2.9%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 1.8%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 31.6

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

   7. Applied rewrites 31.6%

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

   if -1.3999999999999999 < x < 1.30000000000000004

   1. Initial program 9.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 99.9

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), \color{blue}{x \cdot x}, -0.16666666666666666\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.044642857142857144, x \cdot x, 0.075\right), x \cdot x, -0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   if 1.30000000000000004 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 5: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.075, -0.16666666666666666\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35)
   (fma (* (fma (* x x) 0.075 -0.16666666666666666) x) (* x x) x)
   (log (* 2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35) {
		tmp = fma((fma((x * x), 0.075, -0.16666666666666666) * x), (x * x), x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35)
		tmp = fma(Float64(fma(Float64(x * x), 0.075, -0.16666666666666666) * x), Float64(x * x), x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.35], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075 + -0.16666666666666666), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001

   1. Initial program 7.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. +-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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   if 1.3500000000000001 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

Alternative 6: 58.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 7.4%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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. +-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 67.0

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

   5. Applied rewrites 67.0%

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

   7. Applied rewrites 67.0%

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

   if 1.55000000000000004 < x

   1. Initial program 46.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-+.f64 31.5

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

   5. Applied rewrites 31.5%

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

Alternative 7: 52.4% accurate, 24.4× speedup

Math

\[\begin{array}{l} \\ -\left(-x\right) \end{array} \]

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 Float64(-Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = -(-x);
end

Wolfram

code[x_] := (-(-x))

Derivation

1. Initial program 18.3%

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

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. clear-num N/A

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

   6. log-rec N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 6.3%

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

5. Taylor expanded in x around 0

   \[\leadsto -\color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.9

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

7. Applied rewrites 49.9%

   \[\leadsto -\color{blue}{\left(-x\right)} \]
8. Add Preprocessing

Developer Target 1: 29.7% 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 2024331 
(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)))))