Hyperbolic arcsine

Percentage Accurate: 17.7% → 99.3%

Time: 9.2s

Alternatives: 6

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: 17.7% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.95)
   (- (log (- (- (/ -0.5 x) x) x)))
   (if (<= x 1.25)
     (fma x (* -0.16666666666666666 (* x x)) x)
     (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.95) {
		tmp = -log((((-0.5 / x) - x) - x));
	} else if (x <= 1.25) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.95)
		tmp = Float64(-log(Float64(Float64(Float64(-0.5 / x) - x) - x)));
	elseif (x <= 1.25)
		tmp = fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.95], (-N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.94999999999999996

   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 5.7

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

      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 5.7

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

   4. Applied rewrites 5.7%

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*l/ N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-lft-identity N/A

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

      15. times-frac N/A

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

      16. metadata-eval N/A

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

      17. *-inverses N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac2 N/A

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

      20. distribute-neg-frac N/A

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

      21. metadata-eval N/A

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

      22. lower-/.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if -0.94999999999999996 < x < 1.25

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if 1.25 < x

   1. Initial program 50.8%

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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(\left(\frac{-0.5}{x} - x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(\left(-x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (log (- (- x) x)))
   (if (<= x 1.25)
     (fma x (* -0.16666666666666666 (* x x)) x)
     (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = -log((-x - x));
	} else if (x <= 1.25) {
		tmp = fma(x, (-0.16666666666666666 * (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(Float64(-x) - x)));
	elseif (x <= 1.25)
		tmp = fma(x, Float64(-0.16666666666666666 * Float64(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[((-x) - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 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 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 5.7

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

      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 5.7

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

   4. Applied rewrites 5.7%

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   9. Applied rewrites 98.9%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if 1.25 < x

   1. Initial program 50.8%

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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;-\log \left(\left(-x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;-\log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (- (log (- 1.0 x)))
   (if (<= x 1.25)
     (fma x (* -0.16666666666666666 (* x x)) x)
     (log (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = -log((1.0 - x));
	} else if (x <= 1.25) {
		tmp = fma(x, (-0.16666666666666666 * (x * x)), x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(-log(Float64(1.0 - x)));
	elseif (x <= 1.25)
		tmp = fma(x, Float64(-0.16666666666666666 * Float64(x * x)), x);
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], (-N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x * N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55000000000000004

   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 5.7

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

      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 5.7

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

   4. Applied rewrites 5.7%

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 31.4%

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

   if -1.55000000000000004 < x < 1.25

   1. Initial program 7.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-rgt-identity N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. pow-plus N/A

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

      9. lower-pow.f64 N/A

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

      10. metadata-eval 100.0

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   if 1.25 < x

   1. Initial program 50.8%

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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;-\log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x, -0.16666666666666666 \cdot \left(x \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) (- (- x)) (log (* 2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = -(-x);
	} else {
		tmp = log((2.0 * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.25d0) then
        tmp = -(-x)
    else
        tmp = log((2.0d0 * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = -(-x);
	} else {
		tmp = Math.log((2.0 * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.25:
		tmp = -(-x)
	else:
		tmp = math.log((2.0 * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = Float64(-Float64(-x));
	else
		tmp = log(Float64(2.0 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.25)
		tmp = -(-x);
	else
		tmp = log((2.0 * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], (-(-x)), N[Log[N[(2.0 * x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 6.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 6.7

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

      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 6.7

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

   4. Applied rewrites 6.7%

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

   6. Applied rewrites 24.6%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.7

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

   9. Applied rewrites 64.7%

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

   if 1.25 < x

   1. Initial program 50.8%

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

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

   5. Applied rewrites 100.0%

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

Alternative 5: 58.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.56) (- (- x)) (log (+ 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = -(-x);
	} else {
		tmp = log((1.0 + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.56d0) then
        tmp = -(-x)
    else
        tmp = log((1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = -(-x);
	} else {
		tmp = Math.log((1.0 + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.56:
		tmp = -(-x)
	else:
		tmp = math.log((1.0 + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.56)
		tmp = Float64(-Float64(-x));
	else
		tmp = log(Float64(1.0 + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.56)
		tmp = -(-x);
	else
		tmp = log((1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.56], (-(-x)), N[Log[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 6.2%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 6.7

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

      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 6.7

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

   4. Applied rewrites 6.7%

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

   6. Applied rewrites 24.6%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.7

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

   9. Applied rewrites 64.7%

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

   if 1.5600000000000001 < x

   1. Initial program 50.8%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   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 31.5%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;-\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.0% 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 17.5%

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

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

6. Applied rewrites 18.9%

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

7. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 49.6

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

9. Applied rewrites 49.6%

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

Developer Target 1: 29.9% 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 2024283 
(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)))))