Hyperbolic arcsine

Percentage Accurate: 17.4% → 99.4%

Time: 11.1s

Alternatives: 5

Speedup: 122.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 17.4% 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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = 0.0 - log((x * -2.0));
	} else if (x <= 0.95) {
		tmp = fma((x * (x * x)), -0.16666666666666666, x);
	} else {
		tmp = log(fma(x, 2.0, (0.5 / x)));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(0.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.95], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x * 2.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.6%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 99.0

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

   5. Simplified 99.0%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval 99.0

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

   7. Applied egg-rr 99.0%

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

   if -1.25 < x < 0.94999999999999996

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x \]

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

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

      7. cube-mult N/A

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

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

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

      9. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 0.94999999999999996 < x

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*l/ N/A

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

      7. unpow2 N/A

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

      8. associate-/r* N/A

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

      9. associate-/l* N/A

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

      10. *-inverses N/A

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

      11. metadata-eval N/A

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

      12. /-lowering-/.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = 0.0 - log((x * -2.0));
	} else if (x <= 1.25) {
		tmp = fma((x * (x * x)), -0.16666666666666666, x);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(0.0 - N[Log[N[(x * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.6%

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

   3. Taylor expanded in x around -inf

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

      1. /-lowering-/.f64 99.0

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

   5. Simplified 99.0%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

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

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

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

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

      5. div-inv N/A

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

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

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

      7. metadata-eval 99.0

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

   7. Applied egg-rr 99.0%

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

   if -1.25 < x < 1.25

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x \]

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

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

      7. cube-mult N/A

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

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

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

      9. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 1.25 < x

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.2%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;0 - \log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;0 - \log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.52)
   (- 0.0 (log (- 1.0 x)))
   (if (<= x 1.25) (fma (* x (* x x)) -0.16666666666666666 x) (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.52) {
		tmp = 0.0 - log((1.0 - x));
	} else if (x <= 1.25) {
		tmp = fma((x * (x * x)), -0.16666666666666666, x);
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.52)
		tmp = Float64(0.0 - log(Float64(1.0 - x)));
	elseif (x <= 1.25)
		tmp = fma(Float64(x * Float64(x * x)), -0.16666666666666666, x);
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.52], N[(0.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + x), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.52

   1. Initial program 3.6%

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

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. rem-square-sqrt N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 N/A

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

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

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

      9. accelerator-lowering-fma.f64 3.5

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

   4. Applied egg-rr 3.5%

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

      1. clear-num N/A

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

      2. log-rec N/A

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

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

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. +-inverses N/A

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

      8. associate--r+ N/A

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

      9. rem-square-sqrt N/A

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

      10. rem-square-sqrt N/A

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

      11. flip-+ N/A

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

      12. distribute-neg-frac N/A

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

   6. Applied egg-rr 61.9%

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

   7. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. --lowering--.f64 31.2

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

   9. Simplified 31.2%

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

   if -1.52 < x < 1.25

   1. Initial program 7.9%

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

   3. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.16666666666666666, 1\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. cube-mult N/A

         \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x \]

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

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

      7. cube-mult N/A

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

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

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

      9. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 1.25 < x

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.52:\\ \;\;\;\;0 - \log \left(1 - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.25) x (log (+ x x))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.5%

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

   if 1.25 < x

   1. Initial program 50.9%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.2%

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

Alternative 5: 52.5% accurate, 122.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 18.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 51.0%

   \[\leadsto \color{blue}{x} \]
6. 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 2024196 
(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)))))