Hyperbolic arcsine

Percentage Accurate: 17.8% → 75.8%

Time: 8.0s

Alternatives: 5

Speedup: 207.0×

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: 17.8% 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: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.8e-6) x (- (log (/ 1.0 (+ x (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = -Math.log((1.0 / (x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.8e-6:
		tmp = x
	else:
		tmp = -math.log((1.0 / (x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = Float64(-log(Float64(1.0 / Float64(x + hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.8e-6], x, (-N[Log[N[(1.0 / N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000004e-6

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.5%

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.9%

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

   if 8.8000000000000004e-6 < x

   1. Initial program 46.0%

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

      1. sqr-neg 46.0%

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

      2. +-commutative 46.0%

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

      3. sqr-neg 46.0%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 2.5%

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

      2. clear-num 2.5%

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

      3. log-div 2.5%

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

      4. metadata-eval 2.5%

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

      5. pow2 2.5%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{{x}^{2}} - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      6. hypot-1-def 2.5%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}\right) \]

      7. hypot-1-def 2.5%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}}}\right) \]

      8. add-sqr-sqrt 2.5%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(1 + x \cdot x\right)}}\right) \]

      9. +-commutative 2.5%

         \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\left(x \cdot x + 1\right)}}\right) \]

      10. fma-define 2.5%

          \[\leadsto 0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \color{blue}{\mathsf{fma}\left(x, x, 1\right)}}\right) \]

   6. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{0 - \log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right)} \]
   7. Step-by-step derivation

      1. neg-sub0 2.5%

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

      2. div-sub 2.5%

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

      3. fma-undefine 2.5%

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

      4. unpow2 2.5%

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

      5. associate--r+ 2.5%

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

      6. +-inverses 2.5%

         \[\leadsto -\log \left(\frac{x}{\color{blue}{0} - 1} - \frac{\mathsf{hypot}\left(1, x\right)}{{x}^{2} - \mathsf{fma}\left(x, x, 1\right)}\right) \]

      7. metadata-eval 2.5%

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

      8. *-rgt-identity 2.5%

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

      9. associate-/l* 2.5%

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

      10. metadata-eval 2.5%

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

      11. fma-undefine 2.5%

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

      12. unpow2 2.5%

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

      13. associate--r+ 3.8%

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

      14. +-inverses 5.5%

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

      15. metadata-eval 5.5%

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

      16. *-rgt-identity 5.5%

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

      17. associate-/l* 5.5%

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

      18. metadata-eval 5.5%

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

      19. *-commutative 5.5%

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

      20. neg-mul-1 5.5%

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

   8. Simplified 5.5%

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

      1. flip-- 3.8%

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

      2. +-commutative 3.8%

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

      3. expm1-log1p-u 3.8%

         \[\leadsto -\log \left(\frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - x \cdot x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}}\right) \]

      4. div-inv 3.8%

         \[\leadsto -\log \color{blue}{\left(\left(\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right) - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right)} \]

      5. hypot-1-def 4.1%

         \[\leadsto -\log \left(\left(\color{blue}{\sqrt{1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right) - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      6. hypot-1-def 3.8%

         \[\leadsto -\log \left(\left(\sqrt{1 + x \cdot x} \cdot \color{blue}{\sqrt{1 + x \cdot x}} - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      7. add-sqr-sqrt 4.1%

         \[\leadsto -\log \left(\left(\color{blue}{\left(1 + x \cdot x\right)} - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      8. +-commutative 4.1%

         \[\leadsto -\log \left(\left(\color{blue}{\left(x \cdot x + 1\right)} - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      9. fma-define 4.1%

         \[\leadsto -\log \left(\left(\color{blue}{\mathsf{fma}\left(x, x, 1\right)} - x \cdot x\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      10. pow2 4.1%

          \[\leadsto -\log \left(\left(\mathsf{fma}\left(x, x, 1\right) - \color{blue}{{x}^{2}}\right) \cdot \frac{1}{\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]

      11. expm1-log1p-u 4.2%

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

   10. Applied egg-rr 4.2%

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

       1. associate-*r/ 4.2%

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

       2. *-rgt-identity 4.2%

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

       3. remove-double-neg 4.2%

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

       4. distribute-neg-frac2 4.2%

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

       5. distribute-frac-neg 4.2%

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

       6. neg-mul-1 4.2%

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

       7. associate-/r* 4.2%

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

   12. Simplified 100.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.8e-6) x (log (expm1 (log1p (+ x (hypot 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = log(expm1(log1p((x + hypot(1.0, x)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = Math.log(Math.expm1(Math.log1p((x + Math.hypot(1.0, x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.8e-6:
		tmp = x
	else:
		tmp = math.log(math.expm1(math.log1p((x + math.hypot(1.0, x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = log(expm1(log1p(Float64(x + hypot(1.0, x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 8.8e-6], x, N[Log[N[(Exp[N[Log[1 + N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000004e-6

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.5%

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.9%

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

   if 8.8000000000000004e-6 < x

   1. Initial program 46.0%

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

      1. sqr-neg 46.0%

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

      2. +-commutative 46.0%

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

      3. sqr-neg 46.0%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 100.0%

         \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.8e-6) x (+ (+ 1.0 (log (+ x (hypot 1.0 x)))) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = (1.0 + log((x + hypot(1.0, x)))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = (1.0 + Math.log((x + Math.hypot(1.0, x)))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.8e-6:
		tmp = x
	else:
		tmp = (1.0 + math.log((x + math.hypot(1.0, x)))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 + log(Float64(x + hypot(1.0, x)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = (1.0 + log((x + hypot(1.0, x)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.8e-6], x, N[(N[(1.0 + N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000004e-6

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.5%

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.9%

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

   if 8.8000000000000004e-6 < x

   1. Initial program 46.0%

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

      1. sqr-neg 46.0%

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

      2. +-commutative 46.0%

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

      3. sqr-neg 46.0%

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

      4. hypot-1-def 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 98.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\right)} \]

      2. expm1-undefine 98.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)} - 1} \]

      3. log1p-undefine 98.2%

         \[\leadsto e^{\color{blue}{\log \left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)}} - 1 \]

      4. rem-exp-log 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1 + x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.8e-6) x (log (+ x (sqrt (+ 1.0 (* x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = log((x + sqrt((1.0 + (x * x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.8d-6) then
        tmp = x
    else
        tmp = log((x + sqrt((1.0d0 + (x * x)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.8e-6) {
		tmp = x;
	} else {
		tmp = Math.log((x + Math.sqrt((1.0 + (x * x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.8e-6:
		tmp = x
	else:
		tmp = math.log((x + math.sqrt((1.0 + (x * x)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = log(Float64(x + sqrt(Float64(1.0 + Float64(x * x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.8e-6)
		tmp = x;
	else
		tmp = log((x + sqrt((1.0 + (x * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.8e-6], x, N[Log[N[(x + N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.8000000000000004e-6

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.5%

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

   3. Simplified 7.5%

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.9%

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

   if 8.8000000000000004e-6 < x

   1. Initial program 46.0%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{1 + x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.3% accurate, 207.0× 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 17.7%

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

   1. sqr-neg 17.7%

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

   2. +-commutative 17.7%

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

   3. sqr-neg 17.7%

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

   4. hypot-1-def 32.8%

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

3. Simplified 32.8%

   \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 48.6%

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

6. Final simplification 48.6%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 30.0% accurate, 1.0× 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 2024066 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))