Hyperbolic arcsine

Percentage Accurate: 18.1% → 99.4%

Time: 10.2s

Alternatives: 7

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: 18.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \mathsf{fma}\left(x \cdot x, 0.075, -0.16666666666666666\right) \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.35)
   (log (/ -0.5 x))
   (if (<= x 1.3)
     (+ x (* (fma (* x x) 0.075 -0.16666666666666666) (pow x 3.0)))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.35) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.3) {
		tmp = x + (fma((x * x), 0.075, -0.16666666666666666) * pow(x, 3.0));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.35)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.3)
		tmp = Float64(x + Float64(fma(Float64(x * x), 0.075, -0.16666666666666666) * (x ^ 3.0)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.35], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3], N[(x + N[(N[(N[(x * x), $MachinePrecision] * 0.075 + -0.16666666666666666), $MachinePrecision] * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001

   1. Initial program 1.7%

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

      1. sqr-neg 1.7%

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

      2. +-commutative 1.7%

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

      3. sqr-neg 1.7%

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

      4. hypot-1-def 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.3500000000000001 < x < 1.30000000000000004

   1. Initial program 6.6%

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

      1. sqr-neg 6.6%

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

      2. +-commutative 6.6%

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

      3. sqr-neg 6.6%

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

      4. hypot-1-def 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 100.0%

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right) \cdot x} \]

      2. *-lft-identity 100.0%

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(0.075 \cdot {x}^{2} - 0.16666666666666666\right)\right) \cdot x \]

      3. *-commutative 100.0%

         \[\leadsto x + \color{blue}{\left(\left(0.075 \cdot {x}^{2} - 0.16666666666666666\right) \cdot {x}^{2}\right)} \cdot x \]

      4. associate-*l* 100.0%

         \[\leadsto x + \color{blue}{\left(0.075 \cdot {x}^{2} - 0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right)} \]

      5. *-commutative 100.0%

         \[\leadsto x + \left(\color{blue}{{x}^{2} \cdot 0.075} - 0.16666666666666666\right) \cdot \left({x}^{2} \cdot x\right) \]

      6. fmm-def 100.0%

         \[\leadsto x + \color{blue}{\mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right)} \cdot \left({x}^{2} \cdot x\right) \]

      7. metadata-eval 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.075, \color{blue}{-0.16666666666666666}\right) \cdot \left({x}^{2} \cdot x\right) \]

      8. unpow2 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) \]

      9. unpow3 100.0%

         \[\leadsto x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x + \mathsf{fma}\left({x}^{2}, 0.075, -0.16666666666666666\right) \cdot {x}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 1.30000000000000004 < x

   1. Initial program 50.4%

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

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.9e-6)
   (log (/ 1.0 (- (hypot 1.0 x) x)))
   (if (<= x 1.25) x (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -3.9e-6) {
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	} else if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.9e-6) {
		tmp = Math.log((1.0 / (Math.hypot(1.0, x) - x)));
	} else if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.9e-6:
		tmp = math.log((1.0 / (math.hypot(1.0, x) - x)))
	elif x <= 1.25:
		tmp = x
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.9e-6)
		tmp = log(Float64(1.0 / Float64(hypot(1.0, x) - x)));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.9e-6)
		tmp = log((1.0 / (hypot(1.0, x) - x)));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.9e-6], N[Log[N[(1.0 / N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], x, N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8999999999999999e-6

   1. Initial program 3.4%

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

      1. sqr-neg 3.4%

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

      2. +-commutative 3.4%

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

      3. sqr-neg 3.4%

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

      4. hypot-1-def 4.8%

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

   3. Simplified 4.8%

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

      1. flip-+ 3.1%

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

      2. div-sub 3.1%

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

      3. pow2 3.1%

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

      4. hypot-1-def 3.1%

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

      5. hypot-1-def 3.1%

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

      6. add-sqr-sqrt 3.1%

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

      7. +-commutative 3.1%

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

      8. fma-define 3.1%

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

   6. Applied egg-rr 3.1%

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

      1. div-sub 3.1%

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

      2. fma-undefine 3.1%

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

      3. unpow2 3.1%

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

      4. associate--r+ 44.9%

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

      5. +-inverses 99.8%

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

      6. metadata-eval 99.8%

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

      7. metadata-eval 99.8%

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

      8. associate-/r* 99.8%

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

      9. neg-mul-1 99.8%

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

      10. neg-sub0 99.8%

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

      11. associate--r- 99.8%

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

      12. neg-sub0 99.8%

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

      13. +-commutative 99.8%

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

      14. sub-neg 99.8%

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

   8. Simplified 99.8%

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

   if -3.8999999999999999e-6 < x < 1.25

   1. Initial program 5.9%

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

      1. sqr-neg 5.9%

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

      2. +-commutative 5.9%

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

      3. sqr-neg 5.9%

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

      4. hypot-1-def 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1.25 < x

   1. Initial program 50.4%

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

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.7%

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

      1. sqr-neg 1.7%

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

      2. +-commutative 1.7%

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

      3. sqr-neg 1.7%

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

      4. hypot-1-def 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 6.6%

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

      1. sqr-neg 6.6%

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

      2. +-commutative 6.6%

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

      3. sqr-neg 6.6%

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

      4. hypot-1-def 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in 99.8%

         \[\leadsto \color{blue}{1 \cdot x + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x} \]

      2. *-lft-identity 99.8%

         \[\leadsto \color{blue}{x} + \left(-0.16666666666666666 \cdot {x}^{2}\right) \cdot x \]

      3. associate-*l* 99.8%

         \[\leadsto x + \color{blue}{-0.16666666666666666 \cdot \left({x}^{2} \cdot x\right)} \]

      4. unpow2 99.8%

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

      5. unpow3 99.8%

         \[\leadsto x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{x + -0.16666666666666666 \cdot {x}^{3}} \]

   if 1.25 < x

   1. Initial program 50.4%

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

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

Alternative 4: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (log (/ -0.5 x))
   (if (<= x 1.25) (/ (* x (+ x 2.0)) (+ x 2.0)) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x * (x + 2.0)) / (x + 2.0);
	} else {
		tmp = log((x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = (x * (x + 2.0d0)) / (x + 2.0d0)
    else
        tmp = log((x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x * (x + 2.0)) / (x + 2.0);
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = (x * (x + 2.0)) / (x + 2.0)
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(Float64(x * Float64(x + 2.0)) / Float64(x + 2.0));
	else
		tmp = log(Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = (x * (x + 2.0)) / (x + 2.0);
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 1.7%

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

      1. sqr-neg 1.7%

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

      2. +-commutative 1.7%

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

      3. sqr-neg 1.7%

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

      4. hypot-1-def 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around -inf 100.0%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 6.6%

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

      1. sqr-neg 6.6%

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

      2. +-commutative 6.6%

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

      3. sqr-neg 6.6%

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

      4. hypot-1-def 6.6%

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

   3. Simplified 6.6%

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

      1. expm1-log1p-u 6.6%

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

      2. expm1-undefine 6.6%

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

      3. log1p-undefine 6.6%

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

      4. rem-exp-log 6.6%

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

   6. Applied egg-rr 6.6%

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

   7. Taylor expanded in x around 0 6.3%

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

      1. +-commutative 6.3%

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

   9. Simplified 6.3%

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

       1. flip-- 6.3%

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

       2. metadata-eval 6.3%

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

       3. difference-of-sqr-1 6.3%

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

       4. associate-+l+ 6.3%

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

       5. metadata-eval 6.3%

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

       6. associate--l+ 99.6%

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

       7. metadata-eval 99.6%

          \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

       8. +-rgt-identity 99.6%

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

       9. associate-+l+ 99.6%

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

       10. metadata-eval 99.6%

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

   11. Applied egg-rr 99.6%

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

   if 1.25 < x

   1. Initial program 50.4%

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

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = x;
	} else {
		tmp = log((x * 2.0));
	}
	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 * 2.0d0))
    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 * 2.0));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 5.2%

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

      1. sqr-neg 5.2%

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

      2. +-commutative 5.2%

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

      3. sqr-neg 5.2%

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

      4. hypot-1-def 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around 0 72.8%

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

   if 1.25 < x

   1. Initial program 50.4%

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

      1. sqr-neg 50.4%

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

      2. +-commutative 50.4%

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

      3. sqr-neg 50.4%

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

      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. Taylor expanded in x around inf 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

Alternative 6: 52.7% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{x + 2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* x 2.0) (+ x 2.0)))

C

double code(double x) {
	return (x * 2.0) / (x + 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * 2.0d0) / (x + 2.0d0)
end function

Java

public static double code(double x) {
	return (x * 2.0) / (x + 2.0);
}

Python

def code(x):
	return (x * 2.0) / (x + 2.0)

Julia

function code(x)
	return Float64(Float64(x * 2.0) / Float64(x + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (x * 2.0) / (x + 2.0);
end

Wolfram

code[x_] := N[(N[(x * 2.0), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.3%

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

   1. sqr-neg 19.3%

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

   2. +-commutative 19.3%

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

   3. sqr-neg 19.3%

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

   4. hypot-1-def 35.1%

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

3. Simplified 35.1%

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

   1. expm1-log1p-u 34.0%

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

   2. expm1-undefine 34.0%

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

   3. log1p-undefine 34.0%

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

   4. rem-exp-log 35.1%

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

6. Applied egg-rr 35.1%

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

7. Taylor expanded in x around 0 5.7%

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

   1. +-commutative 5.7%

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

9. Simplified 5.7%

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

    1. flip-- 5.5%

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

    2. metadata-eval 5.5%

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

    3. difference-of-sqr-1 5.5%

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

    4. associate-+l+ 5.5%

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

    5. metadata-eval 5.5%

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

    6. associate--l+ 51.5%

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

    7. metadata-eval 51.5%

       \[\leadsto \frac{\left(x + 2\right) \cdot \left(x + \color{blue}{0}\right)}{\left(x + 1\right) + 1} \]

    8. +-rgt-identity 51.5%

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

    9. associate-+l+ 51.5%

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

    10. metadata-eval 51.5%

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

11. Applied egg-rr 51.5%

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

12. Taylor expanded in x around 0 53.6%

    \[\leadsto \frac{\color{blue}{2 \cdot x}}{x + 2} \]
13. Step-by-step derivation

    1. *-commutative 53.6%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]

14. Simplified 53.6%

    \[\leadsto \frac{\color{blue}{x \cdot 2}}{x + 2} \]
15. Add Preprocessing

Alternative 7: 52.0% 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 19.3%

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

   1. sqr-neg 19.3%

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

   2. +-commutative 19.3%

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

   3. sqr-neg 19.3%

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

   4. hypot-1-def 35.1%

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

3. Simplified 35.1%

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

5. Taylor expanded in x around 0 51.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer Target 1: 30.1% 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 2024157 
(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)))))