Hyperbolic arcsine

Percentage Accurate: 18.6% → 99.7%

Time: 9.8s

Alternatives: 9

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: 18.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (log (/ -0.5 x))
   (if (<= x 0.008)
     (+ x (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.075 (pow x 5.0))))
     (* 2.0 (log (sqrt (+ x (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.008) {
		tmp = x + ((-0.16666666666666666 * pow(x, 3.0)) + (0.075 * pow(x, 5.0)));
	} else {
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.008) {
		tmp = x + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (0.075 * Math.pow(x, 5.0)));
	} else {
		tmp = 2.0 * Math.log(Math.sqrt((x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.32:
		tmp = math.log((-0.5 / x))
	elif x <= 0.008:
		tmp = x + ((-0.16666666666666666 * math.pow(x, 3.0)) + (0.075 * math.pow(x, 5.0)))
	else:
		tmp = 2.0 * math.log(math.sqrt((x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.008)
		tmp = Float64(x + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.075 * (x ^ 5.0))));
	else
		tmp = Float64(2.0 * log(sqrt(Float64(x + hypot(1.0, x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.32)
		tmp = log((-0.5 / x));
	elseif (x <= 0.008)
		tmp = x + ((-0.16666666666666666 * (x ^ 3.0)) + (0.075 * (x ^ 5.0)));
	else
		tmp = 2.0 * log(sqrt((x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.008], N[(x + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Log[N[Sqrt[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000006

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.32000000000000006 < x < 0.0080000000000000002

   1. Initial program 7.5%

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

      1. sqr-neg 7.5%

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

      2. +-commutative 7.5%

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

      3. sqr-neg 7.5%

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

      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 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)} \]

   if 0.0080000000000000002 < x

   1. Initial program 39.0%

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

      1. sqr-neg 39.0%

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

      2. +-commutative 39.0%

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

      3. sqr-neg 39.0%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. pow2 99.9%

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

      3. log-pow 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \log \left(\sqrt{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (log (/ -0.5 x))
   (if (<= x 0.008)
     (+ x (+ (* -0.16666666666666666 (pow x 3.0)) (* 0.075 (pow x 5.0))))
     (- (log (/ 1.0 (+ x (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.008) {
		tmp = x + ((-0.16666666666666666 * pow(x, 3.0)) + (0.075 * pow(x, 5.0)));
	} else {
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.008) {
		tmp = x + ((-0.16666666666666666 * Math.pow(x, 3.0)) + (0.075 * Math.pow(x, 5.0)));
	} else {
		tmp = -Math.log((1.0 / (x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.32:
		tmp = math.log((-0.5 / x))
	elif x <= 0.008:
		tmp = x + ((-0.16666666666666666 * math.pow(x, 3.0)) + (0.075 * math.pow(x, 5.0)))
	else:
		tmp = -math.log((1.0 / (x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.008)
		tmp = Float64(x + Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(0.075 * (x ^ 5.0))));
	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 <= -1.32)
		tmp = log((-0.5 / x));
	elseif (x <= 0.008)
		tmp = x + ((-0.16666666666666666 * (x ^ 3.0)) + (0.075 * (x ^ 5.0)));
	else
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.008], N[(x + N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(1.0 / N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.32000000000000006

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.32000000000000006 < x < 0.0080000000000000002

   1. Initial program 7.5%

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

      1. sqr-neg 7.5%

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

      2. +-commutative 7.5%

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

      3. sqr-neg 7.5%

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

      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 0 100.0%

      \[\leadsto \color{blue}{x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)} \]

   if 0.0080000000000000002 < x

   1. Initial program 39.0%

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

      1. sqr-neg 39.0%

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

      2. +-commutative 39.0%

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

      3. sqr-neg 39.0%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

      1. flip-+ 2.9%

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

      2. frac-2neg 2.9%

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

      3. log-div 2.9%

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

      4. pow2 2.9%

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

      5. hypot-1-def 2.9%

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

      6. hypot-1-def 2.9%

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

      7. add-sqr-sqrt 2.9%

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

      8. +-commutative 2.9%

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

      9. fma-define 2.9%

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

   6. Applied egg-rr 2.9%

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

      1. neg-sub0 2.9%

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

      2. associate--r- 2.9%

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

      3. neg-sub0 2.9%

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

      4. +-commutative 2.9%

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

      5. sub-neg 2.9%

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

      6. fma-undefine 2.9%

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

      7. unpow2 2.9%

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

      8. +-commutative 2.9%

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

      9. associate--l+ 4.0%

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

      10. +-inverses 6.0%

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

      11. metadata-eval 6.0%

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

      12. metadata-eval 6.0%

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

      13. neg-sub0 6.0%

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

      14. neg-sub0 6.0%

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

      15. associate--r- 6.0%

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

      16. neg-sub0 6.0%

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

      17. +-commutative 6.0%

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

      18. sub-neg 6.0%

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

   8. Simplified 6.0%

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

      1. flip-- 4.2%

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

      2. +-commutative 4.2%

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

      3. div-inv 4.2%

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

      4. hypot-1-def 4.2%

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

      5. hypot-1-def 4.2%

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

      6. add-sqr-sqrt 4.2%

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

      7. +-commutative 4.2%

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

      8. fma-define 4.2%

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

      9. pow2 4.2%

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

   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%

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

       4. distribute-neg-frac 4.2%

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

       5. distribute-frac-neg2 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%

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

       8. neg-sub0 4.2%

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

       9. associate--r- 4.2%

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

       10. neg-sub0 4.2%

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

       11. +-commutative 4.2%

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

       12. sub-neg 4.2%

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

       13. fma-undefine 4.2%

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

       14. unpow2 4.2%

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

       15. associate--r+ 37.0%

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

       16. div-sub 37.0%

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

       17. +-inverses 99.9%

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

       18. metadata-eval 99.9%

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

       19. metadata-eval 99.9%

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

       20. metadata-eval 99.9%

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

   12. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.008:\\ \;\;\;\;x + \left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.001)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (- (log (/ 1.0 (+ x (hypot 1.0 x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.001) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.001) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = -Math.log((1.0 / (x + Math.hypot(1.0, x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.001:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = -math.log((1.0 / (x + math.hypot(1.0, x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.001)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	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 <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.001)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = -log((1.0 / (x + hypot(1.0, x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.001], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Log[N[(1.0 / N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.25 < x < 1e-3

   1. Initial program 6.9%

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

      1. sqr-neg 6.9%

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

      2. +-commutative 6.9%

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

      3. sqr-neg 6.9%

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

      4. hypot-1-def 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 1e-3 < x

   1. Initial program 39.7%

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

      1. sqr-neg 39.7%

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

      2. +-commutative 39.7%

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

      3. sqr-neg 39.7%

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

      4. hypot-1-def 99.7%

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

   3. Simplified 99.7%

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

      1. flip-+ 4.2%

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

      2. frac-2neg 4.2%

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

      3. log-div 4.3%

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

      4. pow2 4.3%

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

      5. hypot-1-def 4.3%

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

      6. hypot-1-def 4.3%

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

      7. add-sqr-sqrt 4.3%

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

      8. +-commutative 4.3%

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

      9. fma-define 4.3%

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

   6. Applied egg-rr 4.3%

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

      1. neg-sub0 4.3%

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

      2. associate--r- 4.3%

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

      3. neg-sub0 4.3%

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

      4. +-commutative 4.3%

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

      5. sub-neg 4.3%

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

      6. fma-undefine 4.3%

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

      7. unpow2 4.3%

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

      8. +-commutative 4.3%

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

      9. associate--l+ 5.4%

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

      10. +-inverses 7.3%

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

      11. metadata-eval 7.3%

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

      12. metadata-eval 7.3%

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

      13. neg-sub0 7.3%

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

      14. neg-sub0 7.3%

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

      15. associate--r- 7.3%

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

      16. neg-sub0 7.3%

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

      17. +-commutative 7.3%

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

      18. sub-neg 7.3%

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

   8. Simplified 7.3%

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

      1. flip-- 5.5%

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

      2. +-commutative 5.5%

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

      3. div-inv 5.5%

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

      4. hypot-1-def 5.5%

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

      5. hypot-1-def 5.5%

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

      6. add-sqr-sqrt 5.5%

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

      7. +-commutative 5.5%

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

      8. fma-define 5.5%

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

      9. pow2 5.5%

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

   10. Applied egg-rr 5.5%

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

       1. associate-*r/ 5.5%

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

       2. *-rgt-identity 5.5%

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

       3. remove-double-neg 5.5%

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

       4. distribute-neg-frac 5.5%

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

       5. distribute-frac-neg2 5.5%

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

       6. neg-mul-1 5.5%

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

       7. associate-/r* 5.5%

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

       8. neg-sub0 5.5%

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

       9. associate--r- 5.5%

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

       10. neg-sub0 5.5%

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

       11. +-commutative 5.5%

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

       12. sub-neg 5.5%

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

       13. fma-undefine 5.5%

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

       14. unpow2 5.5%

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

       15. associate--r+ 37.8%

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

       16. div-sub 37.8%

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

       17. +-inverses 99.7%

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

       18. metadata-eval 99.7%

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

       19. metadata-eval 99.7%

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

       20. metadata-eval 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.001:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-\log \left(\frac{1}{x + \mathsf{hypot}\left(1, x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00098:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.00098)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00098) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00098) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00098:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00098)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00098)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00098], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.25 < x < 9.7999999999999997e-4

   1. Initial program 6.9%

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

      1. sqr-neg 6.9%

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

      2. +-commutative 6.9%

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

      3. sqr-neg 6.9%

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

      4. hypot-1-def 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   if 9.7999999999999997e-4 < x

   1. Initial program 39.7%

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

      1. sqr-neg 39.7%

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

      2. +-commutative 39.7%

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

      3. sqr-neg 39.7%

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

      4. hypot-1-def 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00098:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.25d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.95d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 * (1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (1.0 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 0.95)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.25 < x < 0.94999999999999996

   1. Initial program 8.1%

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

      1. sqr-neg 8.1%

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

      2. +-commutative 8.1%

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

      3. sqr-neg 8.1%

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

      4. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in x around 0 99.5%

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

   if 0.94999999999999996 < x

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. +-commutative 38.1%

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

      3. sqr-neg 38.1%

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

      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 99.4%

      \[\leadsto \log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (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 <= -1.25) {
		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 <= (-1.25d0)) 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 <= -1.25) {
		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 <= -1.25:
		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 <= -1.25)
		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 <= -1.25)
		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, -1.25], 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 < -1.25

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.25 < x < 1.25

   1. Initial program 8.1%

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

      1. sqr-neg 8.1%

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

      2. +-commutative 8.1%

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

      3. sqr-neg 8.1%

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

      4. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in x around 0 99.5%

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

   if 1.25 < x

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. +-commutative 38.1%

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

      3. sqr-neg 38.1%

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

      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 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.5%

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

Alternative 7: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else 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 = log(((-0.5d0) / x))
    else 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 = Math.log((-0.5 / 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 <= -1.25:
		tmp = math.log((-0.5 / 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 <= -1.25)
		tmp = log(Float64(-0.5 / 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 <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[Log[N[(-0.5 / x), $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 < -1.25

   1. Initial program 2.1%

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

      1. sqr-neg 2.1%

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

      2. +-commutative 2.1%

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

      3. sqr-neg 2.1%

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

      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 -1.25 < x < 1.25

   1. Initial program 8.1%

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

      1. sqr-neg 8.1%

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

      2. +-commutative 8.1%

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

      3. sqr-neg 8.1%

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

      4. hypot-1-def 8.1%

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

   3. Simplified 8.1%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 1.25 < x

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. +-commutative 38.1%

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

      3. sqr-neg 38.1%

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

      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 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.2%

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

Alternative 8: 76.6% 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 6.4%

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

      1. sqr-neg 6.4%

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

      2. +-commutative 6.4%

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

      3. sqr-neg 6.4%

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

      4. hypot-1-def 6.7%

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

   3. Simplified 6.7%

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

   5. Taylor expanded in x around 0 72.6%

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

   if 1.25 < x

   1. Initial program 38.1%

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

      1. sqr-neg 38.1%

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

      2. +-commutative 38.1%

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

      3. sqr-neg 38.1%

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

      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 99.1%

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

      1. *-commutative 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 78.9%

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

Alternative 9: 52.7% 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 13.9%

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

   1. sqr-neg 13.9%

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

   2. +-commutative 13.9%

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

   3. sqr-neg 13.9%

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

   4. hypot-1-def 28.9%

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

3. Simplified 28.9%

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

5. Taylor expanded in x around 0 56.5%

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

6. Final simplification 56.5%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 30.5% 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 2024041 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (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)))))