Hyperbolic arcsine

Percentage Accurate: 17.9% → 99.9%

Time: 8.6s

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: 17.9% 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: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.7e-6)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 1e-5) x (+ (+ 1.0 (log (+ x (hypot 1.0 x)))) -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -5.7e-6) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 1e-5) {
		tmp = x;
	} else {
		tmp = (1.0 + log((x + hypot(1.0, x)))) + -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.7e-6) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 1e-5) {
		tmp = x;
	} else {
		tmp = (1.0 + Math.log((x + Math.hypot(1.0, x)))) + -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.7e-6:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 1e-5:
		tmp = x
	else:
		tmp = (1.0 + math.log((x + math.hypot(1.0, x)))) + -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.7e-6)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 1e-5)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 + log(Float64(x + hypot(1.0, x)))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.7e-6)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 1e-5)
		tmp = x;
	else
		tmp = (1.0 + log((x + hypot(1.0, x)))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.7e-6], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1e-5], x, N[(N[(1.0 + N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6999999999999996e-6

   1. Initial program 5.1%

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

      1. sqr-neg 5.1%

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

      2. +-commutative 5.1%

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

      3. sqr-neg 5.1%

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

      4. hypot-1-def 6.2%

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

   3. Simplified 6.2%

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

      1. flip-+ 4.9%

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

      2. frac-2neg 4.9%

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

      3. log-div 4.9%

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

      4. pow2 4.9%

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

      5. hypot-1-def 5.3%

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

      6. hypot-1-def 4.9%

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

      7. add-sqr-sqrt 5.7%

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

      8. +-commutative 5.7%

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

      9. fma-def 5.7%

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

   5. Applied egg-rr 5.7%

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

      1. neg-sub0 5.7%

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

      2. associate--r- 5.7%

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

      3. neg-sub0 5.7%

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

      4. +-commutative 5.7%

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

      5. sub-neg 5.7%

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

      6. fma-udef 5.7%

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

      7. unpow2 5.7%

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

      8. +-commutative 5.7%

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

      9. associate--l+ 55.4%

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

      10. +-inverses 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-sub0 99.9%

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

      14. neg-sub0 99.9%

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

      15. associate--r- 99.9%

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

      16. neg-sub0 99.9%

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

      17. +-commutative 99.9%

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

      18. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -5.6999999999999996e-6 < x < 1.00000000000000008e-5

   1. Initial program 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 1.00000000000000008e-5 < x

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. sqr-neg 55.9%

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

      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. Step-by-step derivation

      1. expm1-log1p-u 98.3%

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

      2. expm1-udef 98.5%

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

      3. log1p-udef 98.5%

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

      4. rem-exp-log 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 10^{-5}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1\\ \end{array} \]

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00047:\\ \;\;\;\;x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.00047)
     (+ x (+ (+ 1.0 (* -0.16666666666666666 (pow x 3.0))) -1.0))
     (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00047) {
		tmp = x + ((1.0 + (-0.16666666666666666 * pow(x, 3.0))) + -1.0);
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.00047) {
		tmp = x + ((1.0 + (-0.16666666666666666 * Math.pow(x, 3.0))) + -1.0);
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.00047:
		tmp = x + ((1.0 + (-0.16666666666666666 * math.pow(x, 3.0))) + -1.0)
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00047)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 3.0))) + -1.0));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.00047)
		tmp = x + ((1.0 + (-0.16666666666666666 * (x ^ 3.0))) + -1.0);
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00047], N[(x + N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < -1.26000000000000001

   1. Initial program 3.9%

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

      1. sqr-neg 3.9%

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

      2. +-commutative 3.9%

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

      3. sqr-neg 3.9%

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

      4. hypot-1-def 5.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 4.69999999999999986e-4

   1. Initial program 7.1%

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

      1. sqr-neg 7.1%

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

      2. +-commutative 7.1%

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

      3. sqr-neg 7.1%

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

      4. hypot-1-def 7.1%

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

   3. Simplified 7.1%

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

      1. expm1-log1p-u 7.1%

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

      2. expm1-udef 7.1%

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

      3. log1p-udef 7.1%

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

      4. rem-exp-log 7.1%

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

   5. Applied egg-rr 7.1%

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

   6. Taylor expanded in x around 0 6.9%

      \[\leadsto \color{blue}{\left(1 + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\right)} - 1 \]
   7. Step-by-step derivation

      1. associate-+r+ 6.9%

         \[\leadsto \color{blue}{\left(\left(1 + x\right) + -0.16666666666666666 \cdot {x}^{3}\right)} - 1 \]

      2. +-commutative 6.9%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} + -0.16666666666666666 \cdot {x}^{3}\right) - 1 \]

   8. Simplified 6.9%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) + -0.16666666666666666 \cdot {x}^{3}\right)} - 1 \]
   9. Step-by-step derivation

      1. sub-neg 6.9%

         \[\leadsto \color{blue}{\left(\left(x + 1\right) + -0.16666666666666666 \cdot {x}^{3}\right) + \left(-1\right)} \]

      2. associate-+l+ 6.9%

         \[\leadsto \color{blue}{\left(x + \left(1 + -0.16666666666666666 \cdot {x}^{3}\right)\right)} + \left(-1\right) \]

      3. associate-+l+ 99.7%

         \[\leadsto \color{blue}{x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + \left(-1\right)\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + \color{blue}{-1}\right) \]

   10. Applied egg-rr 99.7%

       \[\leadsto \color{blue}{x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)} \]

   if 4.69999999999999986e-4 < x

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. sqr-neg 55.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00047:\\ \;\;\;\;x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.7e-6)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 7.4e-6) x (log (+ x (hypot 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -5.7e-6) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 7.4e-6) {
		tmp = x;
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.7e-6) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 7.4e-6) {
		tmp = x;
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.7e-6:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 7.4e-6:
		tmp = x
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.7e-6)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 7.4e-6)
		tmp = x;
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.7e-6)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 7.4e-6)
		tmp = x;
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.7e-6], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 7.4e-6], x, N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6999999999999996e-6

   1. Initial program 5.1%

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

      1. sqr-neg 5.1%

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

      2. +-commutative 5.1%

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

      3. sqr-neg 5.1%

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

      4. hypot-1-def 6.2%

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

   3. Simplified 6.2%

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

      1. flip-+ 4.9%

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

      2. frac-2neg 4.9%

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

      3. log-div 4.9%

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

      4. pow2 4.9%

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

      5. hypot-1-def 5.3%

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

      6. hypot-1-def 4.9%

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

      7. add-sqr-sqrt 5.7%

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

      8. +-commutative 5.7%

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

      9. fma-def 5.7%

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

   5. Applied egg-rr 5.7%

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

      1. neg-sub0 5.7%

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

      2. associate--r- 5.7%

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

      3. neg-sub0 5.7%

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

      4. +-commutative 5.7%

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

      5. sub-neg 5.7%

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

      6. fma-udef 5.7%

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

      7. unpow2 5.7%

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

      8. +-commutative 5.7%

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

      9. associate--l+ 55.4%

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

      10. +-inverses 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. neg-sub0 99.9%

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

      14. neg-sub0 99.9%

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

      15. associate--r- 99.9%

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

      16. neg-sub0 99.9%

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

      17. +-commutative 99.9%

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

      18. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   if -5.6999999999999996e-6 < x < 7.4000000000000003e-6

   1. Initial program 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.5%

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

   3. Simplified 6.5%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 7.4000000000000003e-6 < x

   1. Initial program 55.9%

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

      1. sqr-neg 55.9%

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

      2. +-commutative 55.9%

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

      3. sqr-neg 55.9%

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

      4. hypot-1-def 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{-6}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 0.95)
     (+ x (+ (+ 1.0 (* -0.16666666666666666 (pow x 3.0))) -1.0))
     (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + ((1.0 + (-0.16666666666666666 * pow(x, 3.0))) + -1.0);
	} 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 <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 0.95d0) then
        tmp = x + ((1.0d0 + ((-0.16666666666666666d0) * (x ** 3.0d0))) + (-1.0d0))
    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 <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 0.95) {
		tmp = x + ((1.0 + (-0.16666666666666666 * Math.pow(x, 3.0))) + -1.0);
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 * (1.0 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 0.95:
		tmp = x + ((1.0 + (-0.16666666666666666 * math.pow(x, 3.0))) + -1.0)
	else:
		tmp = math.log(((x * 2.0) + (0.5 * (1.0 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.95)
		tmp = Float64(x + Float64(Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 3.0))) + -1.0));
	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 <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 0.95)
		tmp = x + ((1.0 + (-0.16666666666666666 * (x ^ 3.0))) + -1.0);
	else
		tmp = log(((x * 2.0) + (0.5 * (1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.95], N[(x + N[(N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $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 < -1.26000000000000001

   1. Initial program 3.9%

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

      1. sqr-neg 3.9%

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

      2. +-commutative 3.9%

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

      3. sqr-neg 3.9%

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

      4. hypot-1-def 5.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 0.94999999999999996

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. hypot-1-def 7.7%

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

   3. Simplified 7.7%

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

      1. expm1-log1p-u 7.7%

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

      2. expm1-udef 7.8%

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

      3. log1p-udef 7.8%

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

      4. rem-exp-log 7.8%

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

   5. Applied egg-rr 7.8%

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

   6. Taylor expanded in x around 0 7.1%

      \[\leadsto \color{blue}{\left(1 + \left(x + -0.16666666666666666 \cdot {x}^{3}\right)\right)} - 1 \]
   7. Step-by-step derivation

      1. associate-+r+ 7.1%

         \[\leadsto \color{blue}{\left(\left(1 + x\right) + -0.16666666666666666 \cdot {x}^{3}\right)} - 1 \]

      2. +-commutative 7.1%

         \[\leadsto \left(\color{blue}{\left(x + 1\right)} + -0.16666666666666666 \cdot {x}^{3}\right) - 1 \]

   8. Simplified 7.1%

      \[\leadsto \color{blue}{\left(\left(x + 1\right) + -0.16666666666666666 \cdot {x}^{3}\right)} - 1 \]
   9. Step-by-step derivation

      1. sub-neg 7.1%

         \[\leadsto \color{blue}{\left(\left(x + 1\right) + -0.16666666666666666 \cdot {x}^{3}\right) + \left(-1\right)} \]

      2. associate-+l+ 7.1%

         \[\leadsto \color{blue}{\left(x + \left(1 + -0.16666666666666666 \cdot {x}^{3}\right)\right)} + \left(-1\right) \]

      3. associate-+l+ 99.2%

         \[\leadsto \color{blue}{x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + \left(-1\right)\right)} \]

      4. metadata-eval 99.2%

         \[\leadsto x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + \color{blue}{-1}\right) \]

   10. Applied egg-rr 99.2%

       \[\leadsto \color{blue}{x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)} \]

   if 0.94999999999999996 < x

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. +-commutative 54.9%

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

      3. sqr-neg 54.9%

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

      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. Taylor expanded in x around inf 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (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 <= -1.26) {
		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 <= (-1.26d0)) 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 <= -1.26) {
		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 <= -1.26:
		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 <= -1.26)
		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 <= -1.26)
		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, -1.26], 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 < -1.26000000000000001

   1. Initial program 3.9%

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

      1. sqr-neg 3.9%

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

      2. +-commutative 3.9%

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

      3. sqr-neg 3.9%

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

      4. hypot-1-def 5.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 0.94999999999999996

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. hypot-1-def 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 0.94999999999999996 < x

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. +-commutative 54.9%

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

      3. sqr-neg 54.9%

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

      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. Taylor expanded in x around inf 98.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right)\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (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 <= -1.26) {
		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 <= (-1.26d0)) 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 <= -1.26) {
		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 <= -1.26:
		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 <= -1.26)
		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 <= -1.26)
		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, -1.26], 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 < -1.26000000000000001

   1. Initial program 3.9%

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

      1. sqr-neg 3.9%

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

      2. +-commutative 3.9%

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

      3. sqr-neg 3.9%

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

      4. hypot-1-def 5.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 1.25

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. hypot-1-def 7.7%

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

   3. Simplified 7.7%

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

   4. Taylor expanded in x around 0 99.2%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. +-commutative 54.9%

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

      3. sqr-neg 54.9%

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

      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. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 7: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.26)
   (log (/ -0.5 x))
   (if (<= x 1.25) (* (* x (+ x 2.0)) (/ 1.0 (+ x 2.0))) (log (* x 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x * (x + 2.0)) * (1.0 / (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 <= (-1.26d0)) then
        tmp = log(((-0.5d0) / x))
    else if (x <= 1.25d0) then
        tmp = (x * (x + 2.0d0)) * (1.0d0 / (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 <= -1.26) {
		tmp = Math.log((-0.5 / x));
	} else if (x <= 1.25) {
		tmp = (x * (x + 2.0)) * (1.0 / (x + 2.0));
	} else {
		tmp = Math.log((x * 2.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.26:
		tmp = math.log((-0.5 / x))
	elif x <= 1.25:
		tmp = (x * (x + 2.0)) * (1.0 / (x + 2.0))
	else:
		tmp = math.log((x * 2.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.25)
		tmp = Float64(Float64(x * Float64(x + 2.0)) * Float64(1.0 / 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 <= -1.26)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = (x * (x + 2.0)) * (1.0 / (x + 2.0));
	else
		tmp = log((x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], N[(N[(x * N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.26000000000000001

   1. Initial program 3.9%

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

      1. sqr-neg 3.9%

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

      2. +-commutative 3.9%

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

      3. sqr-neg 3.9%

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

      4. hypot-1-def 5.0%

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

   3. Simplified 5.0%

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

   4. Taylor expanded in x around -inf 99.0%

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

   if -1.26000000000000001 < x < 1.25

   1. Initial program 7.7%

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

      1. sqr-neg 7.7%

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

      2. +-commutative 7.7%

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

      3. sqr-neg 7.7%

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

      4. hypot-1-def 7.7%

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

   3. Simplified 7.7%

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

      1. expm1-log1p-u 7.7%

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

      2. expm1-udef 7.8%

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

      3. log1p-udef 7.8%

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

      4. rem-exp-log 7.8%

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

   5. Applied egg-rr 7.8%

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

   6. Taylor expanded in x around 0 6.9%

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

      1. flip-- 6.9%

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

      2. div-inv 6.9%

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

      3. +-commutative 6.9%

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

      4. +-commutative 6.9%

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

      5. metadata-eval 6.9%

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

      6. difference-of-sqr-1 6.9%

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

      7. associate-+l+ 6.9%

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

      8. metadata-eval 6.9%

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

      9. +-commutative 6.9%

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

      10. add-exp-log 6.9%

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

      11. log1p-udef 6.9%

          \[\leadsto \left(\left(x + 2\right) \cdot \left(e^{\color{blue}{\mathsf{log1p}\left(x\right)}} - 1\right)\right) \cdot \frac{1}{\left(1 + x\right) + 1} \]

      12. expm1-udef 99.0%

          \[\leadsto \left(\left(x + 2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x\right)\right)}\right) \cdot \frac{1}{\left(1 + x\right) + 1} \]

      13. expm1-log1p-u 99.0%

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

      14. +-commutative 99.0%

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

      15. associate-+l+ 99.1%

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

      16. metadata-eval 99.1%

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

   8. Applied egg-rr 99.1%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. +-commutative 54.9%

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

      3. sqr-neg 54.9%

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

      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. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\left(x \cdot \left(x + 2\right)\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 8: 75.3% accurate, 2.0× 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 6.5%

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

      1. sqr-neg 6.5%

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

      2. +-commutative 6.5%

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

      3. sqr-neg 6.5%

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

      4. hypot-1-def 6.8%

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

   3. Simplified 6.8%

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

   4. Taylor expanded in x around 0 67.9%

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

   if 1.25 < x

   1. Initial program 54.9%

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

      1. sqr-neg 54.9%

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

      2. +-commutative 54.9%

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

      3. sqr-neg 54.9%

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

      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. Taylor expanded in x around inf 97.9%

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

      1. *-commutative 97.9%

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

   6. Simplified 97.9%

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

4. Final simplification 73.0%

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

Alternative 9: 51.4% 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 14.6%

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

   1. sqr-neg 14.6%

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

   2. +-commutative 14.6%

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

   3. sqr-neg 14.6%

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

   4. hypot-1-def 22.5%

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

3. Simplified 22.5%

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

4. Taylor expanded in x around 0 57.5%

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

5. Final simplification 57.5%

   \[\leadsto x \]

Developer target: 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 2023306 
(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)))))