Hyperbolic arcsine

Percentage Accurate: 18.3% → 99.6%

Time: 11.0s

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: 18.3% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -0.008], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.32], 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[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0080000000000000002

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

      1. flip-+ 3.5%

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

      2. div-sub 3.5%

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

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. div-sub 4.3%

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

      2. +-commutative 4.3%

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

      3. associate--r+ 46.3%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. log-div 0.0%

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

       3. sub-neg 0.0%

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

       4. +-commutative 0.0%

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

       5. neg-sub0 0.0%

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

       6. associate-+l- 0.0%

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

       7. log-div 100.0%

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

       8. metadata-eval 100.0%

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

       9. associate-/l* 100.0%

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

       10. neg-sub0 100.0%

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

       11. /-rgt-identity 100.0%

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

       12. *-commutative 100.0%

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

       13. neg-mul-1 100.0%

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

       14. sub-neg 100.0%

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

       15. +-commutative 100.0%

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

       16. distribute-neg-in 100.0%

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

       17. remove-double-neg 100.0%

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

       18. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -0.0080000000000000002 < x < 1.32000000000000006

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   if 1.32000000000000006 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. remove-double-neg 99.9%

         \[\leadsto \log 2 + \color{blue}{\log x} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (- (log (- (* x -2.0) (/ 0.5 x))))
   (if (<= x 1.25)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (+ (log 2.0) (log x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(2.0) + log(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = -log(((x * (-2.0d0)) - (0.5d0 / x)))
    else if (x <= 1.25d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log(2.0d0) + log(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -Math.log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(2.0) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = -math.log(((x * -2.0) - (0.5 / x)))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(2.0) + math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(-log(Float64(Float64(x * -2.0) - Float64(0.5 / x))));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(log(2.0) + log(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = -log(((x * -2.0) - (0.5 / x)));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(2.0) + log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], (-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

      1. flip-+ 3.5%

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

      2. div-sub 3.5%

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

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. div-sub 4.3%

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

      2. +-commutative 4.3%

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

      3. associate--r+ 46.3%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. log-div 0.0%

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

       3. sub-neg 0.0%

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

       4. +-commutative 0.0%

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

       5. neg-sub0 0.0%

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

       6. associate-+l- 0.0%

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

       7. log-div 100.0%

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

       8. metadata-eval 100.0%

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

       9. associate-/l* 100.0%

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

       10. neg-sub0 100.0%

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

       11. /-rgt-identity 100.0%

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

       12. *-commutative 100.0%

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

       13. neg-mul-1 100.0%

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

       14. sub-neg 100.0%

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

       15. +-commutative 100.0%

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

       16. distribute-neg-in 100.0%

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

       17. remove-double-neg 100.0%

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

       18. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around -inf 99.0%

       \[\leadsto -\log \color{blue}{\left(-2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 99.0%

          \[\leadsto -\log \left(-2 \cdot x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]

       2. metadata-eval 99.0%

          \[\leadsto -\log \left(-2 \cdot x - \frac{\color{blue}{0.5}}{x}\right) \]

   14. Simplified 99.0%

       \[\leadsto -\log \color{blue}{\left(-2 \cdot x - \frac{0.5}{x}\right)} \]

   if -0.95999999999999996 < x < 1.25

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. remove-double-neg 99.9%

         \[\leadsto \log 2 + \color{blue}{\log x} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00092:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00092)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 1.25)
     (+ x (* -0.16666666666666666 (pow x 3.0)))
     (+ (log 2.0) (log x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00092) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log(2.0) + log(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.00092) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log(2.0) + Math.log(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00092:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log(2.0) + math.log(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00092)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = Float64(log(2.0) + log(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00092)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log(2.0) + log(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00092], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[Log[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.2000000000000003e-4

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

      1. flip-+ 3.5%

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

      2. div-sub 3.5%

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

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. div-sub 4.3%

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

      2. +-commutative 4.3%

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

      3. associate--r+ 46.3%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. log-div 0.0%

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

       3. sub-neg 0.0%

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

       4. +-commutative 0.0%

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

       5. neg-sub0 0.0%

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

       6. associate-+l- 0.0%

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

       7. log-div 100.0%

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

       8. metadata-eval 100.0%

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

       9. associate-/l* 100.0%

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

       10. neg-sub0 100.0%

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

       11. /-rgt-identity 100.0%

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

       12. *-commutative 100.0%

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

       13. neg-mul-1 100.0%

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

       14. sub-neg 100.0%

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

       15. +-commutative 100.0%

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

       16. distribute-neg-in 100.0%

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

       17. remove-double-neg 100.0%

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

       18. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   if -9.2000000000000003e-4 < x < 1.25

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 99.9%

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

      1. mul-1-neg 99.9%

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

      2. log-rec 99.9%

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

      3. remove-double-neg 99.9%

         \[\leadsto \log 2 + \color{blue}{\log x} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\log 2 + \log x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00092:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \log x\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in x around -inf 98.7%

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

   if -1.30000000000000004 < x < 1.25

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. count-2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 99.0%

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

Alternative 5: 99.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.96)
   (- (log (- (* x -2.0) (/ 0.5 x))))
   (if (<= x 1.25) (+ x (* -0.16666666666666666 (pow x 3.0))) (log (+ x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * pow(x, 3.0));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.96d0)) then
        tmp = -log(((x * (-2.0d0)) - (0.5d0 / x)))
    else if (x <= 1.25d0) then
        tmp = x + ((-0.16666666666666666d0) * (x ** 3.0d0))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.96) {
		tmp = -Math.log(((x * -2.0) - (0.5 / x)));
	} else if (x <= 1.25) {
		tmp = x + (-0.16666666666666666 * Math.pow(x, 3.0));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.96:
		tmp = -math.log(((x * -2.0) - (0.5 / x)))
	elif x <= 1.25:
		tmp = x + (-0.16666666666666666 * math.pow(x, 3.0))
	else:
		tmp = math.log((x + x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.96)
		tmp = Float64(-log(Float64(Float64(x * -2.0) - Float64(0.5 / x))));
	elseif (x <= 1.25)
		tmp = Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)));
	else
		tmp = log(Float64(x + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.96)
		tmp = -log(((x * -2.0) - (0.5 / x)));
	elseif (x <= 1.25)
		tmp = x + (-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = log((x + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.96], (-N[Log[N[(N[(x * -2.0), $MachinePrecision] - N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 1.25], N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.95999999999999996

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

      1. flip-+ 3.5%

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

      2. div-sub 3.5%

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

      3. hypot-udef 3.5%

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

      4. hypot-udef 3.5%

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

      5. add-sqr-sqrt 3.5%

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

      6. metadata-eval 3.5%

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

   5. Applied egg-rr 3.5%

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

      1. div-sub 4.3%

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

      2. +-commutative 4.3%

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

      3. associate--r+ 46.3%

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

      4. +-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

      1. *-un-lft-identity 100.0%

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

      2. *-commutative 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. +-rgt-identity 100.0%

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

       2. log-div 0.0%

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

       3. sub-neg 0.0%

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

       4. +-commutative 0.0%

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

       5. neg-sub0 0.0%

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

       6. associate-+l- 0.0%

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

       7. log-div 100.0%

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

       8. metadata-eval 100.0%

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

       9. associate-/l* 100.0%

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

       10. neg-sub0 100.0%

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

       11. /-rgt-identity 100.0%

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

       12. *-commutative 100.0%

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

       13. neg-mul-1 100.0%

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

       14. sub-neg 100.0%

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

       15. +-commutative 100.0%

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

       16. distribute-neg-in 100.0%

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

       17. remove-double-neg 100.0%

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

       18. sub-neg 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around -inf 99.0%

       \[\leadsto -\log \color{blue}{\left(-2 \cdot x - 0.5 \cdot \frac{1}{x}\right)} \]
   13. Step-by-step derivation

       1. associate-*r/ 99.0%

          \[\leadsto -\log \left(-2 \cdot x - \color{blue}{\frac{0.5 \cdot 1}{x}}\right) \]

       2. metadata-eval 99.0%

          \[\leadsto -\log \left(-2 \cdot x - \frac{\color{blue}{0.5}}{x}\right) \]

   14. Simplified 99.0%

       \[\leadsto -\log \color{blue}{\left(-2 \cdot x - \frac{0.5}{x}\right)} \]

   if -0.95999999999999996 < x < 1.25

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. count-2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

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 + 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 <= 1.25d0) then
        tmp = x
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = log((-0.5 / x));
	elseif (x <= 1.25)
		tmp = x;
	else
		tmp = log((x + 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, 1.25], x, N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.8%

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

      1. sqr-neg 3.8%

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

      2. +-commutative 3.8%

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

      3. sqr-neg 3.8%

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

      4. hypot-1-def 5.2%

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

   3. Simplified 5.2%

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

   4. Taylor expanded in x around -inf 98.7%

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

   if -1.25 < x < 1.25

   1. Initial program 8.6%

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

      1. sqr-neg 8.6%

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

      2. +-commutative 8.6%

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

      3. sqr-neg 8.6%

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

      4. hypot-1-def 8.6%

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

   3. Simplified 8.6%

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

   4. Taylor expanded in x around 0 99.1%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. count-2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 98.8%

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

Alternative 7: 58.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.6) x (log (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x;
	} else {
		tmp = log((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.6d0) then
        tmp = x
    else
        tmp = log((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = x;
	} else {
		tmp = Math.log((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = x
	else:
		tmp = math.log((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = x;
	else
		tmp = log(Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = x;
	else
		tmp = log((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.6], x, N[Log[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in x around 0 66.5%

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

   if 1.6000000000000001 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around 0 31.3%

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

      1. +-commutative 31.3%

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

   6. Simplified 31.3%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 1\right)\\ \end{array} \]

Alternative 8: 75.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 7.0%

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

      1. sqr-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. sqr-neg 7.0%

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

      4. hypot-1-def 7.4%

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

   3. Simplified 7.4%

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

   4. Taylor expanded in x around 0 66.5%

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

   if 1.25 < x

   1. Initial program 58.5%

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

      1. sqr-neg 58.5%

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

      2. +-commutative 58.5%

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

      3. sqr-neg 58.5%

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

      4. hypot-1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in x around inf 98.5%

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

      1. count-2 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 74.4%

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

Alternative 9: 51.7% 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 19.6%

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

   1. sqr-neg 19.6%

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

   2. +-commutative 19.6%

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

   3. sqr-neg 19.6%

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

   4. hypot-1-def 29.8%

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

3. Simplified 29.8%

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

4. Taylor expanded in x around 0 51.5%

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

5. Final simplification 51.5%

   \[\leadsto x \]

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 2023283 
(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)))))