Hyperbolic arcsine

Percentage Accurate: 17.5% → 99.8%

Time: 11.7s

Alternatives: 11

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: 17.5% 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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00095:\\ \;\;\;\;0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00095)
   (- 0.0 (log (- (hypot 1.0 x) x)))
   (if (<= x 0.9)
     (* x (+ 1.0 (* (* x x) -0.16666666666666666)))
     (log (+ (* x 2.0) (/ (+ 0.5 (/ -0.125 (* x x))) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00095) {
		tmp = 0.0 - log((hypot(1.0, x) - x));
	} else if (x <= 0.9) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.00095) {
		tmp = 0.0 - Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.9) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = Math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00095:
		tmp = 0.0 - math.log((math.hypot(1.0, x) - x))
	elif x <= 0.9:
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00095)
		tmp = Float64(0.0 - log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.9)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00095)
		tmp = 0.0 - log((hypot(1.0, x) - x));
	elseif (x <= 0.9)
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00095], N[(0.0 - N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.9], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.49999999999999998e-4

   1. Initial program 5.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 + x \cdot x}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 \cdot 1 + x \cdot x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{x - \sqrt{1 \cdot 1 + x \cdot x}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 \cdot 1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x + 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      13. hypot-undefine N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 5.2%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. Applied egg-rr 5.2%

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

      1. clear-num N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{1}{\frac{x \cdot x - \left(x \cdot x + 1\right)}{x - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right) \]

   6. Applied egg-rr 99.9%

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

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(\left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(\left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right)\right) \]

      4. hypot-undefine N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right)\right) \]

      5. hypot-lowering-hypot.f64 99.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(\mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.9%

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

   if -9.49999999999999998e-4 < x < 0.900000000000000022

   1. Initial program 6.3%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 0.900000000000000022 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00095:\\ \;\;\;\;0 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00105:\\ \;\;\;\;\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)\\ \mathbf{elif}\;x \leq 0.9:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.00105)
   (log (/ -1.0 (- x (hypot 1.0 x))))
   (if (<= x 0.9)
     (* x (+ 1.0 (* (* x x) -0.16666666666666666)))
     (log (+ (* x 2.0) (/ (+ 0.5 (/ -0.125 (* x x))) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	} else if (x <= 0.9) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.00105) {
		tmp = Math.log((-1.0 / (x - Math.hypot(1.0, x))));
	} else if (x <= 0.9) {
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	} else {
		tmp = Math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.00105:
		tmp = math.log((-1.0 / (x - math.hypot(1.0, x))))
	elif x <= 0.9:
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666))
	else:
		tmp = math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.00105)
		tmp = log(Float64(-1.0 / Float64(x - hypot(1.0, x))));
	elseif (x <= 0.9)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666)));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00105)
		tmp = log((-1.0 / (x - hypot(1.0, x))));
	elseif (x <= 0.9)
		tmp = x * (1.0 + ((x * x) * -0.16666666666666666));
	else
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.00105], N[Log[N[(-1.0 / N[(x - N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.9], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00104999999999999994

   1. Initial program 5.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 + x \cdot x}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 \cdot 1 + x \cdot x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{x - \sqrt{1 \cdot 1 + x \cdot x}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 \cdot 1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x + 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      13. hypot-undefine N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 5.2%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. Applied egg-rr 5.2%

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

      1. associate--r+ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\left(x \cdot x - x \cdot x\right) - 1\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

      2. +-inverses N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(0 - 1\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

      3. metadata-eval 99.9%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   6. Applied egg-rr 99.9%

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

   if -0.00104999999999999994 < x < 0.900000000000000022

   1. Initial program 6.3%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2}\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-1}{6}}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-1}{6}\right)\right)\right) \]

      6. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-1}{6}\right)\right)\right) \]

   5. Simplified 100.0%

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

   if 0.900000000000000022 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0 - \log \left(x \cdot \left(\frac{0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (-
    0.0
    (log (* x (+ (/ 0.125 (* x (* x (* x x)))) (- -2.0 (/ 0.5 (* x x)))))))
   (if (<= x 1.0)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ (+ 0.5 (/ -0.125 (* x x))) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0 - log((x * ((0.125 / (x * (x * (x * x)))) + (-2.0 - (0.5 / (x * x))))));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0 - log((x * ((0.125d0 / (x * (x * (x * x)))) + ((-2.0d0) - (0.5d0 / (x * x))))))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + ((0.5d0 + ((-0.125d0) / (x * x))) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0 - Math.log((x * ((0.125 / (x * (x * (x * x)))) + (-2.0 - (0.5 / (x * x))))));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0 - math.log((x * ((0.125 / (x * (x * (x * x)))) + (-2.0 - (0.5 / (x * x))))))
	elif x <= 1.0:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(0.0 - log(Float64(x * Float64(Float64(0.125 / Float64(x * Float64(x * Float64(x * x)))) + Float64(-2.0 - Float64(0.5 / Float64(x * x)))))));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0 - log((x * ((0.125 / (x * (x * (x * x)))) + (-2.0 - (0.5 / (x * x))))));
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(0.0 - N[Log[N[(x * N[(N[(0.125 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 3.7%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 + x \cdot x}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 \cdot 1 + x \cdot x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{x - \sqrt{1 \cdot 1 + x \cdot x}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 \cdot 1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x + 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      13. hypot-undefine N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 3.4%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. Applied egg-rr 3.4%

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

      1. clear-num N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{1}{\frac{x \cdot x - \left(x \cdot x + 1\right)}{x - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(0 - \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right) \]

      4. associate-+l- N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\left(0 - \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right) + \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\frac{\frac{1}{8}}{{x}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\frac{\frac{1}{8}}{{x}^{4}} - \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{1}{8}}{{x}^{4}} - \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{\frac{1}{8}}{{x}^{4}} + \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{{x}^{4}}\right), \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   9. Simplified 99.1%

      \[\leadsto -\log \color{blue}{\left(x \cdot \left(\frac{0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\right)} \]

   if -1 < x < 1

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0 - \log \left(x \cdot \left(\frac{0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{\frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(-0.5 + \frac{0.125}{x \cdot x}\right)}{x}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ (+ (/ -0.0625 (* x (* x (* x x)))) (+ -0.5 (/ 0.125 (* x x)))) x))
   (if (<= x 1.0)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ (+ 0.5 (/ -0.125 (* x x))) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log((((-0.0625 / (x * (x * (x * x)))) + (-0.5 + (0.125 / (x * x)))) / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log(((((-0.0625d0) / (x * (x * (x * x)))) + ((-0.5d0) + (0.125d0 / (x * x)))) / x))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + ((0.5d0 + ((-0.125d0) / (x * x))) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log((((-0.0625 / (x * (x * (x * x)))) + (-0.5 + (0.125 / (x * x)))) / x));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log((((-0.0625 / (x * (x * (x * x)))) + (-0.5 + (0.125 / (x * x)))) / x))
	elif x <= 1.0:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(Float64(-0.0625 / Float64(x * Float64(x * Float64(x * x)))) + Float64(-0.5 + Float64(0.125 / Float64(x * x)))) / x));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log((((-0.0625 / (x * (x * (x * x)))) + (-0.5 + (0.125 / (x * x)))) / x));
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(N[(-0.0625 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 3.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(\left(\frac{1}{2} + \frac{\frac{1}{16}}{{x}^{4}}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right) \]

   5. Simplified 99.1%

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-0.0625}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)} + \left(-0.5 + \frac{0.125}{x \cdot x}\right)}{x}\right)} \]

   if -1.1000000000000001 < x < 1

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(x \cdot \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- 0.0 (log (* x (- -2.0 (/ 0.5 (* x x))))))
   (if (<= x 1.0)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ (+ 0.5 (/ -0.125 (* x x))) x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - log((x * (-2.0 - (0.5 / (x * x)))));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.0d0 - log((x * ((-2.0d0) - (0.5d0 / (x * x)))))
    else if (x <= 1.0d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + ((0.5d0 + ((-0.125d0) / (x * x))) / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - Math.log((x * (-2.0 - (0.5 / (x * x)))));
	} else if (x <= 1.0) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.0 - math.log((x * (-2.0 - (0.5 / (x * x)))))
	elif x <= 1.0:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.0 - log(Float64(x * Float64(-2.0 - Float64(0.5 / Float64(x * x))))));
	elseif (x <= 1.0)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.0 - log((x * (-2.0 - (0.5 / (x * x)))));
	elseif (x <= 1.0)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + ((0.5 + (-0.125 / (x * x))) / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.0 - N[Log[N[(x * N[(-2.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.0], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 3.7%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 + x \cdot x}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 \cdot 1 + x \cdot x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{x - \sqrt{1 \cdot 1 + x \cdot x}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 \cdot 1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x + 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      13. hypot-undefine N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 3.4%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. Applied egg-rr 3.4%

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

      1. clear-num N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{1}{\frac{x \cdot x - \left(x \cdot x + 1\right)}{x - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(2\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(-2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(-2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   9. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right) \]

   5. Simplified 100.0%

      \[\leadsto \log \color{blue}{\left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(x \cdot \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(x \cdot \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- 0.0 (log (* x (- -2.0 (/ 0.5 (* x x))))))
   (if (<= x 1.05)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - log((x * (-2.0 - (0.5 / (x * x)))));
	} else if (x <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = 0.0d0 - log((x * ((-2.0d0) - (0.5d0 / (x * x)))))
    else if (x <= 1.05d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = 0.0 - Math.log((x * (-2.0 - (0.5 / (x * x)))));
	} else if (x <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = 0.0 - math.log((x * (-2.0 - (0.5 / (x * x)))))
	elif x <= 1.05:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(0.0 - log(Float64(x * Float64(-2.0 - Float64(0.5 / Float64(x * x))))));
	elseif (x <= 1.05)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = 0.0 - log((x * (-2.0 - (0.5 / (x * x)))));
	elseif (x <= 1.05)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(0.0 - N[Log[N[(x * N[(-2.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   1. Initial program 3.7%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 + x \cdot x}\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + \sqrt{1 \cdot 1 + x \cdot x}\right)\right) \]

      3. flip-+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}}{x - \sqrt{1 \cdot 1 + x \cdot x}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(x \cdot x - \sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\sqrt{1 \cdot 1 + x \cdot x} \cdot \sqrt{1 \cdot 1 + x \cdot x}\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      7. rem-square-sqrt N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 \cdot 1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(1 + x \cdot x\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot x + 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\left(x \cdot x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \left(x - \sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\sqrt{1 \cdot 1 + x \cdot x}\right)\right)\right)\right) \]

      13. hypot-undefine N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right)\right) \]

      14. hypot-lowering-hypot.f64 3.4%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, x\right), 1\right)\right), \mathsf{\_.f64}\left(x, \mathsf{hypot.f64}\left(1, x\right)\right)\right)\right) \]

   4. Applied egg-rr 3.4%

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

      1. clear-num N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{x - \sqrt{1 \cdot 1 + x \cdot x}}{x \cdot x - \left(x \cdot x + 1\right)}\right)\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{1}{\frac{x \cdot x - \left(x \cdot x + 1\right)}{x - \sqrt{1 \cdot 1 + x \cdot x}}}\right)\right) \]

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      4. distribute-neg-in N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(2\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(-2 + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \left(-2 - \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(-2, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right) \]

   9. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1.05000000000000004

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} \cdot x\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{{x}^{2}} \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{{x}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{x \cdot x}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{x}}{x}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{1 \cdot x}}{x}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2}}{1} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

      15. /-lowering-/.f64 99.9%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right) \]

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;0 - \log \left(x \cdot \left(-2 - \frac{0.5}{x \cdot x}\right)\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1:\\ \;\;\;\;\log \left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.1)
   (log (/ (+ -0.5 (/ 0.125 (* x x))) x))
   (if (<= x 1.05)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.1d0)) then
        tmp = log((((-0.5d0) + (0.125d0 / (x * x))) / x))
    else if (x <= 1.05d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.1) {
		tmp = Math.log(((-0.5 + (0.125 / (x * x))) / x));
	} else if (x <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.1:
		tmp = math.log(((-0.5 + (0.125 / (x * x))) / x))
	elif x <= 1.05:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.1)
		tmp = log(Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x));
	elseif (x <= 1.05)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.1)
		tmp = log(((-0.5 + (0.125 / (x * x))) / x));
	elseif (x <= 1.05)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.1], N[Log[N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.05], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001

   1. Initial program 3.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{-1 \cdot \left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}{x}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)}{x}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)}{x}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right) + \frac{1}{2}\right)\right)}{x}\right)\right) \]

      5. distribute-neg-in N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x}\right)\right) \]

      6. remove-double-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}{x}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\frac{\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}}{x}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} - \frac{1}{2}\right), x\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), x\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \frac{-1}{2}\right), x\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), x\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)\right), x\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{8} \cdot 1}{{x}^{2}}\right)\right), x\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{\frac{1}{8}}{{x}^{2}}\right)\right), x\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \left({x}^{2}\right)\right)\right), x\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \left(x \cdot x\right)\right)\right), x\right)\right) \]

      17. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right)\right), x\right)\right) \]

   5. Simplified 98.9%

      \[\leadsto \log \color{blue}{\left(\frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right)} \]

   if -1.1000000000000001 < x < 1.05000000000000004

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} \cdot x\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{{x}^{2}} \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{{x}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{x \cdot x}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{x}}{x}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{1 \cdot x}}{x}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2}}{1} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

      15. /-lowering-/.f64 99.9%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right) \]

   5. Simplified 99.9%

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

Alternative 8: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.05:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2 + \frac{0.5}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.05)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (log (+ (* x 2.0) (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = log((-0.5 / x));
	} else if (x <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = log(((x * 2.0) + (0.5 / 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 <= 1.05d0) then
        tmp = x * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    else
        tmp = log(((x * 2.0d0) + (0.5d0 / 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 <= 1.05) {
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} else {
		tmp = Math.log(((x * 2.0) + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.log((-0.5 / x))
	elif x <= 1.05:
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	else:
		tmp = math.log(((x * 2.0) + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 1.05)
		tmp = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	else
		tmp = log(Float64(Float64(x * 2.0) + Float64(0.5 / 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 <= 1.05)
		tmp = x * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	else
		tmp = log(((x * 2.0) + (0.5 / 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, 1.05], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right) \]

   5. Simplified 98.6%

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

   if -1.25 < x < 1.05000000000000004

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1.05000000000000004 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(x \cdot \left(2 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{log.f64}\left(\left(2 \cdot x + \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(2 \cdot x\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot x\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}} \cdot x\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{{x}^{2}} \cdot x\right)\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{{x}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot x}{x \cdot x}\right)\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{x}}{x}\right)\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2} \cdot x}{1 \cdot x}}{x}\right)\right)\right) \]

      11. times-frac N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{\frac{1}{2}}{1} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot \frac{x}{x}}{x}\right)\right)\right) \]

      13. *-inverses N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\frac{\frac{1}{2}}{x}\right)\right)\right) \]

      15. /-lowering-/.f64 99.9%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right)\right) \]

   5. Simplified 99.9%

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

Alternative 9: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (log (/ -0.5 x))
   (if (<= x 1.25)
     (*
      x
      (+
       1.0
       (*
        (* x x)
        (+
         -0.16666666666666666
         (* x (* x (+ 0.075 (* (* x x) -0.044642857142857144))))))))
     (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 * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} 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 * (1.0d0 + ((x * x) * ((-0.16666666666666666d0) + (x * (x * (0.075d0 + ((x * x) * (-0.044642857142857144d0))))))))
    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 * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	} 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 * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))))
	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 = Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.075 + Float64(Float64(x * x) * -0.044642857142857144))))))));
	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 * (1.0 + ((x * x) * (-0.16666666666666666 + (x * (x * (0.075 + ((x * x) * -0.044642857142857144)))))));
	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], N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.075 + N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 3.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{-1}{2}}{x}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 98.6%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right) \]

   5. Simplified 98.6%

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

   if -1.25 < x < 1.25

   1. Initial program 6.9%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{6} + \color{blue}{{x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{3}{40}} + \frac{-5}{112} \cdot {x}^{2}\right)\right)\right)\right)\right)\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left(x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \color{blue}{\left(x \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \color{blue}{\left(\frac{-5}{112} \cdot {x}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \left({x}^{2} \cdot \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{-5}{112}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. *-lowering-*.f64 99.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{3}{40}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{-5}{112}\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.075 + \left(x \cdot x\right) \cdot -0.044642857142857144\right)\right)\right)\right)} \]

   if 1.25 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.2%

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

Alternative 10: 76.1% accurate, 1.9× 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 6.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 76.3%

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

   if 1.25 < x

   1. Initial program 44.1%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, \color{blue}{x}\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.2%

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

Alternative 11: 52.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 13.8%

   \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 61.9%

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

Developer Target 1: 29.1% 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 2024161 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1))))))

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