Rust f64::asinh

Percentage Accurate: 30.1% → 99.3%

Time: 10.7s

Alternatives: 15

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 30.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Alternative 1: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \mathsf{fma}\left(\left|x\right| - t\_1, \left|x\right|, x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\left(\log \left({\left(\mathsf{fma}\left(t\_2, t\_2 - 1, 1\right)\right)}^{-1}\right) + \mathsf{log1p}\left({t\_2}^{3}\right)\right) - \left(\mathsf{log1p}\left(t\_2\right) - \log \left(t\_1 + \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (sqrt (fma x x 1.0)))
        (t_2 (fma (- (fabs x) t_1) (fabs x) (* x x))))
   (if (<= t_0 -10.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 10.0)
       (copysign
        (-
         (+ (log (pow (fma t_2 (- t_2 1.0) 1.0) -1.0)) (log1p (pow t_2 3.0)))
         (- (log1p t_2) (log (+ t_1 (fabs x)))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = sqrt(fma(x, x, 1.0));
	double t_2 = fma((fabs(x) - t_1), fabs(x), (x * x));
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 10.0) {
		tmp = copysign(((log(pow(fma(t_2, (t_2 - 1.0), 1.0), -1.0)) + log1p(pow(t_2, 3.0))) - (log1p(t_2) - log((t_1 + fabs(x))))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = sqrt(fma(x, x, 1.0))
	t_2 = fma(Float64(abs(x) - t_1), abs(x), Float64(x * x))
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 10.0)
		tmp = copysign(Float64(Float64(log((fma(t_2, Float64(t_2 - 1.0), 1.0) ^ -1.0)) + log1p((t_2 ^ 3.0))) - Float64(log1p(t_2) - log(Float64(t_1 + abs(x))))), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Abs[x], $MachinePrecision] - t$95$1), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[With[{TMP1 = Abs[N[(N[(N[Log[N[Power[N[(t$95$2 * N[(t$95$2 - 1.0), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] + N[Log[1 + N[Power[t$95$2, 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + t$95$2], $MachinePrecision] - N[Log[N[(t$95$1 + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10

   1. Initial program 52.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x \cdot \frac{\left|x\right|}{x}} + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 10

   1. Initial program 12.3%

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

   4. Applied rewrites 98.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) + \log \left(\frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, \left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \mathsf{fma}\left(x, x, 1\right)\right)}\right)}, x\right) \]

   6. Applied rewrites 98.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right) \cdot 1\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right) \cdot 1\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right) \cdot 1\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \left|x\right| + \mathsf{fma}\left(x, x, 1\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      5. lift-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \left|x\right| + \color{blue}{\left(x \cdot x + 1\right)}\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \left|x\right| + \left(\color{blue}{x \cdot x} + 1\right)\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      7. associate-+r+ N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \left|x\right| + x \cdot x\right) + 1\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)} + 1\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      10. flip3-+ N/A

          \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{1}^{3} + {\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3}}{1 \cdot 1 + \left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right) \cdot \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right) - 1 \cdot \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\mathsf{log1p}\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3}\right) + \log \left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right), \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right) - 1, 1\right)\right)}^{-1}\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

   if 10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\left(\log \left({\left(\mathsf{fma}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right), \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right) - 1, 1\right)\right)}^{-1}\right) + \mathsf{log1p}\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3}\right)\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \left|x\right| - t\_1\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{fma}\left(t\_2, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(t\_2, \left|x\right|, x \cdot x\right)\right) - \log \left(t\_1 + \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (sqrt (fma x x 1.0)))
        (t_2 (- (fabs x) t_1)))
   (if (<= t_0 (- INFINITY))
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 10.0)
       (copysign
        (-
         (log (fma t_2 (fabs x) (fma x x 1.0)))
         (- (log1p (fma t_2 (fabs x) (* x x))) (log (+ t_1 (fabs x)))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = sqrt(fma(x, x, 1.0));
	double t_2 = fabs(x) - t_1;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 10.0) {
		tmp = copysign((log(fma(t_2, fabs(x), fma(x, x, 1.0))) - (log1p(fma(t_2, fabs(x), (x * x))) - log((t_1 + fabs(x))))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = sqrt(fma(x, x, 1.0))
	t_2 = Float64(abs(x) - t_1)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 10.0)
		tmp = copysign(Float64(log(fma(t_2, abs(x), fma(x, x, 1.0))) - Float64(log1p(fma(t_2, abs(x), Float64(x * x))) - log(Float64(t_1 + abs(x))))), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[With[{TMP1 = Abs[N[(N[Log[N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(N[Log[1 + N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[N[(t$95$1 + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -inf.0

   1. Initial program 3.1%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x \cdot \frac{\left|x\right|}{x}} + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -inf.0 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 10

   1. Initial program 26.7%

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

   4. Applied rewrites 98.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) + \log \left(\frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, \left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \mathsf{fma}\left(x, x, 1\right)\right)}\right)}, x\right) \]

   6. Applied rewrites 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right) \cdot 1\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right) \cdot 1\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

      2. *-rgt-identity 98.9

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

   8. Applied rewrites 98.9%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)\right)} - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right) \]

   if 10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -\infty:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)\right) - \left(\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \left|x\right| - t\_1\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{t\_1 + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, t\_2, \mathsf{fma}\left(x, x, 1\right)\right)}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(x, x, t\_2 \cdot \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (sqrt (fma x x 1.0)))
        (t_2 (- (fabs x) t_1)))
   (if (<= t_0 -20.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 10.0)
       (copysign
        (+
         (log (/ (+ t_1 (fabs x)) (fma (fabs x) t_2 (fma x x 1.0))))
         (log1p (fma x x (* t_2 (fabs x)))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = sqrt(fma(x, x, 1.0));
	double t_2 = fabs(x) - t_1;
	double tmp;
	if (t_0 <= -20.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 10.0) {
		tmp = copysign((log(((t_1 + fabs(x)) / fma(fabs(x), t_2, fma(x, x, 1.0)))) + log1p(fma(x, x, (t_2 * fabs(x))))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = sqrt(fma(x, x, 1.0))
	t_2 = Float64(abs(x) - t_1)
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 10.0)
		tmp = copysign(Float64(log(Float64(Float64(t_1 + abs(x)) / fma(abs(x), t_2, fma(x, x, 1.0)))) + log1p(fma(x, x, Float64(t_2 * abs(x))))), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x * x + 1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[x], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$0, -20.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 10.0], N[With[{TMP1 = Abs[N[(N[Log[N[(N[(t$95$1 + N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] * t$95$2 + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Log[1 + N[(x * x + N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -20

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x \cdot \frac{\left|x\right|}{x}} + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -20 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 10

   1. Initial program 12.9%

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

   4. Applied rewrites 98.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right) + \log \left(\frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, \left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \mathsf{fma}\left(x, x, 1\right)\right)}\right)}, x\right) \]

   if 10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 44.9%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, \left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \mathsf{fma}\left(x, x, 1\right)\right)}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := \left|x\right| - -1\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{{t\_1}^{2}} - \frac{-0.125}{-1 - \left|x\right|}\right) \cdot x, x, \frac{0.5}{t\_1}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (- (fabs x) -1.0)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.5)
       (copysign
        (fma
         (fma
          (* (- (/ -0.125 (pow t_1 2.0)) (/ -0.125 (- -1.0 (fabs x)))) x)
          x
          (/ 0.5 t_1))
         (* x x)
         (log1p (fabs x)))
        x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = fabs(x) - -1.0;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(fma(fma((((-0.125 / pow(t_1, 2.0)) - (-0.125 / (-1.0 - fabs(x)))) * x), x, (0.5 / t_1)), (x * x), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = Float64(abs(x) - -1.0)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(fma(fma(Float64(Float64(Float64(-0.125 / (t_1 ^ 2.0)) - Float64(-0.125 / Float64(-1.0 - abs(x)))) * x), x, Float64(0.5 / t_1)), Float64(x * x), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[(N[(N[(N[(N[(-0.125 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - N[(-0.125 / N[(-1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.9

          \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{\frac{-0.5}{x}} - x\right)\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(\frac{-0.5}{x} - x\right)}\right), x\right) \]

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), x\right) \]

      3. remove-double-neg N/A

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

      4. lower-log.f64 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) + \log \left(1 + \left|x\right|\right)}, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right) \cdot {x}^{2}} + \log \left(1 + \left|x\right|\right), x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}, {x}^{2}, \log \left(1 + \left|x\right|\right)\right)}, x\right) \]

   8. Applied rewrites 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{{\left(1 + \left|x\right|\right)}^{2}} + \frac{-0.125}{1 + \left|x\right|}\right) \cdot x, x, \frac{0.5}{1 + \left|x\right|}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x - \frac{-0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{{\left(\left|x\right| - -1\right)}^{2}} - \frac{-0.125}{-1 - \left|x\right|}\right) \cdot x, x, \frac{0.5}{\left|x\right| - -1}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := -1 + \left|x\right|\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\frac{-0.125}{t\_1} - \frac{-0.125}{t\_1 \cdot t\_1}, x \cdot x, \frac{-0.5}{1 - \left|x\right|}\right) \cdot \left(x \cdot x\right) - \mathsf{log1p}\left(-\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (+ -1.0 (fabs x))))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.5)
       (copysign
        (-
         (*
          (fma
           (- (/ -0.125 t_1) (/ -0.125 (* t_1 t_1)))
           (* x x)
           (/ -0.5 (- 1.0 (fabs x))))
          (* x x))
         (log1p (- (fabs x))))
        x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = -1.0 + fabs(x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(((fma(((-0.125 / t_1) - (-0.125 / (t_1 * t_1))), (x * x), (-0.5 / (1.0 - fabs(x)))) * (x * x)) - log1p(-fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = Float64(-1.0 + abs(x))
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(Float64(Float64(fma(Float64(Float64(-0.125 / t_1) - Float64(-0.125 / Float64(t_1 * t_1))), Float64(x * x), Float64(-0.5 / Float64(1.0 - abs(x)))) * Float64(x * x)) - log1p(Float64(-abs(x)))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[(N[(N[(N[(N[(-0.125 / t$95$1), $MachinePrecision] - N[(-0.125 / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(-0.5 / N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-N[Abs[x], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.9

          \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{\frac{-0.5}{x}} - x\right)\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(\frac{-0.5}{x} - x\right)}\right), x\right) \]

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. frac-2neg N/A

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

      5. log-div N/A

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

      6. lower--.f64 N/A

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

   4. Applied rewrites 11.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(-x, x, \mathsf{fma}\left(x, x, 1\right)\right)\right) - \log \left(-\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)}, x\right) \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 - \left|x\right|} + 3 \cdot \frac{1}{{\left(1 - \left|x\right|\right)}^{2}}\right)\right) - \frac{1}{2} \cdot \frac{1}{1 - \left|x\right|}\right) - \log \left(1 - \left|x\right|\right)}, x\right) \]

   7. Applied rewrites 98.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-\left(\frac{-0.125}{{\left(1 - \left|x\right|\right)}^{2}} + \frac{-0.125}{1 - \left|x\right|}\right), x \cdot x, \frac{-0.5}{1 - \left|x\right|}\right) \cdot \left(x \cdot x\right) - \mathsf{log1p}\left(-\left|x\right|\right)}, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 98.3%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-\left(\frac{-0.125}{\left(1 - \left|x\right|\right) \cdot \left(1 - \left|x\right|\right)} + \frac{-0.125}{1 - \left|x\right|}\right), x \cdot x, \frac{-0.5}{1 - \left|x\right|}\right) \cdot \left(x \cdot x\right) - \mathsf{log1p}\left(-\left|x\right|\right), x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x - \frac{-0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\frac{-0.125}{-1 + \left|x\right|} - \frac{-0.125}{\left(-1 + \left|x\right|\right) \cdot \left(-1 + \left|x\right|\right)}, x \cdot x, \frac{-0.5}{1 - \left|x\right|}\right) \cdot \left(x \cdot x\right) - \mathsf{log1p}\left(-\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| - -1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.5)
       (copysign (fma (* 0.5 x) (/ x (- (fabs x) -1.0)) (log1p (fabs x))) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(fma((0.5 * x), (x / (fabs(x) - -1.0)), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(fma(Float64(0.5 * x), Float64(x / Float64(abs(x) - -1.0)), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[(N[(0.5 * x), $MachinePrecision] * N[(x / N[(N[Abs[x], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.9

          \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{\frac{-0.5}{x}} - x\right)\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(\frac{-0.5}{x} - x\right)}\right), x\right) \]

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-fabs.f64 97.5

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\color{blue}{\left|x\right|}\right)\right), x\right) \]

   5. Applied rewrites 97.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x - \frac{-0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| - -1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.5)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.sqrt((1.0 + (x * x))) + Math.abs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(Math.log((((-0.5 / x) - x) + Math.abs(x))), x);
	} else if (t_0 <= 0.5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log(((x - (-0.5 / x)) + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.sqrt((1.0 + (x * x))) + math.fabs(x))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(math.log((((-0.5 / x) - x) + math.fabs(x))), x)
	elif t_0 <= 0.5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log(((x - (-0.5 / x)) + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. distribute-lft-neg-in N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.9

          \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \left(\color{blue}{\frac{-0.5}{x}} - x\right)\right), x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(\frac{-0.5}{x} - x\right)}\right), x\right) \]

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. lower-fabs.f64 95.8

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right|}\right), x\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x - \frac{-0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.5)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.sqrt((1.0 + (x * x))) + Math.abs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log(((x - (-0.5 / x)) + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.sqrt((1.0 + (x * x))) + math.fabs(x))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 0.5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log(((x - (-0.5 / x)) + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x \cdot \frac{\left|x\right|}{x}} + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. lower-fabs.f64 95.8

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right|}\right), x\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(x - \frac{-0.5}{x}\right)}\right), x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.5)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.sqrt((1.0 + (x * x))) + Math.abs(x))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + x)), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.sqrt((1.0 + (x * x))) + math.fabs(x))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 0.5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + x)), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -5

   1. Initial program 53.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\left|x\right|}{x}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x \cdot \frac{\left|x\right|}{x}} + \left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 11.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. lower-fabs.f64 95.8

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right|}\right), x\right) \]

   5. Applied rewrites 95.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x) 0.5)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ (fabs x) x)) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x) <= 0.5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.sqrt((1.0 + (x * x))) + Math.abs(x))), x) <= 0.5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.sqrt((1.0 + (x * x))) + math.fabs(x))), x) <= 0.5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x) <= 0.5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 0.5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 23.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

      2. lower-fabs.f64 77.5

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right|}\right), x\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.8%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 12.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.225:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x}{1 - \left|x\right|} \cdot \left(0.5 \cdot x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.225)
   (copysign (* (/ x (- 1.0 (fabs x))) (* 0.5 x)) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.225) {
		tmp = copysign(((x / (1.0 - fabs(x))) * (0.5 * x)), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.225) {
		tmp = Math.copySign(((x / (1.0 - Math.abs(x))) * (0.5 * x)), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.225:
		tmp = math.copysign(((x / (1.0 - math.fabs(x))) * (0.5 * x)), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.225)
		tmp = copysign(Float64(Float64(x / Float64(1.0 - abs(x))) * Float64(0.5 * x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.225)
		tmp = sign(x) * abs(((x / (1.0 - abs(x))) * (0.5 * x)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.225], N[With[{TMP1 = Abs[N[(N[(x / N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.225000000000000006

   1. Initial program 22.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-fabs.f64 71.5

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\color{blue}{\left|x\right|}\right)\right), x\right) \]

   5. Applied rewrites 71.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 6.7%

      \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{1 + \left|x\right|}}, x\right) \]

   10. Applied rewrites 6.8%

       \[\leadsto \mathsf{copysign}\left(\frac{x}{1 - \left|x\right|} \cdot \left(0.5 \cdot \color{blue}{x}\right), x\right) \]

   if 0.225000000000000006 < x

   1. Initial program 46.7%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\log x\right)\right)}\right), x\right) \]

      3. remove-double-neg N/A

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

      4. lower-log.f64 31.2

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

   5. Applied rewrites 31.2%

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

Alternative 12: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (log1p (fabs x)) x))

C

double code(double x) {
	return copysign(log1p(fabs(x)), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log1p(Math.abs(x)), x);
}

Python

def code(x):
	return math.copysign(math.log1p(math.fabs(x)), x)

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]

   2. lower-fabs.f64 66.9

      \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left|x\right|}\right), x\right) \]

5. Applied rewrites 66.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left|x\right|\right)}, x\right) \]
6. Add Preprocessing

Alternative 13: 6.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\frac{x}{1 - \left|x\right|} \cdot \left(0.5 \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (* (/ x (- 1.0 (fabs x))) (* 0.5 x)) x))

C

double code(double x) {
	return copysign(((x / (1.0 - fabs(x))) * (0.5 * x)), x);
}

Java

public static double code(double x) {
	return Math.copySign(((x / (1.0 - Math.abs(x))) * (0.5 * x)), x);
}

Python

def code(x):
	return math.copysign(((x / (1.0 - math.fabs(x))) * (0.5 * x)), x)

Julia

function code(x)
	return copysign(Float64(Float64(x / Float64(1.0 - abs(x))) * Float64(0.5 * x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((x / (1.0 - abs(x))) * (0.5 * x)));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(x / N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 * x), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-fabs.f64 56.1

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\color{blue}{\left|x\right|}\right)\right), x\right) \]

5. Applied rewrites 56.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 6.4%

   \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{1 + \left|x\right|}}, x\right) \]

10. Applied rewrites 6.5%

    \[\leadsto \mathsf{copysign}\left(\frac{x}{1 - \left|x\right|} \cdot \left(0.5 \cdot \color{blue}{x}\right), x\right) \]
11. Add Preprocessing

Alternative 14: 6.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| - -1} \cdot x\right) \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (* (* (/ 0.5 (- (fabs x) -1.0)) x) x) x))

C

double code(double x) {
	return copysign((((0.5 / (fabs(x) - -1.0)) * x) * x), x);
}

Java

public static double code(double x) {
	return Math.copySign((((0.5 / (Math.abs(x) - -1.0)) * x) * x), x);
}

Python

def code(x):
	return math.copysign((((0.5 / (math.fabs(x) - -1.0)) * x) * x), x)

Julia

function code(x)
	return copysign(Float64(Float64(Float64(0.5 / Float64(abs(x) - -1.0)) * x) * x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((((0.5 / (abs(x) - -1.0)) * x) * x));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(N[(0.5 / N[(N[Abs[x], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-fabs.f64 56.1

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\color{blue}{\left|x\right|}\right)\right), x\right) \]

5. Applied rewrites 56.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 6.4%

   \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{1 + \left|x\right|}}, x\right) \]

9. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
10. Step-by-step derivation

11. Applied rewrites 6.5%

    \[\leadsto \mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| + 1} \cdot x\right) \cdot \color{blue}{x}, x\right) \]

12. Final simplification 6.5%

    \[\leadsto \mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| - -1} \cdot x\right) \cdot x, x\right) \]
13. Add Preprocessing

Alternative 15: 6.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\frac{0.5}{\left|x\right| - -1} \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (* (/ 0.5 (- (fabs x) -1.0)) (* x x)) x))

C

double code(double x) {
	return copysign(((0.5 / (fabs(x) - -1.0)) * (x * x)), x);
}

Java

public static double code(double x) {
	return Math.copySign(((0.5 / (Math.abs(x) - -1.0)) * (x * x)), x);
}

Python

def code(x):
	return math.copysign(((0.5 / (math.fabs(x) - -1.0)) * (x * x)), x)

Julia

function code(x)
	return copysign(Float64(Float64(0.5 / Float64(abs(x) - -1.0)) * Float64(x * x)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((0.5 / (abs(x) - -1.0)) * (x * x)));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(0.5 / N[(N[Abs[x], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 28.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-fabs.f64 56.1

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\color{blue}{\left|x\right|}\right)\right), x\right) \]

5. Applied rewrites 56.1%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\frac{1}{2} \cdot \color{blue}{\frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 6.4%

   \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{1 + \left|x\right|}}, x\right) \]

9. Final simplification 6.4%

   \[\leadsto \mathsf{copysign}\left(\frac{0.5}{\left|x\right| - -1} \cdot \left(x \cdot x\right), x\right) \]
10. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024255 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

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