Rust f64::asinh

Percentage Accurate: 30.1% → 98.6%

Time: 8.5s

Alternatives: 10

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: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \left|x\right| - t\_1\\ t_3 := \mathsf{fma}\left(t\_2, \left|x\right|, x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left({t\_3}^{3} + 1\right) \cdot \frac{t\_1 + \left|x\right|}{\mathsf{fma}\left(t\_2, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\mathsf{fma}\left(t\_3, t\_3, -t\_3\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (sqrt (fma x x 1.0)))
        (t_2 (- (fabs x) t_1))
        (t_3 (fma t_2 (fabs x) (* x x))))
   (if (<= t_0 -20.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.5)
       (copysign
        (-
         (log
          (*
           (+ (pow t_3 3.0) 1.0)
           (/ (+ t_1 (fabs x)) (fma t_2 (fabs x) (fma x x 1.0)))))
         (log1p (fma t_3 t_3 (- t_3))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = sqrt(fma(x, x, 1.0));
	double t_2 = fabs(x) - t_1;
	double t_3 = fma(t_2, fabs(x), (x * x));
	double tmp;
	if (t_0 <= -20.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign((log(((pow(t_3, 3.0) + 1.0) * ((t_1 + fabs(x)) / fma(t_2, fabs(x), fma(x, x, 1.0))))) - log1p(fma(t_3, t_3, -t_3))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = sqrt(fma(x, x, 1.0))
	t_2 = Float64(abs(x) - t_1)
	t_3 = fma(t_2, abs(x), Float64(x * x))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(Float64(log(Float64(Float64((t_3 ^ 3.0) + 1.0) * Float64(Float64(t_1 + abs(x)) / fma(t_2, abs(x), fma(x, x, 1.0))))) - log1p(fma(t_3, t_3, Float64(-t_3)))), 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[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]}, 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]}, Block[{t$95$3 = N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x), $MachinePrecision]), $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, 0.5], N[With[{TMP1 = Abs[N[(N[Log[N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(t$95$1 + N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(t$95$3 * t$95$3 + (-t$95$3)), $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 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. associate-*r/ N/A

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

      12. *-commutative N/A

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

      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) < 0.5

   1. Initial program 9.9%

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

   4. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. sum-log N/A

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

      5. flip3-+ N/A

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

   6. Applied rewrites 98.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} - \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} - \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)}\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} + \left(\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)\right)}\right), x\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2}} + \left(\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.7

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

   8. Applied rewrites 98.7%

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

   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 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

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

Alternative 2: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \left|x\right| - t\_1\\ t_3 := \mathsf{fma}\left(t\_2, \left|x\right|, x \cdot x\right)\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left({t\_3}^{3} + 1\right) \cdot \frac{t\_1 + \left|x\right|}{\mathsf{fma}\left(t\_2, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(t\_3 \cdot \left(t\_3 + -1\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (sqrt (fma x x 1.0)))
        (t_2 (- (fabs x) t_1))
        (t_3 (fma t_2 (fabs x) (* x x))))
   (if (<= t_0 -20.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.5)
       (copysign
        (-
         (log
          (*
           (+ (pow t_3 3.0) 1.0)
           (/ (+ t_1 (fabs x)) (fma t_2 (fabs x) (fma x x 1.0)))))
         (log1p (* t_3 (+ t_3 -1.0))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = sqrt(fma(x, x, 1.0));
	double t_2 = fabs(x) - t_1;
	double t_3 = fma(t_2, fabs(x), (x * x));
	double tmp;
	if (t_0 <= -20.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.5) {
		tmp = copysign((log(((pow(t_3, 3.0) + 1.0) * ((t_1 + fabs(x)) / fma(t_2, fabs(x), fma(x, x, 1.0))))) - log1p((t_3 * (t_3 + -1.0)))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = sqrt(fma(x, x, 1.0))
	t_2 = Float64(abs(x) - t_1)
	t_3 = fma(t_2, abs(x), Float64(x * x))
	tmp = 0.0
	if (t_0 <= -20.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.5)
		tmp = copysign(Float64(log(Float64(Float64((t_3 ^ 3.0) + 1.0) * Float64(Float64(t_1 + abs(x)) / fma(t_2, abs(x), fma(x, x, 1.0))))) - log1p(Float64(t_3 * Float64(t_3 + -1.0)))), 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[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]}, 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]}, Block[{t$95$3 = N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x), $MachinePrecision]), $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, 0.5], N[With[{TMP1 = Abs[N[(N[Log[N[(N[(N[Power[t$95$3, 3.0], $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(t$95$1 + N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[Abs[x], $MachinePrecision] + N[(x * x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[1 + N[(t$95$3 * N[(t$95$3 + -1.0), $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 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. associate-*r/ N/A

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

      12. *-commutative N/A

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

      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) < 0.5

   1. Initial program 9.9%

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

   4. Applied rewrites 98.7%

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

      1. lift-+.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-log.f64 N/A

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

      4. sum-log N/A

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

      5. flip3-+ N/A

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

   6. Applied rewrites 98.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} - \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}, x\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} - \mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)}\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2} + \left(\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)\right)}\right), x\right) \]

      3. lift-pow.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\left({\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{3} + 1\right) \cdot \frac{\sqrt{\mathsf{fma}\left(x, x, 1\right)} + \left|x\right|}{\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, \mathsf{fma}\left(x, x, 1\right)\right)}\right) - \mathsf{log1p}\left(\color{blue}{{\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)}^{2}} + \left(\mathsf{neg}\left(\mathsf{fma}\left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}, \left|x\right|, x \cdot x\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 98.7

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

   8. Applied rewrites 98.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 98.7

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

   10. Applied rewrites 98.7%

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

   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 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

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x, x, 1\right)}\\ t_2 := \left|x\right| - t\_1\\ \mathbf{if}\;t\_0 \leq -20:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot t\_2\right)\right) + \log \left(\frac{t\_1 + \left|x\right|}{\mathsf{fma}\left(\left|x\right|, t\_2, \mathsf{fma}\left(x, x, 1\right)\right)}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) 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 0.5)
       (copysign
        (+
         (log1p (fma x x (* (fabs x) t_2)))
         (log (/ (+ t_1 (fabs x)) (fma (fabs x) t_2 (fma x x 1.0)))))
        x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), 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 <= 0.5) {
		tmp = copysign((log1p(fma(x, x, (fabs(x) * t_2))) + log(((t_1 + fabs(x)) / fma(fabs(x), t_2, fma(x, x, 1.0))))), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), 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 <= 0.5)
		tmp = copysign(Float64(log1p(fma(x, x, Float64(abs(x) * t_2))) + log(Float64(Float64(t_1 + abs(x)) / fma(abs(x), t_2, fma(x, x, 1.0))))), 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[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]}, 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, 0.5], N[With[{TMP1 = Abs[N[(N[Log[1 + N[(x * x + N[(N[Abs[x], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 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]), $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 55.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. associate-*r/ N/A

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

      12. *-commutative N/A

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

      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) < 0.5

   1. Initial program 9.9%

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

   4. Applied rewrites 98.7%

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

   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 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

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - 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((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - 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(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[LessEqual[t$95$0, -2.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[(N[(-0.5 / x), $MachinePrecision] - 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[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) < -2

   1. Initial program 56.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. associate-*l/ N/A

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

      14. lft-mult-inverse N/A

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

      15. associate-*r* N/A

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

      16. neg-mul-1 N/A

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

      17. lower--.f64 N/A

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 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. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. *-rgt-identity N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-fabs.f64 N/A

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

      17. lower-log1p.f64 N/A

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

      18. lower-fabs.f64 98.9

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

   5. Applied rewrites 98.9%

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

   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 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

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

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

Alternative 5: 98.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 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((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / 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.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + ((-0.5 / 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.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -2.0:
		tmp = math.copysign(math.log((math.fabs(x) + ((-0.5 / 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(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -2.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / 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[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]}, If[LessEqual[t$95$0, -2.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[(N[(-0.5 / x), $MachinePrecision] - 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[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) < -2

   1. Initial program 56.6%

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

   3. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. associate-*r* N/A

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

      11. unpow2 N/A

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

      12. associate-/r* N/A

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

      13. associate-*l/ N/A

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

      14. lft-mult-inverse N/A

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

      15. associate-*r* N/A

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

      16. neg-mul-1 N/A

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

      17. lower--.f64 N/A

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

   5. Applied rewrites 98.9%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 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 98.1

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

   5. Applied rewrites 98.1%

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

   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 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

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

Alternative 6: 98.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (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((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		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.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -2.0) {
		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.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -2.0:
		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(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -2.0)
		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[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]}, If[LessEqual[t$95$0, -2.0], 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) < -2

   1. Initial program 56.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. associate-*r/ N/A

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

      12. *-commutative N/A

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

      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 98.2

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

   5. Applied rewrites 98.2%

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

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

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 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 98.1

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

   5. Applied rewrites 98.1%

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

   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 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

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

Alternative 7: 81.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), 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.abs(x) + Math.sqrt(((x * x) + 1.0)))), 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.fabs(x) + math.sqrt(((x * x) + 1.0)))), 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(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), 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[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], 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 25.6%

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

   3. Taylor expanded in x around 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 74.3

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

   5. Applied rewrites 74.3%

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

   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 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

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(1 + \frac{\left|x\right|}{x}\right)\right)}, x\right) \]
   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 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 12.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.106:\\ \;\;\;\;\mathsf{copysign}\left(\frac{0.5 \cdot \left(x \cdot x\right)}{1 + \left|x\right|}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.106)
   (copysign (/ (* 0.5 (* x x)) (+ 1.0 (fabs x))) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.106) {
		tmp = copysign(((0.5 * (x * x)) / (1.0 + fabs(x))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.106) {
		tmp = Math.copySign(((0.5 * (x * x)) / (1.0 + Math.abs(x))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.106:
		tmp = math.copysign(((0.5 * (x * x)) / (1.0 + math.fabs(x))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.106)
		tmp = copysign(Float64(Float64(0.5 * Float64(x * x)) / Float64(1.0 + abs(x))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.106)
		tmp = sign(x) * abs(((0.5 * (x * x)) / (1.0 + abs(x))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.106], N[With[{TMP1 = Abs[N[(N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Abs[x], $MachinePrecision]), $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 < 0.105999999999999997

   1. Initial program 25.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   7. 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-fabs.f64 65.7

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

   8. Applied rewrites 65.7%

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

   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 6.3%

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

   if 0.105999999999999997 < x

   1. Initial program 52.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 31.4

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

   5. Applied rewrites 31.4%

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

Alternative 9: 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 32.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 62.9

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

5. Applied rewrites 62.9%

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

Alternative 10: 6.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 32.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 62.9

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

5. Applied rewrites 62.9%

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

6. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
7. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lower-log1p.f64 N/A

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

   10. lower-fabs.f64 49.4

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

8. Applied rewrites 49.4%

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

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 6.0%

    \[\leadsto \mathsf{copysign}\left(\frac{0.5 \cdot \left(x \cdot x\right)}{\color{blue}{1 + \left|x\right|}}, x\right) \]
12. 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 2024321 
(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))