Rust f64::asinh

Percentage Accurate: 29.6% → 99.4%

Time: 9.6s

Alternatives: 13

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: 29.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left({\left({t\_1}^{2}\right)}^{-1} + {t\_1}^{-1}, -0.125 + \left(0.001388888888888889 \cdot \left(x \cdot x\right)\right) \cdot 45, \left(\frac{30}{{t\_1}^{3}} \cdot 0.001388888888888889\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(0.5 \cdot x, \frac{x}{t\_1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.005)
       (copysign
        (fma
         (pow x 4.0)
         (fma
          (+ (pow (pow t_1 2.0) -1.0) (pow t_1 -1.0))
          (+ -0.125 (* (* 0.001388888888888889 (* x x)) 45.0))
          (* (* (/ 30.0 (pow t_1 3.0)) 0.001388888888888889) (* x x)))
         (fma (* 0.5 x) (/ x t_1) (log1p (fabs x))))
        x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = fabs(x) + 1.0;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(fma(pow(x, 4.0), fma((pow(pow(t_1, 2.0), -1.0) + pow(t_1, -1.0)), (-0.125 + ((0.001388888888888889 * (x * x)) * 45.0)), (((30.0 / pow(t_1, 3.0)) * 0.001388888888888889) * (x * x))), fma((0.5 * x), (x / t_1), log1p(fabs(x)))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 = Float64(abs(x) + 1.0)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(fma((x ^ 4.0), fma(Float64(((t_1 ^ 2.0) ^ -1.0) + (t_1 ^ -1.0)), Float64(-0.125 + Float64(Float64(0.001388888888888889 * Float64(x * x)) * 45.0)), Float64(Float64(Float64(30.0 / (t_1 ^ 3.0)) * 0.001388888888888889) * Float64(x * x))), fma(Float64(0.5 * x), Float64(x / t_1), log1p(abs(x)))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / 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[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], 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, 0.005], N[With[{TMP1 = Abs[N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], -1.0], $MachinePrecision] + N[Power[t$95$1, -1.0], $MachinePrecision]), $MachinePrecision] * N[(-0.125 + N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] * 45.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(30.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * x), $MachinePrecision] * N[(x / t$95$1), $MachinePrecision] + N[Log[1 + N[Abs[x], $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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + \frac{1}{720} \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]

   5. Applied rewrites 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left(\frac{1}{{\left(\left|x\right| + 1\right)}^{2}} + \frac{1}{\left|x\right| + 1}, -0.125 + \left(0.001388888888888889 \cdot \left(x \cdot x\right)\right) \cdot 45, \left(\frac{30}{{\left(\left|x\right| + 1\right)}^{3}} \cdot 0.001388888888888889\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right)}, x\right) \]

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left({x}^{4}, \mathsf{fma}\left({\left({\left(\left|x\right| + 1\right)}^{2}\right)}^{-1} + {\left(\left|x\right| + 1\right)}^{-1}, -0.125 + \left(0.001388888888888889 \cdot \left(x \cdot x\right)\right) \cdot 45, \left(\frac{30}{{\left(\left|x\right| + 1\right)}^{3}} \cdot 0.001388888888888889\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{t\_1}, \mathsf{fma}\left({x}^{4}, \frac{-0.125}{{t\_1}^{2}} + \frac{-0.125}{t\_1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.005)
       (copysign
        (fma
         (* 0.5 x)
         (/ x t_1)
         (fma
          (pow x 4.0)
          (+ (/ -0.125 (pow t_1 2.0)) (/ -0.125 t_1))
          (log1p (fabs x))))
        x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = fabs(x) + 1.0;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(fma((0.5 * x), (x / t_1), fma(pow(x, 4.0), ((-0.125 / pow(t_1, 2.0)) + (-0.125 / t_1)), log1p(fabs(x)))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 = Float64(abs(x) + 1.0)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(fma(Float64(0.5 * x), Float64(x / t_1), fma((x ^ 4.0), Float64(Float64(-0.125 / (t_1 ^ 2.0)) + Float64(-0.125 / t_1)), log1p(abs(x)))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / 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[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], 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, 0.005], N[With[{TMP1 = Abs[N[(N[(0.5 * x), $MachinePrecision] * N[(x / t$95$1), $MachinePrecision] + N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[(-0.125 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.125 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[Abs[x], $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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.7%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{fma}\left({x}^{4}, \frac{-0.125}{{\left(\left|x\right| + 1\right)}^{2}} + \frac{-0.125}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right)\right)}, x\right) \]

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right|\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := \left|x\right| + 1\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{\frac{-0.125}{t\_2}}{t\_2} + \frac{-0.125}{t\_0}\right) \cdot x, x, \frac{0.5}{t\_0}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (fabs x)))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (+ (fabs x) 1.0)))
   (if (<= t_1 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_1 0.005)
       (copysign
        (fma
         (fma (* (+ (/ (/ -0.125 t_2) t_2) (/ -0.125 t_0)) x) x (/ 0.5 t_0))
         (* x x)
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = 1.0 + fabs(x);
	double t_1 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_2 = fabs(x) + 1.0;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_1 <= 0.005) {
		tmp = copysign(fma(fma(((((-0.125 / t_2) / t_2) + (-0.125 / t_0)) * x), x, (0.5 / t_0)), (x * x), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + abs(x))
	t_1 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_2 = Float64(abs(x) + 1.0)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_1 <= 0.005)
		tmp = copysign(fma(fma(Float64(Float64(Float64(Float64(-0.125 / t_2) / t_2) + Float64(-0.125 / t_0)) * x), x, Float64(0.5 / t_0)), Float64(x * x), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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$2 = N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], 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$1, 0.005], N[With[{TMP1 = Abs[N[(N[(N[(N[(N[(N[(-0.125 / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(-0.125 / t$95$0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.5 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 2.8

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.7%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(\frac{0.5}{1 - \left|x\right|} \cdot x, x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.005)
       (copysign
        (- (fma (* (/ 0.5 (- 1.0 (fabs x))) x) x (log1p (- (fabs x)))))
        x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 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 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(-fma(((0.5 / (1.0 - fabs(x))) * x), x, log1p(-fabs(x))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(Float64(-fma(Float64(Float64(0.5 / Float64(1.0 - abs(x))) * x), x, log1p(Float64(-abs(x))))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / 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, -1.0], 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, 0.005], N[With[{TMP1 = Abs[(-N[(N[(N[(0.5 / N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + 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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. log-rec N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 9.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   7. Applied rewrites 99.2%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 5: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(0.5 \cdot x, \frac{x}{\left|x\right| + 1}, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.005)
       (copysign (fma (* 0.5 x) (/ x (+ (fabs x) 1.0)) (log1p (fabs x))) x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 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 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(fma((0.5 * x), (x / (fabs(x) + 1.0)), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_0 <= 0.005)
		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) + Float64(x - Float64(-0.5 / 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, -1.0], 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, 0.005], 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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + \frac{1}{2} \cdot \frac{{x}^{2}}{1 + \left|x\right|}}, x\right) \]
   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 99.2

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

   5. Applied rewrites 99.2%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 6: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(\frac{-0.5}{x} - x\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (fabs x) (- (/ -0.5 x) x))) x)
     (if (<= t_0 0.005)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 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 <= -1.0) {
		tmp = copysign(log((fabs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 <= -1.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + ((-0.5 / x) - x))), x);
	} else if (t_0 <= 0.005) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + (x - (-0.5 / 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 <= -1.0:
		tmp = math.copysign(math.log((math.fabs(x) + ((-0.5 / x) - x))), x)
	elif t_0 <= 0.005:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + (x - (-0.5 / 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 <= -1.0)
		tmp = copysign(log(Float64(abs(x) + Float64(Float64(-0.5 / x) - x))), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / 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, -1.0], 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, 0.005], 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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{-1 \cdot \left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right), x\right) \]
   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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 7: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.005)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ (fabs x) (- x (/ -0.5 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 <= -1.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + (x - (-0.5 / 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 <= -1.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + (x - (-0.5 / 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 <= -1.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 0.005:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + (x - (-0.5 / 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 <= -1.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.005)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x - Float64(-0.5 / 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, -1.0], 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, 0.005], 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] + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. 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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

      13. associate-*r* N/A

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

      14. neg-mul-1 N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. distribute-neg-frac N/A

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

      19. metadata-eval N/A

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

      20. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

Alternative 8: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.005)
       (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 <= -1.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		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 <= -1.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 0.005) {
		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 <= -1.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 0.005:
		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 <= -1.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.005)
		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, -1.0], 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, 0.005], 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) < -1

   1. Initial program 49.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. 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 -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 97.3

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

   5. Applied rewrites 97.3%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 9: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.005)
   (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) <= 0.005) {
		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) <= 0.005) {
		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) <= 0.005:
		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) <= 0.005)
		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], 0.005], 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) < 0.0050000000000000001

   1. Initial program 23.0%

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

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

Alternative 10: 12.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\left(\frac{0.5}{\left|x\right| + 1} \cdot x\right) \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.005)
   (copysign (* (* (/ 0.5 (+ (fabs x) 1.0)) x) x) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 0.005) {
		tmp = copysign((((0.5 / (fabs(x) + 1.0)) * x) * x), x);
	} else {
		tmp = copysign(log(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) <= 0.005) {
		tmp = Math.copySign((((0.5 / (Math.abs(x) + 1.0)) * x) * x), x);
	} else {
		tmp = Math.copySign(Math.log(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) <= 0.005:
		tmp = math.copysign((((0.5 / (math.fabs(x) + 1.0)) * x) * x), x)
	else:
		tmp = math.copysign(math.log(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) <= 0.005)
		tmp = copysign(Float64(Float64(Float64(0.5 / Float64(abs(x) + 1.0)) * x) * x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))))) <= 0.005)
		tmp = sign(x) * abs((((0.5 / (abs(x) + 1.0)) * x) * x));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = 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], 0.005], N[With[{TMP1 = Abs[N[(N[(N[(0.5 / N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 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 (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 23.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 1.8

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

   5. Applied rewrites 1.8%

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

   6. Taylor expanded in x around 0

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

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-fabs.f64 N/A

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

      15. lower-log1p.f64 N/A

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

      16. lower-fabs.f64 67.0

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

   8. Applied rewrites 67.0%

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

   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%

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

   13. Applied rewrites 6.5%

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

   if 0.0050000000000000001 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 47.6%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
   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 31.6

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

   5. Applied rewrites 31.6%

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

Alternative 11: 64.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in x around 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 64.4

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

5. Applied rewrites 64.4%

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

Alternative 12: 6.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
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 8.9

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

5. Applied rewrites 8.9%

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

6. Taylor expanded in x around 0

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

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower-fabs.f64 N/A

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

   15. lower-log1p.f64 N/A

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

   16. lower-fabs.f64 52.3

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

8. Applied rewrites 52.3%

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

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%

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

13. Applied rewrites 6.2%

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

Alternative 13: 6.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 28.9%

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

3. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}, x\right) \]
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 8.9

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

5. Applied rewrites 8.9%

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

6. Taylor expanded in x around 0

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

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 N/A

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

   13. lower-+.f64 N/A

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

   14. lower-fabs.f64 N/A

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

   15. lower-log1p.f64 N/A

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

   16. lower-fabs.f64 52.3

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

8. Applied rewrites 52.3%

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

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%

    \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{1 + \left|x\right|}}, x\right) \]
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 2024308 
(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))