Rust f64::asinh

Percentage Accurate: 31.3% → 99.7%

Time: 9.5s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 31.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot 0.001388888888888889\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := \left|x\right| + 1\\ t_3 := t\_2 \cdot t\_2\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\frac{1}{t\_2} + \frac{1}{t\_3}, -0.125 + t\_0 \cdot 45, \frac{t\_0 \cdot 30}{t\_2 \cdot t\_3}\right), \frac{0.5}{t\_2}\right), \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) 0.001388888888888889))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (+ (fabs x) 1.0))
        (t_3 (* t_2 t_2)))
   (if (<= t_1 -1.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_1 1.0)
       (copysign
        (fma
         (* x x)
         (fma
          (* x x)
          (fma
           (+ (/ 1.0 t_2) (/ 1.0 t_3))
           (+ -0.125 (* t_0 45.0))
           (/ (* t_0 30.0) (* t_2 t_3)))
          (/ 0.5 t_2))
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = (x * x) * 0.001388888888888889;
	double t_1 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_2 = fabs(x) + 1.0;
	double t_3 = t_2 * t_2;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = copysign(log(((fabs(x) - x) + (-0.5 / x))), x);
	} else if (t_1 <= 1.0) {
		tmp = copysign(fma((x * x), fma((x * x), fma(((1.0 / t_2) + (1.0 / t_3)), (-0.125 + (t_0 * 45.0)), ((t_0 * 30.0) / (t_2 * t_3))), (0.5 / t_2)), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * 0.001388888888888889)
	t_1 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_2 = Float64(abs(x) + 1.0)
	t_3 = Float64(t_2 * t_2)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = copysign(log(Float64(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_1 <= 1.0)
		tmp = copysign(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(Float64(1.0 / t_2) + Float64(1.0 / t_3)), Float64(-0.125 + Float64(t_0 * 45.0)), Float64(Float64(t_0 * 30.0) / Float64(t_2 * t_3))), Float64(0.5 / t_2)), 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[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $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]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[With[{TMP1 = Abs[N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(N[(1.0 / t$95$2), $MachinePrecision] + N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision] * N[(-0.125 + N[(t$95$0 * 45.0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 30.0), $MachinePrecision] / N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t$95$2), $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 40.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.3%

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

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

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 99.0%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.4%

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

4. Final simplification 98.9%

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

Alternative 2: 99.6% accurate, 0.3× speedup

Math

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

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) x) (/ -0.5 x))) x)
     (if (<= t_0 1.0)
       (copysign (log1p (fma x (* x 0.5) (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) + (-0.5 / x))), x);
	} else if (t_0 <= 1.0) {
		tmp = copysign(log1p(fma(x, (x * 0.5), 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(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 1.0)
		tmp = copysign(log1p(fma(x, Float64(x * 0.5), 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[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision] + N[(-0.5 / x), $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[1 + N[(x * N[(x * 0.5), $MachinePrecision] + 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 40.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.3%

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

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

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 6.9

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

   5. Simplified 6.9%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. fabs-lowering-fabs.f64 98.5

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

   7. Applied egg-rr 98.5%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.4%

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

Alternative 3: 99.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -2.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 1.0)
       (copysign (log1p (fma x (* x 0.5) (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 <= -2.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 1.0) {
		tmp = copysign(log1p(fma(x, (x * 0.5), 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 <= -2.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 1.0)
		tmp = copysign(log1p(fma(x, Float64(x * 0.5), 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, -2.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1.0], N[With[{TMP1 = Abs[N[Log[1 + N[(x * N[(x * 0.5), $MachinePrecision] + 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) < -2

   1. Initial program 39.8%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. fabs-lowering-fabs.f64 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 7.1

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

   5. Simplified 7.1%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. fabs-lowering-fabs.f64 97.9

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

   7. Applied egg-rr 97.9%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-+r+ N/A

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

      2. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. distribute-lft-in N/A

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. distribute-rgt-in N/A

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

      14. *-lft-identity N/A

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

      15. associate-*l* N/A

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

      16. unpow2 N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. lft-mult-inverse N/A

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

   5. Simplified 99.4%

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

Alternative 4: 99.3% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 39.8%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. fabs-lowering-fabs.f64 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 9.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

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

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

      6. *-lowering-*.f64 7.1

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

   5. Simplified 7.1%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-log1p.f64 N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. fabs-lowering-fabs.f64 97.9

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

   7. Applied egg-rr 97.9%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

4. Final simplification 98.5%

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

Alternative 5: 98.6% accurate, 0.3× speedup

Math

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

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 1.0)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x (fabs 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 <= 1.0) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + fabs(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 <= 1.0) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(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 <= 1.0:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(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 <= 1.0)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + abs(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, 1.0], 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[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 40.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(-1 \cdot \left(x \cdot \left(1 + -1 \cdot \frac{\left|x\right|}{x}\right)\right)\right)}, x\right) \]
   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-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

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

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

      16. fabs-lowering-fabs.f64 97.9

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

   5. Simplified 97.9%

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

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

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. accelerator-lowering-log1p.f64 N/A

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

      2. fabs-lowering-fabs.f64 97.8

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

   5. Simplified 97.8%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

4. Final simplification 98.2%

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

Alternative 6: 81.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 1.0) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + fabs(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) <= 1.0) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(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) <= 1.0:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(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) <= 1.0)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + abs(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], 1.0], 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[(x + N[Abs[x], $MachinePrecision]), $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) < 1

   1. Initial program 19.7%

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

   3. Taylor expanded in x around 0

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

      1. accelerator-lowering-log1p.f64 N/A

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

      2. fabs-lowering-fabs.f64 74.5

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

   5. Simplified 74.5%

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

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

   1. Initial program 56.8%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 99.1%

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

4. Final simplification 80.7%

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

Alternative 7: 12.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \frac{x \cdot 0.5}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.11)
   (copysign (* x (/ (* x 0.5) (+ (fabs x) 1.0))) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.11) {
		tmp = copysign((x * ((x * 0.5) / (fabs(x) + 1.0))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.11) {
		tmp = Math.copySign((x * ((x * 0.5) / (Math.abs(x) + 1.0))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.11:
		tmp = math.copysign((x * ((x * 0.5) / (math.fabs(x) + 1.0))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.11)
		tmp = copysign(Float64(x * Float64(Float64(x * 0.5) / Float64(abs(x) + 1.0))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sign(x) * abs((x * ((x * 0.5) / (abs(x) + 1.0))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.11], N[With[{TMP1 = Abs[N[(x * N[(N[(x * 0.5), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 18.8%

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

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

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. accelerator-lowering-log1p.f64 N/A

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

      15. fabs-lowering-fabs.f64 66.1

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

   5. Simplified 66.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. fabs-lowering-fabs.f64 6.1

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

   8. Simplified 6.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

      7. fabs-lowering-fabs.f64 6.3

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

   10. Applied egg-rr 6.3%

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

   if 0.110000000000000001 < x

   1. Initial program 58.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. log-lowering-log.f64 30.8

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

   5. Simplified 30.8%

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

4. Final simplification 12.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \frac{x \cdot 0.5}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 29.1%

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

3. Taylor expanded in x around 0

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

   1. accelerator-lowering-log1p.f64 N/A

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

   2. fabs-lowering-fabs.f64 63.5

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

5. Simplified 63.5%

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

Alternative 9: 6.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. accelerator-lowering-log1p.f64 N/A

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

   15. fabs-lowering-fabs.f64 50.4

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

5. Simplified 50.4%

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

6. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. fabs-lowering-fabs.f64 6.0

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

8. Simplified 6.0%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

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

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

   7. fabs-lowering-fabs.f64 6.2

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

10. Applied egg-rr 6.2%

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

11. Final simplification 6.2%

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

Alternative 10: 6.1% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 29.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. accelerator-lowering-log1p.f64 N/A

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

   15. fabs-lowering-fabs.f64 50.4

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

5. Simplified 50.4%

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

6. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

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

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

   3. *-commutative N/A

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

   4. unpow2 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. fabs-lowering-fabs.f64 6.0

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

8. Simplified 6.0%

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

   1. associate-*r* N/A

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

   2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

   7. fabs-lowering-fabs.f64 6.0

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

10. Applied egg-rr 6.0%

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

11. Final simplification 6.0%

    \[\leadsto \mathsf{copysign}\left(\left(x \cdot x\right) \cdot \frac{0.5}{\left|x\right| + 1}, x\right) \]
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 2024199 
(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))