Rust f64::asinh

Percentage Accurate: 29.8% → 99.1%

Time: 8.5s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.8% 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: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \left|x\right| + 1\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.125}{t\_1}, x \cdot \left(x + \frac{x}{t\_1}\right), \frac{0.5}{t\_1}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (+ (fabs x) 1.0)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 2e-8)
       (copysign
        (fma
         (fma (/ -0.125 t_1) (* x (+ x (/ x t_1))) (/ 0.5 t_1))
         (* x x)
         (log1p (fabs x)))
        x)
       (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = fabs(x) + 1.0;
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log(((fabs(x) - x) + (-0.5 / x))), x);
	} else if (t_0 <= 2e-8) {
		tmp = copysign(fma(fma((-0.125 / t_1), (x * (x + (x / t_1))), (0.5 / t_1)), (x * x), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = Float64(abs(x) + 1.0)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 2e-8)
		tmp = copysign(fma(fma(Float64(-0.125 / t_1), Float64(x * Float64(x + Float64(x / t_1))), Float64(0.5 / t_1)), Float64(x * x), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x + Float64(0.5 / x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], 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, 2e-8], N[With[{TMP1 = Abs[N[(N[(N[(-0.125 / t$95$1), $MachinePrecision] * N[(x * N[(x + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[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) < -5

   1. Initial program 59.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2e-8

   1. Initial program 8.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 8.1

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 8.1

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

   4. Applied egg-rr 8.1%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 100.0%

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

   9. Applied egg-rr 100.0%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 98.7

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

   5. Simplified 98.7%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| - x\right) + \frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, x \cdot x, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 2e-8)
       (copysign (log1p (fma 0.5 (* x x) (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 <= -5.0) {
		tmp = copysign(log(((fabs(x) - x) + (-0.5 / x))), x);
	} else if (t_0 <= 2e-8) {
		tmp = copysign(log1p(fma(0.5, (x * x), 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 <= -5.0)
		tmp = copysign(log(Float64(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 2e-8)
		tmp = copysign(log1p(fma(0.5, Float64(x * x), 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, -5.0], 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, 2e-8], N[With[{TMP1 = Abs[N[Log[1 + N[(0.5 * N[(x * x), $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) < -5

   1. Initial program 59.4%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2e-8

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 8.0

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

   5. Simplified 8.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 99.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 98.7

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

   5. Simplified 98.7%

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

Alternative 3: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, x \cdot x, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 2e-8)
       (copysign (log1p (fma 0.5 (* x x) (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 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 2e-8) {
		tmp = copysign(log1p(fma(0.5, (x * x), 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 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 2e-8)
		tmp = copysign(log1p(fma(0.5, Float64(x * x), 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, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[With[{TMP1 = Abs[N[Log[1 + N[(0.5 * N[(x * x), $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) < -5

   1. Initial program 59.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-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. lower--.f64 N/A

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

      16. lower-fabs.f64 99.4

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

   5. Simplified 99.4%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2e-8

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 8.0

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

   5. Simplified 8.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 99.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l* N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. lower-+.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 98.7

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

   5. Simplified 98.7%

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

Alternative 4: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5, x \cdot x, \left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 2e-8)
       (copysign (log1p (fma 0.5 (* x x) (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 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 2e-8) {
		tmp = copysign(log1p(fma(0.5, (x * x), 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 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 2e-8)
		tmp = copysign(log1p(fma(0.5, Float64(x * x), 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, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 2e-8], N[With[{TMP1 = Abs[N[Log[1 + N[(0.5 * N[(x * x), $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) < -5

   1. Initial program 59.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-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. lower--.f64 N/A

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

      16. lower-fabs.f64 99.4

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

   5. Simplified 99.4%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2e-8

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-fabs.f64 8.0

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

   5. Simplified 8.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-+l+ N/A

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

      7. lower-log1p.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 99.7

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

   7. Applied egg-rr 99.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log1p.f64 99.7

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.7

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

   9. Applied egg-rr 99.7%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 98.1

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

   5. Simplified 98.1%

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

Alternative 5: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 2e-8)
       (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 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 2e-8) {
		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 <= -5.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 2e-8) {
		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 <= -5.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 2e-8:
		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 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 2e-8)
		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, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 2e-8], 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) < -5

   1. Initial program 59.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-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. lower--.f64 N/A

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

      16. lower-fabs.f64 99.4

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

   5. Simplified 99.4%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2e-8

   1. Initial program 8.1%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-fabs.f64 98.2

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

   5. Simplified 98.2%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 98.1

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

   5. Simplified 98.1%

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

Alternative 6: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 2e-8)
   (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) <= 2e-8) {
		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) <= 2e-8) {
		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) <= 2e-8:
		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) <= 2e-8)
		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], 2e-8], 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) < 2e-8

   1. Initial program 24.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-fabs.f64 76.8

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

   5. Simplified 76.8%

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

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 54.7%

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

   3. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fabs.f64 98.1

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

   5. Simplified 98.1%

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

Alternative 7: 11.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ \mathbf{if}\;x \leq 2.05 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{t\_0}, -0.125\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{t\_0}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (fabs x) 1.0)))
   (if (<= x 2.05e-153)
     (copysign
      (/ (* (fma -0.125 (/ 1.0 t_0) -0.125) (* (* x x) (* x x))) t_0)
      x)
     (copysign (log x) x))))

C

double code(double x) {
	double t_0 = fabs(x) + 1.0;
	double tmp;
	if (x <= 2.05e-153) {
		tmp = copysign(((fma(-0.125, (1.0 / t_0), -0.125) * ((x * x) * (x * x))) / t_0), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(abs(x) + 1.0)
	tmp = 0.0
	if (x <= 2.05e-153)
		tmp = copysign(Float64(Float64(fma(-0.125, Float64(1.0 / t_0), -0.125) * Float64(Float64(x * x) * Float64(x * x))) / t_0), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, 2.05e-153], N[With[{TMP1 = Abs[N[(N[(N[(-0.125 * N[(1.0 / t$95$0), $MachinePrecision] + -0.125), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $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 < 2.05e-153

   1. Initial program 27.1%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 27.1

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 27.1

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

   4. Applied egg-rr 27.1%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 64.4%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)}{1 + \left|x\right|}}, x\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)\right)}{1 + \left|x\right|}}, x\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)\right)}{1 + \left|x\right|}}, x\right) \]

   10. Simplified 5.5%

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

   if 2.05e-153 < x

   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

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f64 23.9

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

   5. Simplified 23.9%

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

4. Final simplification 11.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.05 \cdot 10^{-153}:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{\left|x\right| + 1}, -0.125\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.0% 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 31.8%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-fabs.f64 65.8

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

5. Simplified 65.8%

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

Alternative 9: 5.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ \mathsf{copysign}\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{t\_0}, -0.125\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{t\_0}, x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (fabs x) 1.0)))
   (copysign
    (/ (* (fma -0.125 (/ 1.0 t_0) -0.125) (* (* x x) (* x x))) t_0)
    x)))

C

double code(double x) {
	double t_0 = fabs(x) + 1.0;
	return copysign(((fma(-0.125, (1.0 / t_0), -0.125) * ((x * x) * (x * x))) / t_0), x);
}

Julia

function code(x)
	t_0 = Float64(abs(x) + 1.0)
	return copysign(Float64(Float64(fma(-0.125, Float64(1.0 / t_0), -0.125) * Float64(Float64(x * x) * Float64(x * x))) / t_0), x)
end

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-+.f64 31.8

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

   7. lift-+.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 31.8

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

4. Applied egg-rr 31.8%

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

5. Taylor expanded in x around 0

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

7. Simplified 53.5%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{-1}{8} \cdot \frac{{x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)}{1 + \left|x\right|}}, x\right) \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)\right)}{1 + \left|x\right|}}, x\right) \]

   2. lower-/.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\frac{\frac{-1}{8} \cdot \left({x}^{4} \cdot \left(1 + \frac{1}{1 + \left|x\right|}\right)\right)}{1 + \left|x\right|}}, x\right) \]

10. Simplified 5.1%

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

11. Final simplification 5.1%

    \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(-0.125, \frac{1}{\left|x\right| + 1}, -0.125\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}{\left|x\right| + 1}, x\right) \]
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 2024219 
(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))