Rust f64::asinh

Percentage Accurate: 30.1% → 99.5%

Time: 9.8s

Alternatives: 11

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.0002:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (log (+ (fabs x) (hypot 1.0 x))) x)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.0002)
       (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
       t_1))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = copysign(log((fabs(x) + hypot(1.0, x))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0002) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = t_1;
	}
	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 t_1 = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, x))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.0002) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	t_1 = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, x))), x)
	tmp = 0
	if t_0 <= -10.0:
		tmp = t_1
	elif t_0 <= 0.0002:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = copysign(log(Float64(abs(x) + hypot(1.0, x))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.0002)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	t_1 = sign(x) * abs(log((abs(x) + hypot(1.0, x))));
	tmp = 0.0;
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.0002)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = t_1;
	end
	tmp_2 = 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[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.0002], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$1]]]]

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) < -10 or 2.0000000000000001e-4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 53.6%

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

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), \color{blue}{x}\right) \]

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

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

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

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

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{x \cdot x + 1}\right)\right)\right), x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\sqrt{1 + x \cdot x}\right)\right)\right), x\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{hypot}\left(1, x\right)\right)\right)\right), x\right) \]

      7. hypot-lowering-hypot.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{hypot.f64}\left(1, x\right)\right)\right), x\right) \]

   3. Simplified 100.0%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 2.0000000000000001e-4

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - \left(x + x\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.88:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x} + \left(x + \left|x\right|\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (- 0.0 (+ x x))) x)
   (if (<= x 0.88)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ (/ (+ 0.5 (/ -0.125 (* x x))) x) (+ x (fabs x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((0.0 - (x + x))), x);
	} else if (x <= 0.88) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((((0.5 + (-0.125 / (x * x))) / x) + (x + fabs(x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((0.0 - (x + x))), x);
	} else if (x <= 0.88) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((((0.5 + (-0.125 / (x * x))) / x) + (x + Math.abs(x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((0.0 - (x + x))), x)
	elif x <= 0.88:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((((0.5 + (-0.125 / (x * x))) / x) + (x + math.fabs(x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(0.0 - Float64(x + x))), x);
	elseif (x <= 0.88)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(Float64(Float64(0.5 + Float64(-0.125 / Float64(x * x))) / x) + Float64(x + abs(x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((0.0 - (x + x))));
	elseif (x <= 0.88)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((((0.5 + (-0.125 / (x * x))) / x) + (x + abs(x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(0.0 - N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.88], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(0.5 + N[(-0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 99.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]

   if -1.25 < x < 0.880000000000000004

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 0.880000000000000004 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right)\right)\right), x\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot {x}^{3}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left({x}^{3}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      20. cube-mult N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      24. *-lowering-*.f64 99.0%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left(x + \left(\left|x\right| + \frac{1}{2} \cdot \frac{1}{x}\right)\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right), x\right)} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\frac{0.5 + \frac{-0.125}{x \cdot x}}{x} + \left(x + \left|x\right|\right)\right), x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - \left(x + x\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (- 0.0 (+ x x))) x)
   (if (<= x 0.96)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ (fabs x) (+ x (/ 0.5 x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((0.0 - (x + x))), x);
	} else if (x <= 0.96) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((fabs(x) + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((0.0 - (x + x))), x);
	} else if (x <= 0.96) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + (x + (0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((0.0 - (x + x))), x)
	elif x <= 0.96:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + (x + (0.5 / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(0.0 - Float64(x + x))), x);
	elseif (x <= 0.96)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(abs(x) + Float64(x + Float64(0.5 / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((0.0 - (x + x))));
	elseif (x <= 0.96)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((abs(x) + (x + (0.5 / x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(0.0 - N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 0.96], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[(x + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 99.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]

   if -1.25 < x < 0.95999999999999996

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 0.95999999999999996 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(x \cdot \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 98.8%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(x \cdot \left(1 + \frac{\frac{1}{2}}{x \cdot x}\right) + \left|x\right|\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x \cdot \left(1 + \frac{\frac{1}{2}}{x \cdot x}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(1 \cdot x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x + \frac{\frac{1}{2}}{x \cdot x} \cdot x\right), \left(\left|x\right|\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{\frac{1}{2}}{x \cdot x} \cdot x\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      6. div-inv N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\left(\frac{1}{2} \cdot \frac{1}{x \cdot x}\right) \cdot x\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \left(\frac{1}{x \cdot x} \cdot x\right)\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \left(\frac{1}{{x}^{2}} \cdot x\right)\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      9. pow-flip N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \left({x}^{\left(\mathsf{neg}\left(2\right)\right)} \cdot x\right)\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      10. pow-plus N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\left(\left(\mathsf{neg}\left(2\right)\right) + 1\right)}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{\left(-2 + 1\right)}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot {x}^{-1}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      13. inv-pow N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{1}{2} \cdot \frac{1}{x}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      14. div-inv N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \left(\frac{\frac{1}{2}}{x}\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \left(\left|x\right|\right)\right)\right), x\right) \]

      16. fabs-lowering-fabs.f64 98.8%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(x, \mathsf{/.f64}\left(\frac{1}{2}, x\right)\right), \mathsf{fabs.f64}\left(x\right)\right)\right), x\right) \]

   7. Applied egg-rr 98.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - \left(x + x\right)\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - \left(x + x\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.1:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (- 0.0 (+ x x))) x)
   (if (<= x 1.1)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ (/ 0.5 x) (/ -0.125 (* x (* x x))))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((0.0 - (x + x))), x);
	} else if (x <= 1.1) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log(((0.5 / x) + (-0.125 / (x * (x * x))))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((0.0 - (x + x))), x);
	} else if (x <= 1.1) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log(((0.5 / x) + (-0.125 / (x * (x * x))))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((0.0 - (x + x))), x)
	elif x <= 1.1:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log(((0.5 / x) + (-0.125 / (x * (x * x))))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(0.0 - Float64(x + x))), x);
	elseif (x <= 1.1)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(Float64(0.5 / x) + Float64(-0.125 / Float64(x * Float64(x * x))))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((0.0 - (x + x))));
	elseif (x <= 1.1)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log(((0.5 / x) + (-0.125 / (x * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(0.0 - N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.1], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(0.5 / x), $MachinePrecision] + N[(-0.125 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 99.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]

   if -1.25 < x < 1.1000000000000001

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 1.1000000000000001 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(x \cdot \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) - \frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right) + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot 1}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2}}{{x}^{2}}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left({x}^{2}\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \left(x \cdot x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{4}}\right)\right)\right)\right)\right)\right), x\right) \]

      10. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{4}}\right)\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{\left(2 \cdot 2\right)}}\right)\right)\right)\right)\right), x\right) \]

      12. pow-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{\frac{-1}{8}}{{x}^{2} \cdot {x}^{2}}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{2} \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(\left(x \cdot x\right) \cdot {x}^{2}\right)\right)\right)\right)\right)\right), x\right) \]

      16. associate-*l* N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      18. cube-mult N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot {x}^{3}\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left({x}^{3}\right)\right)\right)\right)\right)\right)\right), x\right) \]

      20. cube-mult N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      21. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right)\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

      24. *-lowering-*.f64 99.0%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right)\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{\frac{1}{2} \cdot {x}^{2} - \frac{1}{8}}{{x}^{3}}\right)}\right), x\right) \]
   7. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} - \frac{\frac{1}{8}}{{x}^{3}}\right)\right), x\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      3. unpow3 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{\left(x \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{\frac{1}{2} \cdot {x}^{2}}{{x}^{2} \cdot x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{{x}^{2} \cdot \frac{1}{2}}{{x}^{2} \cdot x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      6. times-frac N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{{x}^{2}}{{x}^{2}} \cdot \frac{\frac{1}{2}}{x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      7. *-inverses N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(1 \cdot \frac{\frac{1}{2}}{x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot \frac{1}{2}}{x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      9. associate-*l/ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{x} \cdot \frac{1}{2} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \frac{1}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2} \cdot 1}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{2}}{x}\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8} \cdot 1}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{8}}{{x}^{3}}\right)\right)\right)\right), x\right) \]

      19. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{8}\right)}{{x}^{3}}\right)\right)\right), x\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \left(\frac{\frac{-1}{8}}{{x}^{3}}\right)\right)\right), x\right) \]

      21. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left({x}^{3}\right)\right)\right)\right), x\right) \]

      22. cube-mult N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\right), x\right) \]

      23. unpow2 N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{/.f64}\left(\frac{-1}{8}, \left(x \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{2}, x\right), \mathsf{/.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right)\right)\right), x\right) \]

   8. Simplified 98.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.5}{x} + \frac{-0.125}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - \left(x + x\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (copysign (log (- 0.0 (+ x x))) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = copysign(log((0.0 - (x + x))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = Math.copySign(Math.log((0.0 - (x + x))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.25:
		tmp = math.copysign(math.log((0.0 - (x + x))), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = copysign(log(Float64(0.0 - Float64(x + x))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.25)
		tmp = sign(x) * abs(log((0.0 - (x + x))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[With[{TMP1 = Abs[N[Log[N[(0.0 - N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 99.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - \left(x + x\right)\right)}, x\right) \]

   if -1.25 < x < 1.25

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 1.25 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 3.1%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 3.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \left(0 - x\right)\right), \color{blue}{x}\right) \]

   7. Applied egg-rr 98.1%

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

Alternative 6: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.3)
   (copysign (- 0.0 (log (- 1.0 x))) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.3) {
		tmp = copysign((0.0 - log((1.0 - x))), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.3) {
		tmp = Math.copySign((0.0 - Math.log((1.0 - x))), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.3:
		tmp = math.copysign((0.0 - math.log((1.0 - x))), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.3)
		tmp = copysign(Float64(0.0 - log(Float64(1.0 - x))), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.3)
		tmp = sign(x) * abs((0.0 - log((1.0 - x))));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.3], N[With[{TMP1 = Abs[N[(0.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2999999999999998

   1. Initial program 51.6%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), x\right) \]

   5. Simplified 31.8%

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

   7. Applied egg-rr 0.0%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + -1 \cdot x\right)}\right)\right), x\right) \]
   9. Step-by-step derivation

      1. neg-mul-1 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(1 + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), x\right) \]

      2. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(1 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, x\right)\right)\right), x\right) \]

   10. Simplified 31.8%

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

   if -3.2999999999999998 < x < 1.25

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 1.25 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 3.1%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 3.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \left(0 - x\right)\right), \color{blue}{x}\right) \]

   7. Applied egg-rr 98.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3:\\ \;\;\;\;\mathsf{copysign}\left(0 - \log \left(1 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.5)
   (copysign (log (- 0.0 x)) x)
   (if (<= x 1.25)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (+ x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = copysign(log((0.0 - x)), x);
	} else if (x <= 1.25) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.5) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else if (x <= 1.25) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.5:
		tmp = math.copysign(math.log((0.0 - x)), x)
	elif x <= 1.25:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.5)
		tmp = copysign(log(Float64(0.0 - x)), x);
	elseif (x <= 1.25)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.5)
		tmp = sign(x) * abs(log((0.0 - x)));
	elseif (x <= 1.25)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.5], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.25], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), x\right) \]

      3. --lowering--.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), x\right) \]

   5. Simplified 31.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x\right)}, x\right) \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), x\right) \]

   7. Applied egg-rr 31.8%

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

   if -3.5 < x < 1.25

   1. Initial program 8.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 8.9%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 8.9%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 100.0%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 1.25 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \color{blue}{\left(-1 \cdot x\right)}\right)\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(x\right)\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(0 - x\right)\right)\right), x\right) \]

      3. --lowering--.f64 3.1%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{\_.f64}\left(0, x\right)\right)\right), x\right) \]

   5. Simplified 3.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\left(0 - x\right)}\right), x\right) \]
   6. Step-by-step derivation

      1. copysign-lowering-copysign.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left|x\right| + \left(0 - x\right)\right), \color{blue}{x}\right) \]

   7. Applied egg-rr 98.1%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) (copysign (log (- 0.0 x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = copysign(log((0.0 - x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.copysign(math.log((0.0 - x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = copysign(log(Float64(0.0 - x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 51.6%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(0 - x\right)\right), x\right) \]

      3. --lowering--.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right), x\right) \]

   5. Simplified 31.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x\right)}, x\right) \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right), x\right) \]

      2. neg-lowering-neg.f64 31.8%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{neg.f64}\left(x\right)\right), x\right) \]

   7. Applied egg-rr 31.8%

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

   if -1 < x

   1. Initial program 23.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f64 77.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), x\right) \]

   5. Simplified 77.3%

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

   7. Applied egg-rr 25.1%

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

   9. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55)
   (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
   (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 21.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 9.5%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 9.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 72.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 72.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 1.55000000000000004 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   4. Step-by-step derivation

      1. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

      2. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(\left|x\right|\right)\right), x\right) \]

      3. fabs-lowering-fabs.f64 31.1%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), x\right) \]

   5. Simplified 31.1%

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

   7. Applied egg-rr 7.2%

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

   9. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.0)
   (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $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 < 2

   1. Initial program 21.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(1 + \left|x\right|\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left(\left|x\right| + 1\right) + \frac{1}{2} \cdot {x}^{2}\right)\right), x\right) \]

      3. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\left(\left|x\right| + \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(1 + \frac{1}{2} \cdot {x}^{2}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot {x}^{2}\right)\right)\right)\right), x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right)\right)\right), x\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

      11. *-lowering-*.f64 9.5%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{1}{2}\right)\right)\right)\right)\right), x\right) \]

   5. Simplified 9.5%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(1 + x \cdot \left(x \cdot \frac{1}{2}\right)\right) + \left|x\right|\right), x\right) \]

      2. associate-+l+ N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(1 + \left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|x\right|\right)\right), x\right) \]

      5. neg-fabs N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\mathsf{neg}\left(x\right)\right|\right)\right), x\right) \]

      6. sub0-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|0 - x\right|\right)\right), x\right) \]

      7. flip3-- N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \left|\frac{{0}^{3} - {x}^{3}}{0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)}\right|\right)\right), x\right) \]

      8. fabs-div N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 \cdot 0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|0 + \left(x \cdot x + 0 \cdot x\right)\right|}\right)\right), x\right) \]

      10. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x + 0 \cdot x\right|}\right)\right), x\right) \]

      11. distribute-rgt-out N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(x + 0\right)\right|}\right)\right), x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot \left(0 + x\right)\right|}\right)\right), x\right) \]

      13. +-lft-identity N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{\left|x \cdot x\right|}\right)\right), x\right) \]

      14. fabs-sqr N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(x \cdot \left(x \cdot \frac{1}{2}\right) + \frac{\left|{0}^{3} - {x}^{3}\right|}{x \cdot x}\right)\right), x\right) \]

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right)}, x\right) \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {x}^{2}\right)\right)\right), x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left({x}^{2} \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(\left(x \cdot x\right) \cdot \frac{-1}{6}\right)\right)\right), x\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right), x\right) \]

      7. *-lowering-*.f64 72.3%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right), x\right) \]

   10. Simplified 72.3%

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right)}, x\right) \]

   if 2 < x

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{x}\right)\right)\right), x\right) \]

      2. log-rec N/A

         \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log x\right)\right)\right)\right), x\right) \]

      3. remove-double-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\log x, x\right) \]

      4. log-lowering-log.f64 31.0%

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(x\right), x\right) \]

   5. Simplified 31.0%

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

Alternative 11: 52.2% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Wolfram

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

Derivation

1. Initial program 29.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
4. Step-by-step derivation

   1. log1p-define N/A

      \[\leadsto \mathsf{copysign.f64}\left(\left(\mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   2. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\left(\left|x\right|\right)\right), x\right) \]

   3. fabs-lowering-fabs.f64 67.4%

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\mathsf{fabs.f64}\left(x\right)\right), x\right) \]

5. Simplified 67.4%

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

7. Applied egg-rr 19.6%

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

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{x}, x\right) \]
9. Step-by-step derivation

10. Simplified 56.3%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
11. 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 2024138 
(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))