Rust f64::asinh

Percentage Accurate: 29.2% → 99.5%

Time: 9.7s

Alternatives: 10

Speedup: 2.0×

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: 29.2% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| + 1\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_2 := {t\_0}^{2}\\ \mathbf{if}\;t\_1 \leq -4:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right) + \left(x \cdot x\right) \cdot \left(\frac{0.5}{t\_0} + x \cdot \left(x \cdot \left(\left(\frac{-0.125}{t\_0} + \frac{-0.125}{t\_2}\right) + 0.001388888888888889 \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{45}{t\_0} + \left(\frac{45}{t\_2} + \frac{30}{{t\_0}^{3}}\right)\right)\right)\right)\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (fabs x) 1.0))
        (t_1 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_2 (pow t_0 2.0)))
   (if (<= t_1 -4.0)
     (copysign (log (+ (fabs x) (hypot 1.0 x))) x)
     (if (<= t_1 0.005)
       (copysign
        (+
         (log1p (fabs x))
         (*
          (* x x)
          (+
           (/ 0.5 t_0)
           (*
            x
            (*
             x
             (+
              (+ (/ -0.125 t_0) (/ -0.125 t_2))
              (*
               0.001388888888888889
               (*
                (* x x)
                (+
                 (/ 45.0 t_0)
                 (+ (/ 45.0 t_2) (/ 30.0 (pow t_0 3.0))))))))))))
        x)
       (copysign (log (+ x (fabs x))) x)))))

C

double code(double x) {
	double t_0 = fabs(x) + 1.0;
	double t_1 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_2 = pow(t_0, 2.0);
	double tmp;
	if (t_1 <= -4.0) {
		tmp = copysign(log((fabs(x) + hypot(1.0, x))), x);
	} else if (t_1 <= 0.005) {
		tmp = copysign((log1p(fabs(x)) + ((x * x) * ((0.5 / t_0) + (x * (x * (((-0.125 / t_0) + (-0.125 / t_2)) + (0.001388888888888889 * ((x * x) * ((45.0 / t_0) + ((45.0 / t_2) + (30.0 / pow(t_0, 3.0)))))))))))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) + 1.0;
	double t_1 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double t_2 = Math.pow(t_0, 2.0);
	double tmp;
	if (t_1 <= -4.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, x))), x);
	} else if (t_1 <= 0.005) {
		tmp = Math.copySign((Math.log1p(Math.abs(x)) + ((x * x) * ((0.5 / t_0) + (x * (x * (((-0.125 / t_0) + (-0.125 / t_2)) + (0.001388888888888889 * ((x * x) * ((45.0 / t_0) + ((45.0 / t_2) + (30.0 / Math.pow(t_0, 3.0)))))))))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) + 1.0
	t_1 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	t_2 = math.pow(t_0, 2.0)
	tmp = 0
	if t_1 <= -4.0:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, x))), x)
	elif t_1 <= 0.005:
		tmp = math.copysign((math.log1p(math.fabs(x)) + ((x * x) * ((0.5 / t_0) + (x * (x * (((-0.125 / t_0) + (-0.125 / t_2)) + (0.001388888888888889 * ((x * x) * ((45.0 / t_0) + ((45.0 / t_2) + (30.0 / math.pow(t_0, 3.0)))))))))))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) + 1.0)
	t_1 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_2 = t_0 ^ 2.0
	tmp = 0.0
	if (t_1 <= -4.0)
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, x))), x);
	elseif (t_1 <= 0.005)
		tmp = copysign(Float64(log1p(abs(x)) + Float64(Float64(x * x) * Float64(Float64(0.5 / t_0) + Float64(x * Float64(x * Float64(Float64(Float64(-0.125 / t_0) + Float64(-0.125 / t_2)) + Float64(0.001388888888888889 * Float64(Float64(x * x) * Float64(Float64(45.0 / t_0) + Float64(Float64(45.0 / t_2) + Float64(30.0 / (t_0 ^ 3.0)))))))))))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -4.0], 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$1, 0.005], N[With[{TMP1 = Abs[N[(N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(N[(0.5 / t$95$0), $MachinePrecision] + N[(x * N[(x * N[(N[(N[(-0.125 / t$95$0), $MachinePrecision] + N[(-0.125 / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[(N[(x * x), $MachinePrecision] * N[(N[(45.0 / t$95$0), $MachinePrecision] + N[(N[(45.0 / t$95$2), $MachinePrecision] + N[(30.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 57.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.0050000000000000001

   1. Initial program 6.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-1}{24} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right) + \frac{1}{720} \cdot \left({x}^{2} \cdot \left(45 \cdot \frac{1}{1 + \left|x\right|} + \left(45 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}} + 30 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{3}}\right)\right)\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)\right)}, x\right) \]

   7. Simplified 100.0%

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

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

   1. Initial program 52.4%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \mathbf{if}\;t\_0 \leq -4:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -4.0)
     t_1
     (if (<= t_0 5e-14)
       (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) 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 <= -4.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-14) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * 0.5))), 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 <= -4.0) {
		tmp = t_1;
	} else if (t_0 <= 5e-14) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * 0.5))), 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 <= -4.0:
		tmp = t_1
	elif t_0 <= 5e-14:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * 0.5))), 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 <= -4.0)
		tmp = t_1;
	elseif (t_0 <= 5e-14)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * 0.5))), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[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, -4.0], t$95$1, If[LessEqual[t$95$0, 5e-14], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $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) < -4 or 5.0000000000000002e-14 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 55.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 99.9%

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

   3. Simplified 99.9%

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

   if -4 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 5.0000000000000002e-14

   1. Initial program 5.6%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   6. 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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 5.6%

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

   7. Simplified 5.6%

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. log1p-define N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.9)
   (copysign (log (- (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x)) x)
   (if (<= x 1.5)
     (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.9) {
		tmp = copysign(log(((fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) - x)), x);
	} else if (x <= 1.5) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.9) {
		tmp = Math.copySign(Math.log(((Math.abs(x) + ((-0.5 + (0.125 / (x * x))) / x)) - x)), x);
	} else if (x <= 1.5) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.9:
		tmp = math.copysign(math.log(((math.fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) - x)), x)
	elif x <= 1.5:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * 0.5))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.9)
		tmp = copysign(log(Float64(Float64(abs(x) + Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x)) - x)), x);
	elseif (x <= 1.5)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * 0.5))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.9], N[With[{TMP1 = Abs[N[Log[N[(N[(N[Abs[x], $MachinePrecision] + N[(N[(-0.5 + N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.5], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.900000000000000022

   1. Initial program 57.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + \left(\mathsf{neg}\left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 - \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x}\right)\right)}, x\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{copysign.f64}\left(\log \left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{\frac{1}{2} + \frac{\frac{-1}{8}}{x \cdot x}}{x}}{x}\right)\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 99.8%

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

   if -0.900000000000000022 < x < 1.5

   1. Initial program 6.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   6. 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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 6.1%

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

   7. Simplified 6.1%

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. log1p-define N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 1.5 < x

   1. Initial program 52.4%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (copysign (log (* x (+ -1.0 (/ (+ (fabs x) (/ -0.5 x)) x)))) x)
   (if (<= x 1.5)
     (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = copysign(log((x * (-1.0 + ((fabs(x) + (-0.5 / x)) / x)))), x);
	} else if (x <= 1.5) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = Math.copySign(Math.log((x * (-1.0 + ((Math.abs(x) + (-0.5 / x)) / x)))), x);
	} else if (x <= 1.5) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = math.copysign(math.log((x * (-1.0 + ((math.fabs(x) + (-0.5 / x)) / x)))), x)
	elif x <= 1.5:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * 0.5))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = copysign(log(Float64(x * Float64(-1.0 + Float64(Float64(abs(x) + Float64(-0.5 / x)) / x)))), x);
	elseif (x <= 1.5)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * 0.5))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[Log[N[(x * N[(-1.0 + N[(N[(N[Abs[x], $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.5], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 57.5%

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

   5. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| - \frac{1}{2} \cdot \frac{1}{x}\right), x\right)\right)\right)\right)\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left|x\right|\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right)\right)\right)\right), x\right) \]

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

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{x}\right)\right)\right), x\right)\right)\right)\right)\right), x\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{x}\right)\right)\right), x\right)\right)\right)\right)\right), x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{x}\right)\right)\right), x\right)\right)\right)\right)\right), x\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{x}\right)\right), x\right)\right)\right)\right)\right), x\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \left(\frac{\frac{-1}{2}}{x}\right)\right), x\right)\right)\right)\right)\right), x\right) \]

      16. /-lowering-/.f64 99.3%

          \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{fabs.f64}\left(x\right), \mathsf{/.f64}\left(\frac{-1}{2}, x\right)\right), x\right)\right)\right)\right)\right), x\right) \]

   7. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x \cdot \left(1 - \frac{\left|x\right| + \frac{-0.5}{x}}{x}\right)\right)}, x\right) \]

   if -1 < x < 1.5

   1. Initial program 6.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   6. 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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 6.1%

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

   7. Simplified 6.1%

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. log1p-define N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 1.5 < x

   1. Initial program 52.4%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (copysign (log (/ (- (/ 0.125 (* x x)) 0.5) x)) x)
   (if (<= x 1.5)
     (copysign (log1p (+ (fabs x) (* (* x x) 0.5))) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = copysign(log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else if (x <= 1.5) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = Math.copySign(Math.log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else if (x <= 1.5) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * 0.5))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.2:
		tmp = math.copysign(math.log((((0.125 / (x * x)) - 0.5) / x)), x)
	elif x <= 1.5:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * 0.5))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = copysign(log(Float64(Float64(Float64(0.125 / Float64(x * x)) - 0.5) / x)), x);
	elseif (x <= 1.5)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * 0.5))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.5], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996

   1. Initial program 57.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + \left(\mathsf{neg}\left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 - \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. unsub-neg N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 39.5%

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

   10. Simplified 39.5%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.125 - x \cdot \left(x \cdot 0.5\right)}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
   11. Step-by-step derivation

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

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

   12. Applied egg-rr 99.3%

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

   if -1.19999999999999996 < x < 1.5

   1. Initial program 6.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   6. 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. *-lowering-*.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 6.1%

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

   7. Simplified 6.1%

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

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

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

      2. associate-+r+ N/A

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

      3. +-commutative N/A

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

      4. associate-+l+ N/A

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

      5. log1p-define N/A

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 1.5 < x

   1. Initial program 52.4%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot 0.5\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.88)
   (copysign (log (/ (- (/ 0.125 (* x x)) 0.5) x)) x)
   (if (<= x 1.0)
     (copysign (log1p (fabs x)) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.88) {
		tmp = copysign(log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else if (x <= 1.0) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.88) {
		tmp = Math.copySign(Math.log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else if (x <= 1.0) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.88:
		tmp = math.copysign(math.log((((0.125 / (x * x)) - 0.5) / x)), x)
	elif x <= 1.0:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.88)
		tmp = copysign(log(Float64(Float64(Float64(0.125 / Float64(x * x)) - 0.5) / x)), x);
	elseif (x <= 1.0)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -0.880000000000000004

   1. Initial program 57.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + \left(\mathsf{neg}\left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 - \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. unsub-neg N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 39.5%

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

   10. Simplified 39.5%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.125 - x \cdot \left(x \cdot 0.5\right)}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
   11. Step-by-step derivation

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

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

   12. Applied egg-rr 99.3%

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

   if -0.880000000000000004 < x < 1

   1. Initial program 6.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 6.3%

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

   3. Simplified 6.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. 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 99.3%

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

   7. Simplified 99.3%

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

   if 1 < x

   1. Initial program 52.4%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

      8. *-rgt-identity N/A

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

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

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

      10. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{\frac{0.125}{x \cdot x} - 0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.88)
   (copysign (log (/ (- (/ 0.125 (* x x)) 0.5) x)) x)
   (copysign (log1p (fabs x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.88) {
		tmp = copysign(log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else {
		tmp = copysign(log1p(fabs(x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.88) {
		tmp = Math.copySign(Math.log((((0.125 / (x * x)) - 0.5) / x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.88:
		tmp = math.copysign(math.log((((0.125 / (x * x)) - 0.5) / x)), x)
	else:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.88)
		tmp = copysign(log(Float64(Float64(Float64(0.125 / Float64(x * x)) - 0.5) / x)), x);
	else
		tmp = copysign(log1p(abs(x)), x);
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.880000000000000004

   1. Initial program 57.5%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + \left(\mathsf{neg}\left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 - \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. unsub-neg N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 39.5%

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

   10. Simplified 39.5%

       \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{0.125 - x \cdot \left(x \cdot 0.5\right)}{x \cdot \left(x \cdot x\right)}\right)}, x\right) \]
   11. Step-by-step derivation

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

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

   12. Applied egg-rr 99.3%

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

   if -0.880000000000000004 < x

   1. Initial program 21.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 37.6%

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

   3. Simplified 37.6%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   6. 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 76.7%

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

   7. Simplified 76.7%

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

Alternative 8: 35.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309) (copysign (log (/ -0.5 x)) x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = copysign(log((-0.5 / x)), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = Math.copySign(Math.log((-0.5 / x)), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.copysign(math.log((-0.5 / x)), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = copysign(log(Float64(-0.5 / x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = sign(x) * abs(log((-0.5 / x)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], N[With[{TMP1 = Abs[N[Log[N[(-0.5 / x), $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 < -1.000000000000002e-309

   1. Initial program 36.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

      4. neg-sub0 N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + -1 \cdot \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 + \left(\mathsf{neg}\left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(1 - \frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \left(\frac{\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}}{x}\right)\right)\right)\right), x\right) \]

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

         \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\left|x\right| + -1 \cdot \frac{\frac{1}{2} - \frac{1}{8} \cdot \frac{1}{{x}^{2}}}{x}\right), x\right)\right)\right)\right), x\right) \]

   7. Simplified 58.3%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(0 - x\right) \cdot \left(1 - \frac{\left|x\right| - \frac{0.5 + \frac{-0.125}{x \cdot x}}{x}}{x}\right)\right)}, x\right) \]

   8. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. unsub-neg N/A

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

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

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

      6. unpow2 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. cube-mult N/A

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 23.1%

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

   10. Simplified 23.1%

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

   11. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 59.4%

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

   13. Simplified 59.4%

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

   if -1.000000000000002e-309 < x

   1. Initial program 27.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. 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 17.7%

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

   7. Simplified 17.7%

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

Alternative 9: 18.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1e-309) (copysign (log (- 0.0 x)) x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = copysign(log((0.0 - x)), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1e-309) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1e-309:
		tmp = math.copysign(math.log((0.0 - x)), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1e-309)
		tmp = copysign(log(Float64(0.0 - x)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1e-309)
		tmp = sign(x) * abs(log((0.0 - x)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1e-309], 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[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.000000000000002e-309

   1. Initial program 36.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 60.9%

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

   3. Simplified 60.9%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   6. 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 20.2%

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

   7. Simplified 20.2%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(0 - x\right)}, x\right) \]
   8. 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 20.2%

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

   9. Applied egg-rr 20.2%

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

   if -1.000000000000002e-309 < x

   1. Initial program 27.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   6. 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 17.7%

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

   7. Simplified 17.7%

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

4. Final simplification 18.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.9%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 55.4%

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

3. Simplified 55.4%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
6. 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 9.1%

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

7. Simplified 9.1%

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

Developer Target 1: 100.0% 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 2024159 
(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))