Rust f64::asinh

Percentage Accurate: 29.7% → 99.0%

Time: 8.5s

Alternatives: 13

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.7% 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: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign
      (log (* x (+ -1.0 (/ (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x))))
      x)
     (if (<= t_0 1e-7)
       (copysign (log1p (+ x (* x (* x 0.5)))) x)
       (copysign
        (log
         (*
          x
          (+
           (+ 1.0 (+ (/ (fabs x) x) (/ 0.5 (* x x))))
           (/ -0.125 (* x (* x (* x x)))))))
        x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((x * (-1.0 + ((fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x);
	} else if (t_0 <= 1e-7) {
		tmp = copysign(log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = copysign(log((x * ((1.0 + ((fabs(x) / x) + (0.5 / (x * x)))) + (-0.125 / (x * (x * (x * x))))))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(Math.log((x * (-1.0 + ((Math.abs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x);
	} else if (t_0 <= 1e-7) {
		tmp = Math.copySign(Math.log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = Math.copySign(Math.log((x * ((1.0 + ((Math.abs(x) / x) + (0.5 / (x * x)))) + (-0.125 / (x * (x * (x * x))))))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(math.log((x * (-1.0 + ((math.fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x)
	elif t_0 <= 1e-7:
		tmp = math.copysign(math.log1p((x + (x * (x * 0.5)))), x)
	else:
		tmp = math.copysign(math.log((x * ((1.0 + ((math.fabs(x) / x) + (0.5 / (x * x)))) + (-0.125 / (x * (x * (x * x))))))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(x * Float64(-1.0 + Float64(Float64(abs(x) + Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x)) / x)))), x);
	elseif (t_0 <= 1e-7)
		tmp = copysign(log1p(Float64(x + Float64(x * Float64(x * 0.5)))), x);
	else
		tmp = copysign(log(Float64(x * Float64(Float64(1.0 + Float64(Float64(abs(x) / x) + Float64(0.5 / Float64(x * x)))) + Float64(-0.125 / Float64(x * Float64(x * Float64(x * x))))))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(x * N[(-1.0 + 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]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-7], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x * N[(N[(1.0 + N[(N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.125 / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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 100.0%

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

   8. Taylor expanded in x around -inf

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

   10. Simplified 100.0%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 9.9999999999999995e-8

   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 9.9999999999999995e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 46.2%

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

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. +-commutative N/A

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. pow-sqr N/A

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

      18. /-lowering-/.f64 N/A

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

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot \left(\left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-7}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(\left(1 + \left(\frac{\left|x\right|}{x} + \frac{0.5}{x \cdot x}\right)\right) + \frac{-0.125}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(\left|x\right| + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.75)
   (copysign
    (log (* x (+ -1.0 (/ (+ (fabs x) (/ (+ -0.5 (/ 0.125 (* x x))) x)) x))))
    x)
   (if (<= x 1.0)
     (copysign (log1p (+ x (* x (* x 0.5)))) x)
     (copysign (log (+ x (+ (fabs x) (/ 0.5 x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.75) {
		tmp = copysign(log((x * (-1.0 + ((fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x);
	} else if (x <= 1.0) {
		tmp = copysign(log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = copysign(log((x + (fabs(x) + (0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.75) {
		tmp = Math.copySign(Math.log((x * (-1.0 + ((Math.abs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x);
	} else if (x <= 1.0) {
		tmp = Math.copySign(Math.log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + (Math.abs(x) + (0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.75:
		tmp = math.copysign(math.log((x * (-1.0 + ((math.fabs(x) + ((-0.5 + (0.125 / (x * x))) / x)) / x)))), x)
	elif x <= 1.0:
		tmp = math.copysign(math.log1p((x + (x * (x * 0.5)))), x)
	else:
		tmp = math.copysign(math.log((x + (math.fabs(x) + (0.5 / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.75)
		tmp = copysign(log(Float64(x * Float64(-1.0 + Float64(Float64(abs(x) + Float64(Float64(-0.5 + Float64(0.125 / Float64(x * x))) / x)) / x)))), x);
	elseif (x <= 1.0)
		tmp = copysign(log1p(Float64(x + Float64(x * Float64(x * 0.5)))), x);
	else
		tmp = copysign(log(Float64(x + Float64(abs(x) + Float64(0.5 / x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.75], N[With[{TMP1 = Abs[N[Log[N[(x * N[(-1.0 + 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]), $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[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[(N[Abs[x], $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.75

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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 100.0%

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

   8. Taylor expanded in x around -inf

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

   10. Simplified 100.0%

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

   if -0.75 < 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

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. *-rgt-identity N/A

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

      14. unpow2 N/A

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

      15. sqr-abs N/A

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

      16. associate-/r* N/A

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

      17. *-rgt-identity N/A

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

      18. associate-*r/ N/A

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

      19. rgt-mult-inverse N/A

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

      20. *-commutative N/A

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

      21. associate-*r/ N/A

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

   10. Simplified 99.8%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.75:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \left(-1 + \frac{\left|x\right| + \frac{-0.5 + \frac{0.125}{x \cdot x}}{x}}{x}\right)\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(\left|x\right| + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(\left|x\right| + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.78)
   (copysign (log (- (+ (fabs x) (/ -0.5 x)) x)) x)
   (if (<= x 1.0)
     (copysign (log1p (+ x (* x (* x 0.5)))) x)
     (copysign (log (+ x (+ (fabs x) (/ 0.5 x)))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = copysign(log(((fabs(x) + (-0.5 / x)) - x)), x);
	} else if (x <= 1.0) {
		tmp = copysign(log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = copysign(log((x + (fabs(x) + (0.5 / x)))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.78) {
		tmp = Math.copySign(Math.log(((Math.abs(x) + (-0.5 / x)) - x)), x);
	} else if (x <= 1.0) {
		tmp = Math.copySign(Math.log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = Math.copySign(Math.log((x + (Math.abs(x) + (0.5 / x)))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.78:
		tmp = math.copysign(math.log(((math.fabs(x) + (-0.5 / x)) - x)), x)
	elif x <= 1.0:
		tmp = math.copysign(math.log1p((x + (x * (x * 0.5)))), x)
	else:
		tmp = math.copysign(math.log((x + (math.fabs(x) + (0.5 / x)))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.78)
		tmp = copysign(log(Float64(Float64(abs(x) + Float64(-0.5 / x)) - x)), x);
	elseif (x <= 1.0)
		tmp = copysign(log1p(Float64(x + Float64(x * Float64(x * 0.5)))), x);
	else
		tmp = copysign(log(Float64(x + Float64(abs(x) + Float64(0.5 / x)))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.78], N[With[{TMP1 = Abs[N[Log[N[(N[(N[Abs[x], $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $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[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[(N[Abs[x], $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.78000000000000003

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. fabs-mul N/A

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

      7. *-rgt-identity N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. sub-neg N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. /-lowering-/.f64 99.9%

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

   10. Simplified 99.9%

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

   if -0.78000000000000003 < 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

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. *-rgt-identity N/A

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

      14. unpow2 N/A

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

      15. sqr-abs N/A

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

      16. associate-/r* N/A

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

      17. *-rgt-identity N/A

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

      18. associate-*r/ N/A

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

      19. rgt-mult-inverse N/A

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

      20. *-commutative N/A

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

      21. associate-*r/ N/A

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

   10. Simplified 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.78:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\left|x\right| + \frac{-0.5}{x}\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(\left|x\right| + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around inf

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

      1. fabs-lowering-fabs.f64 99.1%

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

   10. Simplified 99.1%

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

   if -1 < 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

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(1 + \left(\left|x\right| + \frac{1}{2} \cdot {x}^{2}\right)\right)}\right), x\right) \]
   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf

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

      1. distribute-rgt-in N/A

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

      2. *-lft-identity N/A

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

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

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

      4. *-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. unpow2 N/A

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

      7. sqr-abs N/A

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

      8. metadata-eval N/A

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

      9. unpow2 N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r* N/A

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

      12. associate-/l* N/A

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

      13. *-rgt-identity N/A

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

      14. unpow2 N/A

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

      15. sqr-abs N/A

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

      16. associate-/r* N/A

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

      17. *-rgt-identity N/A

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

      18. associate-*r/ N/A

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

      19. rgt-mult-inverse N/A

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

      20. *-commutative N/A

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

      21. associate-*r/ N/A

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

   10. Simplified 99.8%

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

4. Final simplification 99.7%

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

Alternative 5: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.5], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $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 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around inf

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

      1. fabs-lowering-fabs.f64 99.1%

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

   10. Simplified 99.1%

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

   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1.5 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 99.6%

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

Alternative 6: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq -0.92:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + \left(x \cdot x\right) \cdot -0.5}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.5e+102)
   (copysign (log (- 0.0 x)) x)
   (if (<= x -0.92)
     (copysign (log (/ (+ 0.125 (* (* x x) -0.5)) (* x (* x x)))) x)
     (if (<= x 1.5)
       (copysign (log1p (+ x (* x (* x 0.5)))) x)
       (copysign (log (/ 0.5 x)) x)))))

C

double code(double x) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = copysign(log((0.0 - x)), x);
	} else if (x <= -0.92) {
		tmp = copysign(log(((0.125 + ((x * x) * -0.5)) / (x * (x * x)))), x);
	} else if (x <= 1.5) {
		tmp = copysign(log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.5e+102) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else if (x <= -0.92) {
		tmp = Math.copySign(Math.log(((0.125 + ((x * x) * -0.5)) / (x * (x * x)))), x);
	} else if (x <= 1.5) {
		tmp = Math.copySign(Math.log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.5e+102:
		tmp = math.copysign(math.log((0.0 - x)), x)
	elif x <= -0.92:
		tmp = math.copysign(math.log(((0.125 + ((x * x) * -0.5)) / (x * (x * x)))), x)
	elif x <= 1.5:
		tmp = math.copysign(math.log1p((x + (x * (x * 0.5)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.5e+102)
		tmp = copysign(log(Float64(0.0 - x)), x);
	elseif (x <= -0.92)
		tmp = copysign(log(Float64(Float64(0.125 + Float64(Float64(x * x) * -0.5)) / Float64(x * Float64(x * x)))), x);
	elseif (x <= 1.5)
		tmp = copysign(log1p(Float64(x + Float64(x * Float64(x * 0.5)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5.5e+102], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, -0.92], N[With[{TMP1 = Abs[N[Log[N[(N[(0.125 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * 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[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.49999999999999981e102

   1. Initial program 32.0%

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

      \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right), x\right) \]
   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 32.4%

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

   7. Simplified 32.4%

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

   if -5.49999999999999981e102 < x < -0.92000000000000004

   1. Initial program 100.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 100.0%

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

   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. +-lowering-+.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. cube-mult N/A

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.5%

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

   10. Simplified 99.5%

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

   if -0.92000000000000004 < 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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1.5 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq -0.92:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.125 + \left(x \cdot x\right) \cdot -0.5}{x \cdot \left(x \cdot x\right)}\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -10.0)
   (copysign (log (- (/ -0.5 x) x)) x)
   (if (<= x 1.5)
     (copysign (log1p (+ x (* x (* x 0.5)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -10.0) {
		tmp = copysign(log(((-0.5 / x) - x)), x);
	} else if (x <= 1.5) {
		tmp = copysign(log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -10.0) {
		tmp = Math.copySign(Math.log(((-0.5 / x) - x)), x);
	} else if (x <= 1.5) {
		tmp = Math.copySign(Math.log1p((x + (x * (x * 0.5)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -10.0:
		tmp = math.copysign(math.log(((-0.5 / x) - x)), x)
	elif x <= 1.5:
		tmp = math.copysign(math.log1p((x + (x * (x * 0.5)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -10.0)
		tmp = copysign(log(Float64(Float64(-0.5 / x) - x)), x);
	elseif (x <= 1.5)
		tmp = copysign(log1p(Float64(x + Float64(x * Float64(x * 0.5)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[(-0.5 / x), $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[(x + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -10

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 31.2%

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

   10. Simplified 31.2%

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

   if -10 < 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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   if 1.5 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + x \cdot \left(x \cdot 0.5\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.9)
   (copysign (log (- (/ -0.5 x) x)) x)
   (if (<= x 1.26)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = copysign(log(((-0.5 / x) - x)), x);
	} else if (x <= 1.26) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.9) {
		tmp = Math.copySign(Math.log(((-0.5 / x) - x)), x);
	} else if (x <= 1.26) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.9:
		tmp = math.copysign(math.log(((-0.5 / x) - x)), x)
	elif x <= 1.26:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.9)
		tmp = copysign(log(Float64(Float64(-0.5 / x) - x)), x);
	elseif (x <= 1.26)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.9)
		tmp = sign(x) * abs(log(((-0.5 / x) - x)));
	elseif (x <= 1.26)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.9], N[With[{TMP1 = Abs[N[Log[N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.26], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-lft-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

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

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

   7. Simplified 51.4%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 31.2%

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

   10. Simplified 31.2%

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

   if -1.8999999999999999 < x < 1.26000000000000001

   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

       7. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if 1.26000000000000001 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

Alternative 9: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + x \cdot \left(x \cdot -0.16666666666666666\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.95)
   (copysign (log (- 0.0 x)) x)
   (if (<= x 1.26)
     (copysign (* x (+ 1.0 (* x (* x -0.16666666666666666)))) x)
     (copysign (log (/ 0.5 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = copysign(log((0.0 - x)), x);
	} else if (x <= 1.26) {
		tmp = copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = copysign(log((0.5 / x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.95) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else if (x <= 1.26) {
		tmp = Math.copySign((x * (1.0 + (x * (x * -0.16666666666666666)))), x);
	} else {
		tmp = Math.copySign(Math.log((0.5 / x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.95:
		tmp = math.copysign(math.log((0.0 - x)), x)
	elif x <= 1.26:
		tmp = math.copysign((x * (1.0 + (x * (x * -0.16666666666666666)))), x)
	else:
		tmp = math.copysign(math.log((0.5 / x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.95)
		tmp = copysign(log(Float64(0.0 - x)), x);
	elseif (x <= 1.26)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(x * Float64(x * -0.16666666666666666)))), x);
	else
		tmp = copysign(log(Float64(0.5 / x)), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.95)
		tmp = sign(x) * abs(log((0.0 - x)));
	elseif (x <= 1.26)
		tmp = sign(x) * abs((x * (1.0 + (x * (x * -0.16666666666666666)))));
	else
		tmp = sign(x) * abs(log((0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.95], N[With[{TMP1 = Abs[N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.26], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(x * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.94999999999999996

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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 31.2%

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

   7. Simplified 31.2%

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

   if -1.94999999999999996 < x < 1.26000000000000001

   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 6.3%

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

   7. Simplified 6.3%

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. log1p-define N/A

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

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

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

      5. neg-fabs N/A

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

      6. sub0-neg N/A

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

      7. flip3-- N/A

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

      8. metadata-eval N/A

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

      9. sub0-neg N/A

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

      10. cube-neg N/A

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

      11. metadata-eval N/A

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

      12. +-lft-identity N/A

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

      13. distribute-rgt-out N/A

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

      14. +-commutative N/A

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

      15. +-lft-identity N/A

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

      16. div-fabs N/A

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0

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

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

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

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

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

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

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

       7. *-lowering-*.f64 100.0%

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

   12. Simplified 100.0%

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

   if 1.26000000000000001 < x

   1. Initial program 46.2%

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

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

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

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

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

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

      3. +-commutative N/A

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 99.3%

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

   10. Simplified 99.3%

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

Alternative 10: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(0 - x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.5) (copysign (log (- 0.0 x)) x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = copysign(log((0.0 - x)), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.5) {
		tmp = Math.copySign(Math.log((0.0 - x)), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.5:
		tmp = math.copysign(math.log((0.0 - x)), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.5

   1. Initial program 51.6%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   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 31.2%

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

   7. Simplified 31.2%

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

   if -0.5 < x

   1. Initial program 19.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in x around 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%

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

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

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

   9. Applied egg-rr 76.7%

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

Alternative 11: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.56) (copysign x x) (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.56) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.56:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.56)
		tmp = copysign(x, x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.56], N[With[{TMP1 = Abs[x], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 21.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 38.3%

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

   3. Simplified 38.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 76.0%

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

   7. Simplified 76.0%

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

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

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

   9. Applied egg-rr 65.3%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 67.6%

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

   if 1.5600000000000001 < x

   1. Initial program 46.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 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 31.3%

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

   7. Simplified 31.3%

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

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

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

   9. Applied egg-rr 31.3%

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

Alternative 12: 58.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2) (copysign x x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = copysign(x, x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.2) {
		tmp = Math.copySign(x, x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.2:
		tmp = math.copysign(x, x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2)
		tmp = copysign(x, x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.2)
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.2], N[With[{TMP1 = Abs[x], 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 < 3.2000000000000002

   1. Initial program 21.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 38.3%

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

   3. Simplified 38.3%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 76.0%

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

   7. Simplified 76.0%

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

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

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

   9. Applied egg-rr 65.3%

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

   10. Taylor expanded in x around 0

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

   12. Simplified 67.6%

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

   if 3.2000000000000002 < x

   1. Initial program 46.2%

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

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   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 31.3%

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

   7. Simplified 31.3%

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

Alternative 13: 52.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.8%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 53.5%

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

3. Simplified 53.5%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
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 65.0%

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

7. Simplified 65.0%

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

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

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

9. Applied egg-rr 56.9%

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

10. Taylor expanded in x around 0

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

12. Simplified 52.3%

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

Developer Target 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

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

Reproduce

herbie shell --seed 2024161 
(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))