Rust f64::asinh

Percentage Accurate: 30.0% → 99.4%

Time: 8.4s

Alternatives: 9

Speedup: 4.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.0% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \frac{\left|x\right|}{x} - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -10.0)
     (copysign (log (- (* x (/ (fabs x) x)) x)) x)
     (if (<= t_0 2e-6)
       (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
       (copysign (log (+ (fabs x) (hypot 1.0 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 <= -10.0) {
		tmp = copysign(log(((x * (fabs(x) / x)) - x)), x);
	} else if (t_0 <= 2e-6) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = copysign(log((fabs(x) + hypot(1.0, 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 <= -10.0) {
		tmp = Math.copySign(Math.log(((x * (Math.abs(x) / x)) - x)), x);
	} else if (t_0 <= 2e-6) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + Math.hypot(1.0, 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 <= -10.0:
		tmp = math.copysign(math.log(((x * (math.fabs(x) / x)) - x)), x)
	elif t_0 <= 2e-6:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + math.hypot(1.0, 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 <= -10.0)
		tmp = copysign(log(Float64(Float64(x * Float64(abs(x) / x)) - x)), x);
	elseif (t_0 <= 2e-6)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * -0.125))))), x);
	else
		tmp = copysign(log(Float64(abs(x) + hypot(1.0, 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, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[(x * N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 2e-6], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[1.0 ^ 2 + x ^ 2], $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) < -10

   1. Initial program 44.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. remove-double-neg N/A

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

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

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

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

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

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

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

      14. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.99999999999999991e-6

   1. Initial program 8.5%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 8.6%

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

   3. Simplified 8.6%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 8.6%

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

   7. Simplified 8.6%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 100.0%

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

   if 1.99999999999999991e-6 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 44.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \frac{\left|x\right|}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.5)
   (copysign (log (- (* x (/ (fabs x) x)) x)) x)
   (if (<= x 1.45)
     (copysign (log1p (+ (fabs x) (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = copysign(log(((x * (fabs(x) / x)) - x)), x);
	} else if (x <= 1.45) {
		tmp = copysign(log1p((fabs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.5) {
		tmp = Math.copySign(Math.log(((x * (Math.abs(x) / x)) - x)), x);
	} else if (x <= 1.45) {
		tmp = Math.copySign(Math.log1p((Math.abs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.5:
		tmp = math.copysign(math.log(((x * (math.fabs(x) / x)) - x)), x)
	elif x <= 1.45:
		tmp = math.copysign(math.log1p((math.fabs(x) + ((x * x) * (0.5 + ((x * x) * -0.125))))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.5)
		tmp = copysign(log(Float64(Float64(x * Float64(abs(x) / x)) - x)), x);
	elseif (x <= 1.45)
		tmp = copysign(log1p(Float64(abs(x) + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * -0.125))))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.5], N[With[{TMP1 = Abs[N[Log[N[(N[(x * N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.45], N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5

   1. Initial program 44.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. remove-double-neg N/A

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

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

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

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

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

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

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

      14. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

   if -1.5 < x < 1.44999999999999996

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 9.3%

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

   3. Simplified 9.3%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 8.9%

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

   7. Simplified 8.9%

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

   8. Taylor expanded in x around 0

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

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

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

   10. Simplified 99.5%

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

   if 1.44999999999999996 < x

   1. Initial program 43.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

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

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

      9. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \frac{\left|x\right|}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \frac{\left|x\right|}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.16)
   (copysign (log (- (* x (/ (fabs x) x)) x)) x)
   (if (<= x 1.45)
     (copysign (log1p (+ x (* (* x x) (+ 0.5 (* (* x x) -0.125))))) x)
     (copysign (log (+ x (fabs x))) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.16) {
		tmp = copysign(log(((x * (fabs(x) / x)) - x)), x);
	} else if (x <= 1.45) {
		tmp = copysign(log1p((x + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.16) {
		tmp = Math.copySign(Math.log(((x * (Math.abs(x) / x)) - x)), x);
	} else if (x <= 1.45) {
		tmp = Math.copySign(Math.log1p((x + ((x * x) * (0.5 + ((x * x) * -0.125))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.16:
		tmp = math.copysign(math.log(((x * (math.fabs(x) / x)) - x)), x)
	elif x <= 1.45:
		tmp = math.copysign(math.log1p((x + ((x * x) * (0.5 + ((x * x) * -0.125))))), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.16)
		tmp = copysign(log(Float64(Float64(x * Float64(abs(x) / x)) - x)), x);
	elseif (x <= 1.45)
		tmp = copysign(log1p(Float64(x + Float64(Float64(x * x) * Float64(0.5 + Float64(Float64(x * x) * -0.125))))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.16], N[With[{TMP1 = Abs[N[Log[N[(N[(x * N[(N[Abs[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[x, 1.45], N[With[{TMP1 = Abs[N[Log[1 + N[(x + N[(N[(x * x), $MachinePrecision] * N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15999999999999992

   1. Initial program 44.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 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

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. associate--r+ N/A

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

      7. neg-sub0 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. remove-double-neg N/A

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

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

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

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

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

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

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

      14. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

   if -1.15999999999999992 < x < 1.44999999999999996

   1. Initial program 9.2%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 9.3%

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

   3. Simplified 9.3%

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

   5. Taylor expanded in x around 0

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

      1. associate-+r+ N/A

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

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

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

      3. +-commutative N/A

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

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

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

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

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 8.9%

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

   7. Simplified 8.9%

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

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

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

   9. Applied egg-rr 99.5%

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

   if 1.44999999999999996 < x

   1. Initial program 43.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

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

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

      9. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot \frac{\left|x\right|}{x} - x\right), x\right)\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x + \left(x \cdot x\right) \cdot \left(0.5 + \left(x \cdot x\right) \cdot -0.125\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 20.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 39.2%

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

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 75.7%

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

   7. Simplified 75.7%

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

   if 1 < x

   1. Initial program 43.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

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

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

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

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

      4. fabs-lowering-fabs.f64 100.0%

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

   7. Simplified 100.0%

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. lft-mult-inverse N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

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

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

      9. fabs-lowering-fabs.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 81.9%

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

Alternative 5: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (log1p (fabs x)) x))

C

double code(double x) {
	return copysign(log1p(fabs(x)), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log1p(Math.abs(x)), x);
}

Python

def code(x):
	return math.copysign(math.log1p(math.fabs(x)), x)

Julia

function code(x)
	return copysign(log1p(abs(x)), x)
end

Wolfram

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

Derivation

1. Initial program 26.6%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in x around 0

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

   1. log1p-define N/A

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

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

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

   3. fabs-lowering-fabs.f64 64.6%

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

7. Simplified 64.6%

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

Alternative 6: 65.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 44.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 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 31.6%

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

   7. Simplified 31.6%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 31.6%

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

   9. Applied egg-rr 31.6%

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

   if -1 < x

   1. Initial program 20.7%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 39.8%

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

   3. Simplified 39.8%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 9.0%

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

   7. Simplified 9.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}, x\right) \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      8. fabs-lowering-fabs.f64 69.0%

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

   9. Applied egg-rr 69.0%

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

       1. neg-fabs N/A

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

       2. sub0-neg N/A

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

       3. flip3-- N/A

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

       4. fabs-div N/A

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

       5. metadata-eval N/A

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

       6. cube-unmult N/A

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

       7. neg-sub0 N/A

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

       8. fabs-neg N/A

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

       9. mul-fabs N/A

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

       10. mul-fabs N/A

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

       11. cube-mult N/A

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

       12. metadata-eval N/A

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

       13. pow-pow N/A

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

       14. pow2 N/A

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

       15. sqr-abs N/A

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

       16. pow2 N/A

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

       17. pow-pow N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. +-lft-identity N/A

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

       21. distribute-rgt-out N/A

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

       22. +-commutative N/A

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

       23. +-lft-identity N/A

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

       24. fabs-sqr N/A

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

   11. Applied egg-rr 69.0%

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

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\color{blue}{x}\right), x\right) \]
   13. Step-by-step derivation

   14. Simplified 75.4%

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

4. Final simplification 64.6%

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

Alternative 7: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\mathsf{copysign}\left(x \cdot \left(1 + \left(x \cdot x\right) \cdot -0.16666666666666666\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5)
   (copysign (* x (+ 1.0 (* (* x x) -0.16666666666666666))) x)
   (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = Math.copySign((x * (1.0 + ((x * x) * -0.16666666666666666))), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = math.copysign((x * (1.0 + ((x * x) * -0.16666666666666666))), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = copysign(Float64(x * Float64(1.0 + Float64(Float64(x * x) * -0.16666666666666666))), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[1 + x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 20.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 39.2%

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

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 9.0%

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

   7. Simplified 9.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}, x\right) \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      8. fabs-lowering-fabs.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. neg-fabs N/A

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

       2. sub0-neg N/A

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

       3. flip3-- N/A

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

       4. fabs-div N/A

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

       5. metadata-eval N/A

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

       6. cube-unmult N/A

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

       7. neg-sub0 N/A

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

       8. fabs-neg N/A

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

       9. mul-fabs N/A

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

       10. mul-fabs N/A

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

       11. cube-mult N/A

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

       12. metadata-eval N/A

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

       13. pow-pow N/A

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

       14. pow2 N/A

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

       15. sqr-abs N/A

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

       16. pow2 N/A

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

       17. pow-pow N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. +-lft-identity N/A

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

       21. distribute-rgt-out N/A

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

       22. +-commutative N/A

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

       23. +-lft-identity N/A

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

       24. fabs-sqr N/A

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

   11. Applied egg-rr 69.7%

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

   12. Taylor expanded in x around 0

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 67.9%

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

   14. Simplified 67.9%

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

   if 1.5 < x

   1. Initial program 43.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 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 9.6%

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

   7. Simplified 9.6%

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

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

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

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

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

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

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

      8. fabs-lowering-fabs.f64 9.6%

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

   9. Applied egg-rr 9.6%

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

       1. neg-fabs N/A

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

       2. sub0-neg N/A

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

       3. flip3-- N/A

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

       4. fabs-div N/A

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

       5. metadata-eval N/A

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

       6. cube-unmult N/A

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

       7. neg-sub0 N/A

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

       8. fabs-neg N/A

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

       9. mul-fabs N/A

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

       10. mul-fabs N/A

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

       11. cube-mult N/A

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

       12. metadata-eval N/A

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

       13. pow-pow N/A

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

       14. pow2 N/A

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

       15. sqr-abs N/A

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

       16. pow2 N/A

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

       17. pow-pow N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. +-lft-identity N/A

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

       21. distribute-rgt-out N/A

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

       22. +-commutative N/A

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

       23. +-lft-identity N/A

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

       24. fabs-sqr N/A

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

   11. Applied egg-rr 9.6%

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

   12. Taylor expanded in x around 0

       \[\leadsto \mathsf{copysign.f64}\left(\mathsf{log1p.f64}\left(\color{blue}{x}\right), x\right) \]
   13. Step-by-step derivation

   14. Simplified 32.0%

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

Alternative 8: 59.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[With[{TMP1 = Abs[N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 20.9%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 39.2%

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

   3. Simplified 39.2%

      \[\leadsto \color{blue}{\mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right)} \]
   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 9.0%

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

   7. Simplified 9.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(1 + \left(x \cdot x\right) \cdot 0.5\right)\right)}, x\right) \]
   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. +-lowering-+.f64 N/A

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

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

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

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

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

      8. fabs-lowering-fabs.f64 69.7%

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

   9. Applied egg-rr 69.7%

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

       1. neg-fabs N/A

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

       2. sub0-neg N/A

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

       3. flip3-- N/A

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

       4. fabs-div N/A

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

       5. metadata-eval N/A

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

       6. cube-unmult N/A

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

       7. neg-sub0 N/A

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

       8. fabs-neg N/A

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

       9. mul-fabs N/A

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

       10. mul-fabs N/A

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

       11. cube-mult N/A

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

       12. metadata-eval N/A

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

       13. pow-pow N/A

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

       14. pow2 N/A

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

       15. sqr-abs N/A

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

       16. pow2 N/A

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

       17. pow-pow N/A

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

       18. metadata-eval N/A

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

       19. metadata-eval N/A

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

       20. +-lft-identity N/A

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

       21. distribute-rgt-out N/A

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

       22. +-commutative N/A

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

       23. +-lft-identity N/A

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

       24. fabs-sqr N/A

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

   11. Applied egg-rr 69.7%

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

   12. Taylor expanded in x around 0

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 67.9%

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

   14. Simplified 67.9%

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

   if 2 < x

   1. Initial program 43.4%

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

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

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

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

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

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

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

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

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

      5. +-commutative N/A

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

      6. hypot-1-def N/A

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

      7. hypot-lowering-hypot.f64 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{copysign.f64}\left(\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right)\right)}, x\right) \]
   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 32.0%

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

   7. Simplified 32.0%

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

Alternative 9: 52.8% 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 26.6%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
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 54.6%

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

3. Simplified 54.6%

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

5. Taylor expanded in x around 0

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

   1. log1p-define N/A

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

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

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

   3. fabs-lowering-fabs.f64 64.6%

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

7. Simplified 64.6%

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

9. Applied egg-rr 3.2%

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

10. Taylor expanded in x around 0

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

12. Simplified 51.9%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x}, x\right) \]
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 2024160 
(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))