Rust f64::asinh

Percentage Accurate: 29.7% → 99.1%

Time: 12.2s

Alternatives: 15

Speedup: 2.1×

• Could not converge on a ground truth

  1. x = -2.3018748230803487e+288

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log 3 + \left(\log \left(\frac{-1}{x}\right) + \log 0.16666666666666666\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x + x}\right), x\right)\\ \end{array} \end{array} \]

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 3.0) (+ (log (/ -1.0 x)) (log 0.16666666666666666))) x)
     (if (<= t_0 0.05)
       (copysign
        (fma
         (* x (* x x))
         (fma
          (* x x)
          (fma (* x x) -0.044642857142857144 0.075)
          -0.16666666666666666)
         x)
        x)
       (copysign (log (/ 1.0 (+ x x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign((log(3.0) + (log((-1.0 / x)) + log(0.16666666666666666))), x);
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma((x * (x * x)), fma((x * x), fma((x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	} else {
		tmp = copysign(log((1.0 / (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(Float64(log(3.0) + Float64(log(Float64(-1.0 / x)) + log(0.16666666666666666))), x);
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(Float64(x * Float64(x * x)), fma(Float64(x * x), fma(Float64(x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	else
		tmp = copysign(log(Float64(1.0 / Float64(x + x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[(N[Log[3.0], $MachinePrecision] + N[(N[Log[N[(-1.0 / x), $MachinePrecision]], $MachinePrecision] + N[Log[0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(1.0 / N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.9%

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

   4. Applied egg-rr 45.1%

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

   6. Applied egg-rr 3.7%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \frac{1}{6} + \left(\log 3 + \left(-2 \cdot \log \left(\frac{-1}{x}\right) + 3 \cdot \log \left(\frac{-1}{x}\right)\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \frac{1}{6} + \log 3\right) + \left(-2 \cdot \log \left(\frac{-1}{x}\right) + 3 \cdot \log \left(\frac{-1}{x}\right)\right)}, x\right) \]

      2. distribute-rgt-out N/A

         \[\leadsto \mathsf{copysign}\left(\left(\log \frac{1}{6} + \log 3\right) + \color{blue}{\log \left(\frac{-1}{x}\right) \cdot \left(-2 + 3\right)}, x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{copysign}\left(\left(\log \frac{1}{6} + \log 3\right) + \log \left(\frac{-1}{x}\right) \cdot \color{blue}{1}, x\right) \]

      4. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(\left(\log \frac{1}{6} + \log 3\right) + \color{blue}{\log \left(\frac{-1}{x}\right)}, x\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\frac{-1}{x}\right) + \left(\log \frac{1}{6} + \log 3\right)}, x\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\log \left(\frac{-1}{x}\right) + \color{blue}{\left(\log 3 + \log \frac{1}{6}\right)}, x\right) \]

      7. associate-+l+ N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log 3\right) + \log \frac{1}{6}}, x\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\log 3 + \log \left(\frac{-1}{x}\right)\right)} + \log \frac{1}{6}, x\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 3 + \left(\log \left(\frac{-1}{x}\right) + \log \frac{1}{6}\right)}, x\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 3 + \left(\log \left(\frac{-1}{x}\right) + \log \frac{1}{6}\right)}, x\right) \]

      11. lower-log.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 3} + \left(\log \left(\frac{-1}{x}\right) + \log \frac{1}{6}\right), x\right) \]

      12. lower-+.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\log 3 + \color{blue}{\left(\log \left(\frac{-1}{x}\right) + \log \frac{1}{6}\right)}, x\right) \]

      13. lower-log.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\log 3 + \left(\color{blue}{\log \left(\frac{-1}{x}\right)} + \log \frac{1}{6}\right), x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\log 3 + \left(\log \color{blue}{\left(\frac{-1}{x}\right)} + \log \frac{1}{6}\right), x\right) \]

      15. lower-log.f64 98.9

          \[\leadsto \mathsf{copysign}\left(\log 3 + \left(\log \left(\frac{-1}{x}\right) + \color{blue}{\log 0.16666666666666666}\right), x\right) \]

   9. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log 3 + \left(\log \left(\frac{-1}{x}\right) + \log 0.16666666666666666\right)}, x\right) \]

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x\right)}, x\right) \]

   9. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right)}, x\right) \]

   if 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 44.9

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

   5. Simplified 44.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square N/A

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

      3. lift-fabs.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-fabs.f64 N/A

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

      11. rem-sqrt-square N/A

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

      12. sqrt-prod N/A

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

      13. rem-square-sqrt N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. lift-fabs.f64 N/A

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

      20. sqr-abs N/A

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

      21. lift-*.f64 N/A

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

      22. lift-fabs.f64 N/A

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

      23. lift-fabs.f64 N/A

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

      24. sqr-abs N/A

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

      25. lift-*.f64 N/A

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

      26. +-inverses N/A

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

      27. +-inverses N/A

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

   7. Applied egg-rr 99.4%

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

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x + x}\right), x\right)\\ \end{array} \end{array} \]

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 (- (fabs x) x)) x)
     (if (<= t_0 0.05)
       (copysign
        (fma
         (* x (* x x))
         (fma
          (* x x)
          (fma (* x x) -0.044642857142857144 0.075)
          -0.16666666666666666)
         x)
        x)
       (copysign (log (/ 1.0 (+ x x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma((x * (x * x)), fma((x * x), fma((x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	} else {
		tmp = copysign(log((1.0 / (x + x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(Float64(x * Float64(x * x)), fma(Float64(x * x), fma(Float64(x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	else
		tmp = copysign(log(Float64(1.0 / Float64(x + x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(1.0 / N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in x around -inf

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

      1. lower-copysign.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 97.6

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

   8. Simplified 97.6%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x\right)}, x\right) \]

   9. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right)}, x\right) \]

   if 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 44.9

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

   5. Simplified 44.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square N/A

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

      3. lift-fabs.f64 N/A

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

      4. flip-+ N/A

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

      5. clear-num N/A

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

      6. lift-fabs.f64 N/A

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

      7. rem-sqrt-square N/A

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

      8. sqrt-prod N/A

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

      9. rem-square-sqrt N/A

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

      10. lift-fabs.f64 N/A

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

      11. rem-sqrt-square N/A

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

      12. sqrt-prod N/A

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

      13. rem-square-sqrt N/A

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

      14. +-inverses N/A

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

      15. +-inverses N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fabs.f64 N/A

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

      19. lift-fabs.f64 N/A

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

      20. sqr-abs N/A

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

      21. lift-*.f64 N/A

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

      22. lift-fabs.f64 N/A

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

      23. lift-fabs.f64 N/A

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

      24. sqr-abs N/A

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

      25. lift-*.f64 N/A

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

      26. +-inverses N/A

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

      27. +-inverses N/A

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

   7. Applied egg-rr 99.4%

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

Alternative 3: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

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 (- (fabs x) x)) x)
     (if (<= t_0 0.05)
       (copysign
        (fma
         (* x (* x x))
         (fma
          (* x x)
          (fma (* x x) -0.044642857142857144 0.075)
          -0.16666666666666666)
         x)
        x)
       (copysign (log (+ x x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma((x * (x * x)), fma((x * x), fma((x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(Float64(x * Float64(x * x)), fma(Float64(x * x), fma(Float64(x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in x around -inf

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

      1. lower-copysign.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-fabs.f64 97.6

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

   8. Simplified 97.6%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x\right)}, x\right) \]

   9. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right)}, x\right) \]

   if 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 44.9

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

   5. Simplified 44.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-copysign.f64 44.9

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

      7. lift-fabs.f64 N/A

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

      8. rem-sqrt-square N/A

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

      9. sqrt-prod N/A

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

      10. rem-square-sqrt 44.9

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 4: 56.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (/ (fabs x) (- x)) x)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.05)
       (copysign
        (fma
         (* x (* x x))
         (fma
          (* x x)
          (fma (* x x) -0.044642857142857144 0.075)
          -0.16666666666666666)
         x)
        x)
       t_1))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = copysign((fabs(x) / -x), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma((x * (x * x)), fma((x * x), fma((x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = copysign(Float64(abs(x) / Float64(-x)), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(Float64(x * Float64(x * x)), fma(Float64(x * x), fma(Float64(x * x), -0.044642857142857144, 0.075), -0.16666666666666666), x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[(N[Abs[x], $MachinePrecision] / (-x)), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044642857142857144 + 0.075), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10 or 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 46.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.5

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 16.5

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

   8. Simplified 16.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 14.1

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

   11. Simplified 14.1%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      5. cube-mult N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

      6. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x\right)}, x\right) \]

   9. Simplified 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.044642857142857144, 0.075\right), -0.16666666666666666\right), x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 56.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.075, -0.16666666666666666\right)\right), x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (/ (fabs x) (- x)) x)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.05)
       (copysign
        (fma x (* x (* x (fma (* x x) 0.075 -0.16666666666666666))) x)
        x)
       t_1))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = copysign((fabs(x) / -x), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma(x, (x * (x * fma((x * x), 0.075, -0.16666666666666666))), x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = copysign(Float64(abs(x) / Float64(-x)), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(x, Float64(x * Float64(x * fma(Float64(x * x), 0.075, -0.16666666666666666))), x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[(N[Abs[x], $MachinePrecision] / (-x)), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(x * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.075 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10 or 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 46.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.5

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 16.5

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

   8. Simplified 16.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 14.1

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

   11. Simplified 14.1%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}, x\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \mathsf{copysign}\left(x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) + \color{blue}{x}, x\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right), x\right)}, x\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right), x\right), x\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, x\right), x\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, x\right), x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, x\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\left(\frac{3}{40} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)}\right), x\right), x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{3}{40}} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right), x\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \left({x}^{2} \cdot \frac{3}{40} + \color{blue}{\frac{-1}{6}}\right)\right), x\right), x\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{3}{40}, \frac{-1}{6}\right)}\right), x\right), x\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{3}{40}, \frac{-1}{6}\right)\right), x\right), x\right) \]

      14. lower-*.f64 99.8

          \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.075, -0.16666666666666666\right)\right), x\right), x\right) \]

   9. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.075, -0.16666666666666666\right)\right), x\right)}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.16666666666666666, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (/ (fabs x) (- x)) x)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.05)
       (copysign (fma x (* (* x x) -0.16666666666666666) x) x)
       t_1))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = copysign((fabs(x) / -x), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma(x, ((x * x) * -0.16666666666666666), x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = copysign(Float64(abs(x) / Float64(-x)), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(x, Float64(Float64(x * x) * -0.16666666666666666), x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[(N[Abs[x], $MachinePrecision] / (-x)), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(x * N[(N[(x * x), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10 or 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 46.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.5

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 16.5

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

   8. Simplified 16.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 14.1

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

   11. Simplified 14.1%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, \frac{-1}{6} \cdot {x}^{2}, x\right)}, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right), x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{-1}{6}}, x\right), x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{-1}{6}, x\right), x\right) \]

      8. lower-*.f64 99.6

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

   9. Simplified 99.6%

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

Alternative 7: 56.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ t_1 := \mathsf{copysign}\left(\frac{\left|x\right|}{-x}, x\right)\\ \mathbf{if}\;t\_0 \leq -10:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(x, x \cdot x, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))
        (t_1 (copysign (/ (fabs x) (- x)) x)))
   (if (<= t_0 -10.0)
     t_1
     (if (<= t_0 0.05) (copysign (fma x (* x x) x) x) t_1))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double t_1 = copysign((fabs(x) / -x), x);
	double tmp;
	if (t_0 <= -10.0) {
		tmp = t_1;
	} else if (t_0 <= 0.05) {
		tmp = copysign(fma(x, (x * x), x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	t_1 = copysign(Float64(abs(x) / Float64(-x)), x)
	tmp = 0.0
	if (t_0 <= -10.0)
		tmp = t_1;
	elseif (t_0 <= 0.05)
		tmp = copysign(fma(x, Float64(x * x), x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[(N[Abs[x], $MachinePrecision] / (-x)), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -10.0], t$95$1, If[LessEqual[t$95$0, 0.05], N[With[{TMP1 = Abs[N[(x * N[(x * x), $MachinePrecision] + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10 or 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 46.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.5

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

   5. Simplified 45.5%

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

   6. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 16.5

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

   8. Simplified 16.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 14.1

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

   11. Simplified 14.1%

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

   if -10 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 9.2%

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

   4. Applied egg-rr 98.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)}, x\right) \]
   5. Step-by-step derivation

      1. rem-sqrt-square N/A

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

      2. sqrt-prod N/A

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

      3. lift-fabs.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift--.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 98.2

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

      10. lift-fabs.f64 N/A

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

      11. rem-sqrt-square N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x \cdot x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

      12. sqrt-prod N/A

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

      13. rem-square-sqrt 98.6

          \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{x} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   6. Applied egg-rr 98.6%

      \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right)\right), x\right) \]

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 97.3

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

   9. Simplified 97.3%

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

   10. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. *-rgt-identity N/A

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

       3. cube-mult N/A

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

       4. unpow2 N/A

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

       5. cube-mult N/A

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

       6. unpow2 N/A

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

       7. associate-*l* N/A

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

       8. rgt-mult-inverse N/A

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

       9. *-rgt-identity N/A

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

       10. lower-fma.f64 N/A

           \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, {x}^{2}, x\right)}, x\right) \]

       11. unpow2 N/A

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

       12. lower-*.f64 98.7

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

   12. Simplified 98.7%

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

Alternative 8: 82.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 0.05)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x x)) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 0.05) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= 0.05) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= 0.05:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= 0.05)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 0.05], 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 + x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.050000000000000003

   1. Initial program 21.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-fabs.f64 76.1

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

   5. Simplified 76.1%

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

   if 0.050000000000000003 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 45.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 44.9

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

   5. Simplified 44.9%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. lift-copysign.f64 44.9

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

      7. lift-fabs.f64 N/A

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

      8. rem-sqrt-square N/A

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

      9. sqrt-prod N/A

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

      10. rem-square-sqrt 44.9

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

      11. lift-sqrt.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. sqrt-prod N/A

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

      14. rem-square-sqrt 99.4

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

   7. Applied egg-rr 99.4%

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

Alternative 9: 60.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) -10.0)
   (copysign (/ (fabs x) (- x)) x)
   (copysign (log1p x) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= -10.0) {
		tmp = copysign((fabs(x) / -x), x);
	} else {
		tmp = copysign(log1p(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= -10.0) {
		tmp = Math.copySign((Math.abs(x) / -x), x);
	} else {
		tmp = Math.copySign(Math.log1p(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= -10.0:
		tmp = math.copysign((math.fabs(x) / -x), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= -10.0)
		tmp = copysign(Float64(abs(x) / Float64(-x)), x);
	else
		tmp = copysign(log1p(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], -10.0], N[With[{TMP1 = Abs[N[(N[Abs[x], $MachinePrecision] / (-x)), $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 (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < -10

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 46.1

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

   5. Simplified 46.1%

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

   6. Taylor expanded in x around -inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fabs.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-/.f64 33.4

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

   8. Simplified 33.4%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. distribute-neg-frac2 N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-fabs.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 14.1

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

   11. Simplified 14.1%

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

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

   1. Initial program 21.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-fabs.f64 75.6

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

   5. Simplified 75.6%

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

      1. lift-fabs.f64 N/A

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

      2. lift-log1p.f64 N/A

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

      3. lift-copysign.f64 75.6

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

      4. lift-fabs.f64 N/A

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

      5. rem-sqrt-square N/A

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

      6. sqrt-prod N/A

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

      7. rem-square-sqrt 75.6

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

   7. Applied egg-rr 75.6%

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

Alternative 10: 64.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-fabs.f64 64.9

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

5. Simplified 64.9%

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

Alternative 11: 51.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return copysign(fma(x, Float64(x * x), x), x)
end

Wolfram

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

Derivation

1. Initial program 27.4%

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

4. Applied egg-rr 59.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)}, x\right) \]
5. Step-by-step derivation

   1. rem-sqrt-square N/A

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

   2. sqrt-prod N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 59.1

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

   10. lift-fabs.f64 N/A

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

   11. rem-sqrt-square N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x \cdot x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   12. sqrt-prod N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   13. rem-square-sqrt 59.7

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{x} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

6. Applied egg-rr 59.7%

   \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right)\right), x\right) \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 51.6

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

9. Simplified 51.6%

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

10. Taylor expanded in x around inf

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

    1. distribute-lft-in N/A

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

    2. *-rgt-identity N/A

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

    3. cube-mult N/A

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

    4. unpow2 N/A

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

    5. cube-mult N/A

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

    6. unpow2 N/A

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

    7. associate-*l* N/A

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

    8. rgt-mult-inverse N/A

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

    9. *-rgt-identity N/A

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

    10. lower-fma.f64 N/A

        \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, {x}^{2}, x\right)}, x\right) \]

    11. unpow2 N/A

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

    12. lower-*.f64 52.1

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

12. Simplified 52.1%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot x, x\right)}, x\right) \]
13. Add Preprocessing

Alternative 12: 50.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(-0.5, x \cdot x, x\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (fma -0.5 (* x x) x) x))

C

double code(double x) {
	return copysign(fma(-0.5, (x * x), x), x);
}

Julia

function code(x)
	return copysign(fma(-0.5, Float64(x * x), x), x)
end

Wolfram

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

Derivation

1. Initial program 27.4%

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

3. Taylor expanded in x around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-fabs.f64 64.9

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

5. Simplified 64.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-log1p.f64 N/A

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

   3. lift-copysign.f64 64.9

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

   4. lift-fabs.f64 N/A

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

   5. rem-sqrt-square N/A

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt 57.3

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

7. Applied egg-rr 57.3%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. associate-*l* N/A

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

   4. unpow2 N/A

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

   5. *-lft-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 51.6

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

10. Simplified 51.6%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, x\right)}, x\right) \]
11. Add Preprocessing

Alternative 13: 50.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (copysign (* x (+ x 1.0)) x))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

4. Applied egg-rr 59.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)}, x\right) \]
5. Step-by-step derivation

   1. rem-sqrt-square N/A

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

   2. sqrt-prod N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 59.1

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

   10. lift-fabs.f64 N/A

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

   11. rem-sqrt-square N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x \cdot x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   12. sqrt-prod N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   13. rem-square-sqrt 59.7

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{x} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

6. Applied egg-rr 59.7%

   \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right)\right), x\right) \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 51.6

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

9. Simplified 51.6%

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

    1. distribute-lft1-in N/A

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

    2. lower-*.f64 N/A

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

    3. lower-+.f64 51.6

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

11. Applied egg-rr 51.6%

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

12. Final simplification 51.6%

    \[\leadsto \mathsf{copysign}\left(x \cdot \left(x + 1\right), x\right) \]
13. Add Preprocessing

Alternative 14: 50.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.4%

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

4. Applied egg-rr 59.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)}, x\right) \]
5. Step-by-step derivation

   1. rem-sqrt-square N/A

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

   2. sqrt-prod N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 59.1

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

   10. lift-fabs.f64 N/A

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

   11. rem-sqrt-square N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x \cdot x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   12. sqrt-prod N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   13. rem-square-sqrt 59.7

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{x} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

6. Applied egg-rr 59.7%

   \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right)\right), x\right) \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 51.6

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

9. Simplified 51.6%

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

Alternative 15: 12.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x \cdot 2, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* x 2.0) x))

C

double code(double x) {
	return copysign((x * 2.0), x);
}

Java

public static double code(double x) {
	return Math.copySign((x * 2.0), x);
}

Python

def code(x):
	return math.copysign((x * 2.0), x)

Julia

function code(x)
	return copysign(Float64(x * 2.0), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs((x * 2.0));
end

Wolfram

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

Derivation

1. Initial program 27.4%

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

4. Applied egg-rr 59.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left|x\right| \cdot \left(\left|x\right| - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\right)}, x\right) \]
5. Step-by-step derivation

   1. rem-sqrt-square N/A

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

   2. sqrt-prod N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift--.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 59.1

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

   10. lift-fabs.f64 N/A

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

   11. rem-sqrt-square N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x \cdot x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   12. sqrt-prod N/A

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

   13. rem-square-sqrt 59.7

       \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \left(\color{blue}{x} - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x\right)\right), x\right) \]

6. Applied egg-rr 59.7%

   \[\leadsto \mathsf{copysign}\left(\log \left(\mathsf{fma}\left(x, x \cdot x, \mathsf{fma}\left(x, x, 1\right) \cdot \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)\right) - \mathsf{log1p}\left(\mathsf{fma}\left(x, x, \color{blue}{\left(x - \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right) \cdot x}\right)\right), x\right) \]

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 51.6

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

9. Simplified 51.6%

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

10. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 2}, x\right) \]

    2. lower-*.f64 12.2

       \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 2}, x\right) \]

12. Simplified 12.2%

    \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot 2}, x\right) \]
13. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024214 
(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))