Rust f64::asinh

Percentage Accurate: 29.2% → 99.4%

Time: 8.4s

Alternatives: 14

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 29.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \left|x\right|\\ t_1 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_2 := {t\_0}^{2}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_1 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), \frac{45}{t\_0} + \left(\frac{30}{{t\_0}^{3}} + \frac{45}{t\_2}\right), \frac{-0.125}{t\_0} + \frac{-0.125}{t\_2}\right), x \cdot x, \frac{0.5}{t\_0}\right), x \cdot x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- 1.0 (fabs x)))
        (t_1 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_2 (pow t_0 2.0)))
   (if (<= t_1 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_1 0.001)
       (copysign
        (-
         (fma
          (fma
           (fma
            (* 0.001388888888888889 (* x x))
            (+ (/ 45.0 t_0) (+ (/ 30.0 (pow t_0 3.0)) (/ 45.0 t_2)))
            (+ (/ -0.125 t_0) (/ -0.125 t_2)))
           (* x x)
           (/ 0.5 t_0))
          (* x x)
          (log1p (- (fabs x)))))
        x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = 1.0 - fabs(x);
	double t_1 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_2 = pow(t_0, 2.0);
	double tmp;
	if (t_1 <= -1.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_1 <= 0.001) {
		tmp = copysign(-fma(fma(fma((0.001388888888888889 * (x * x)), ((45.0 / t_0) + ((30.0 / pow(t_0, 3.0)) + (45.0 / t_2))), ((-0.125 / t_0) + (-0.125 / t_2))), (x * x), (0.5 / t_0)), (x * x), log1p(-fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 - abs(x))
	t_1 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_2 = t_0 ^ 2.0
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_1 <= 0.001)
		tmp = copysign(Float64(-fma(fma(fma(Float64(0.001388888888888889 * Float64(x * x)), Float64(Float64(45.0 / t_0) + Float64(Float64(30.0 / (t_0 ^ 3.0)) + Float64(45.0 / t_2))), Float64(Float64(-0.125 / t_0) + Float64(-0.125 / t_2))), Float64(x * x), Float64(0.5 / t_0)), Float64(x * x), log1p(Float64(-abs(x))))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$1, 0.001], N[With[{TMP1 = Abs[(-N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(45.0 / t$95$0), $MachinePrecision] + N[(N[(30.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] + N[(45.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.125 / t$95$0), $MachinePrecision] + N[(-0.125 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(0.5 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + (-N[Abs[x], $MachinePrecision])], $MachinePrecision]), $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 8.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 8.5

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

   4. Applied rewrites 8.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. /-rgt-identity N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 8.6

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 8.5%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 100.0%

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

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), \frac{45}{1 - \left|x\right|} + \left(\frac{30}{{\left(1 - \left|x\right|\right)}^{3}} + \frac{45}{{\left(1 - \left|x\right|\right)}^{2}}\right), \frac{-0.125}{1 - \left|x\right|} + \frac{-0.125}{{\left(1 - \left|x\right|\right)}^{2}}\right), x \cdot x, \frac{0.5}{1 - \left|x\right|}\right), x \cdot x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := 1 - \left|x\right|\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.125}{t\_1} + \frac{-0.125}{{t\_1}^{2}}, x \cdot x, \frac{0.5}{t\_1}\right), x \cdot x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (- 1.0 (fabs x))))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.001)
       (copysign
        (-
         (fma
          (fma (+ (/ -0.125 t_1) (/ -0.125 (pow t_1 2.0))) (* x x) (/ 0.5 t_1))
          (* x x)
          (log1p (- (fabs x)))))
        x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = 1.0 - fabs(x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(-fma(fma(((-0.125 / t_1) + (-0.125 / pow(t_1, 2.0))), (x * x), (0.5 / t_1)), (x * x), log1p(-fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = Float64(1.0 - abs(x))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(Float64(-fma(fma(Float64(Float64(-0.125 / t_1) + Float64(-0.125 / (t_1 ^ 2.0))), Float64(x * x), Float64(0.5 / t_1)), Float64(x * x), log1p(Float64(-abs(x))))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[(-N[(N[(N[(N[(-0.125 / t$95$1), $MachinePrecision] + N[(-0.125 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + (-N[Abs[x], $MachinePrecision])], $MachinePrecision]), $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 8.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 8.5

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

   4. Applied rewrites 8.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-+.f64 N/A

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

      5. /-rgt-identity N/A

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

      6. clear-num N/A

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

      7. lift-/.f64 N/A

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

      8. neg-log N/A

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

      9. lift-log.f64 N/A

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

      10. lift-neg.f64 8.6

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

      11. lift-/.f64 N/A

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

   6. Applied rewrites 8.5%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(1 - \left|x\right|\right) + {x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 - \left|x\right|} + 3 \cdot \frac{1}{{\left(1 - \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 - \left|x\right|}\right)\right)}, x\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   9. Applied rewrites 100.0%

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

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(-\mathsf{fma}\left(\mathsf{fma}\left(\frac{-0.125}{1 - \left|x\right|} + \frac{-0.125}{{\left(1 - \left|x\right|\right)}^{2}}, x \cdot x, \frac{0.5}{1 - \left|x\right|}\right), x \cdot x, \mathsf{log1p}\left(-\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ t_1 := 1 + \left|x\right|\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{t\_1} + \frac{-0.125}{\left(1 + x\right) \cdot \left(1 + x\right)}\right) \cdot x, x, \frac{0.5}{t\_1}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x))
        (t_1 (+ 1.0 (fabs x))))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.001)
       (copysign
        (fma
         (fma
          (* (+ (/ -0.125 t_1) (/ -0.125 (* (+ 1.0 x) (+ 1.0 x)))) x)
          x
          (/ 0.5 t_1))
         (* x x)
         (log1p (fabs x)))
        x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double t_1 = 1.0 + fabs(x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(fma(fma((((-0.125 / t_1) + (-0.125 / ((1.0 + x) * (1.0 + x)))) * x), x, (0.5 / t_1)), (x * x), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	t_1 = Float64(1.0 + abs(x))
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(fma(fma(Float64(Float64(Float64(-0.125 / t_1) + Float64(-0.125 / Float64(Float64(1.0 + x) * Float64(1.0 + x)))) * x), x, Float64(0.5 / t_1)), Float64(x * x), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[N[(N[(N[(N[(N[(-0.125 / t$95$1), $MachinePrecision] + N[(-0.125 / N[(N[(1.0 + x), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 8.5

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 8.5

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right) + {x}^{2} \cdot \left(\frac{-1}{24} \cdot \left({x}^{2} \cdot \left(3 \cdot \frac{1}{1 + \left|x\right|} + 3 \cdot \frac{1}{{\left(1 + \left|x\right|\right)}^{2}}\right)\right) + \frac{1}{2} \cdot \frac{1}{1 + \left|x\right|}\right)}, x\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{\left(x + 1\right) \cdot \left(x + 1\right)} + \frac{-0.125}{1 + \left|x\right|}\right) \cdot x, x, \frac{0.5}{1 + \left|x\right|}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right) \]

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\frac{-0.125}{1 + \left|x\right|} + \frac{-0.125}{\left(1 + x\right) \cdot \left(1 + x\right)}\right) \cdot x, x, \frac{0.5}{1 + \left|x\right|}\right), x \cdot x, \mathsf{log1p}\left(\left|x\right|\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\frac{0.5}{1 + x}, x \cdot x, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.001)
       (copysign (fma (/ 0.5 (+ 1.0 x)) (* x x) (log1p x)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(fma((0.5 / (1.0 + x)), (x * x), log1p(x)), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(fma(Float64(0.5 / Float64(1.0 + x)), Float64(x * x), log1p(x)), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[N[(N[(0.5 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[Log[1 + x], $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. difference-of-sqr-1 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. flip-+ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 8.5

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

   4. Applied rewrites 8.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

   6. Applied rewrites 8.6%

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

   7. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 99.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{0.5}{1 + \left|x\right|} \cdot x, x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   11. Applied rewrites 99.5%

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\frac{0.5}{x + 1}, \color{blue}{x \cdot x}, \mathsf{log1p}\left(x\right)\right), x\right) \]

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\frac{0.5}{1 + x}, x \cdot x, \mathsf{log1p}\left(x\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (+ (- (/ -0.5 x) x) (fabs x))) x)
     (if (<= t_0 0.001)
       (copysign (/ (* (fma 0.25 (* x x) -1.0) x) (fma -0.5 x -1.0)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((((-0.5 / x) - x) + fabs(x))), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(((fma(0.25, (x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(Float64(Float64(-0.5 / x) - x) + abs(x))), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(Float64(Float64(fma(0.25, Float64(x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[(-0.5 / x), $MachinePrecision] - x), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[N[(N[(N[(0.25 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(-0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. sub-neg N/A

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

      7. associate-*l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

      11. associate-*l/ N/A

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

      12. lft-mult-inverse N/A

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

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

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. distribute-neg-frac N/A

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

      18. metadata-eval N/A

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

      19. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.9%

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

   12. Applied rewrites 97.9%

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, -1\right)}, x\right) \]

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(\frac{-0.5}{x} - x\right) + \left|x\right|\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.001)
       (copysign (/ (* (fma 0.25 (* x x) -1.0) x) (fma -0.5 x -1.0)) x)
       (copysign (log (+ (- x (/ -0.5 x)) (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(((fma(0.25, (x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	} else {
		tmp = copysign(log(((x - (-0.5 / x)) + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(Float64(Float64(fma(0.25, Float64(x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	else
		tmp = copysign(log(Float64(Float64(x - Float64(-0.5 / x)) + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[N[(N[(N[(0.25 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(-0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.9%

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

   12. Applied rewrites 97.9%

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, -1\right)}, x\right) \]

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. distribute-lft-in N/A

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

      2. *-rgt-identity N/A

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

      3. cancel-sign-sub N/A

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

      4. mul-1-neg N/A

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/l* N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. distribute-lft-neg-out N/A

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

      15. rgt-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. neg-mul-1 N/A

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

      18. lower--.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left(x - \frac{-0.5}{x}\right) + \left|x\right|\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right)\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;t\_0 \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x)))
   (if (<= t_0 -1.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 0.001)
       (copysign (/ (* (fma 0.25 (* x x) -1.0) x) (fma -0.5 x -1.0)) x)
       (copysign (log (+ (fabs x) x)) x)))))

C

double code(double x) {
	double t_0 = copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 0.001) {
		tmp = copysign(((fma(0.25, (x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 0.001)
		tmp = copysign(Float64(Float64(fma(0.25, Float64(x * x), -1.0) * x) / fma(-0.5, x, -1.0)), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.001], N[With[{TMP1 = Abs[N[(N[(N[(0.25 * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision] / N[(-0.5 * x + -1.0), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + 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) < -1

   1. Initial program 61.7%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-lft-identity N/A

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

      5. distribute-neg-in N/A

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

      6. *-commutative N/A

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

      7. mul-1-neg N/A

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

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

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

      9. remove-double-neg N/A

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

      10. sub-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*l/ N/A

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

      13. associate-/l* N/A

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

      14. *-inverses N/A

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

      15. *-rgt-identity N/A

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

      16. lower--.f64 N/A

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

      17. lower-fabs.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1e-3

   1. Initial program 8.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 97.9

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

   5. Applied rewrites 97.9%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.9%

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

   12. Applied rewrites 97.9%

       \[\leadsto \mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, \color{blue}{x}, -1\right)}, x\right) \]

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\frac{\mathsf{fma}\left(0.25, x \cdot x, -1\right) \cdot x}{\mathsf{fma}\left(-0.5, x, -1\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (sqrt (+ 1.0 (* x x))) (fabs x))) x) 0.001)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ (fabs x) x)) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x) <= 0.001) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((fabs(x) + x)), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.sqrt((1.0 + (x * x))) + Math.abs(x))), x) <= 0.001) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((Math.abs(x) + x)), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.sqrt((1.0 + (x * x))) + math.fabs(x))), x) <= 0.001:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((math.fabs(x) + x)), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x) <= 0.001)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(abs(x) + x)), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 0.001], 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[(N[Abs[x], $MachinePrecision] + 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) < 1e-3

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

   if 1e-3 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   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

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-inverses N/A

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

      6. *-rgt-identity N/A

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

      7. *-lft-identity N/A

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

      8. lower-+.f64 N/A

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

      9. lower-fabs.f64 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\sqrt{1 + x \cdot x} + \left|x\right|\right), x\right) \leq 0.001:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\left|x\right| + x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (copysign(log((sqrt((1.0 + (x * x))) + fabs(x))), x) <= -1.0) {
		tmp = copysign((((0.5 / (1.0 + fabs(x))) * 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.sqrt((1.0 + (x * x))) + Math.abs(x))), x) <= -1.0) {
		tmp = Math.copySign((((0.5 / (1.0 + Math.abs(x))) * 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.sqrt((1.0 + (x * x))) + math.fabs(x))), x) <= -1.0:
		tmp = math.copysign((((0.5 / (1.0 + math.fabs(x))) * x) * x), x)
	else:
		tmp = math.copysign(math.log1p(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(sqrt(Float64(1.0 + Float64(x * x))) + abs(x))), x) <= -1.0)
		tmp = copysign(Float64(Float64(Float64(0.5 / Float64(1.0 + abs(x))) * x) * 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[Sqrt[N[(1.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], -1.0], N[With[{TMP1 = Abs[N[(N[(N[(0.5 / N[(1.0 + N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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) < -1

   1. Initial program 61.7%

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

      1. lift-+.f64 N/A

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

      2. flip-+ N/A

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

      3. metadata-eval N/A

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

      4. difference-of-sqr-1 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. difference-of-sqr-1 N/A

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

      8. times-frac N/A

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

      9. lift-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. flip-+ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 61.8

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

   4. Applied rewrites 61.8%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

   6. Applied rewrites 61.8%

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

   7. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   9. Applied rewrites 5.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\frac{0.5}{1 + \left|x\right|} \cdot x, x, \mathsf{log1p}\left(\left|x\right|\right)\right)}, x\right) \]

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 5.8%

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

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

   1. Initial program 21.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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 75.0

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

   5. Applied rewrites 75.0%

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

   7. Applied rewrites 75.0%

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

4. Final simplification 54.5%

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

Alternative 10: 58.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.46:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(-0.5 \cdot x, x, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.46) (copysign (fma (* -0.5 x) x x) x) (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.46) {
		tmp = copysign(fma((-0.5 * x), x, x), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.46)
		tmp = copysign(fma(Float64(-0.5 * x), x, x), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.46], N[With[{TMP1 = Abs[N[(N[(-0.5 * x), $MachinePrecision] * x + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.46

   1. Initial program 29.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 59.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 61.3%

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

   12. Applied rewrites 61.3%

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

   if 1.46 < x

   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

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log.f64 31.5

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

   5. Applied rewrites 31.5%

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

Alternative 11: 65.4% 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 33.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
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 62.0

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

5. Applied rewrites 62.0%

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

Alternative 12: 52.1% accurate, 2.0× speedup

Math

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

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(Float64(-0.5 * x), x, x), x)
end

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
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 62.0

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

5. Applied rewrites 62.0%

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

7. Applied rewrites 52.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 47.5%

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

12. Applied rewrites 47.5%

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

Alternative 13: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (* (fma -0.5 x 1.0) x) x))

C

double code(double x) {
	return copysign((fma(-0.5, x, 1.0) * x), x);
}

Julia

function code(x)
	return copysign(Float64(fma(-0.5, x, 1.0) * x), x)
end

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
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 62.0

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

5. Applied rewrites 62.0%

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

7. Applied rewrites 52.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 47.5%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.5, x, 1\right) \cdot \color{blue}{x}, x\right) \]
11. Add Preprocessing

Alternative 14: 5.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.copySign(((-0.5 * x) * x), x);
}

Python

def code(x):
	return math.copysign(((-0.5 * x) * x), x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((-0.5 * x) * x));
end

Wolfram

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

Derivation

1. Initial program 33.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(1 + \left|x\right|\right)}, x\right) \]
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 62.0

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

5. Applied rewrites 62.0%

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

7. Applied rewrites 52.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 47.5%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 5.6%

    \[\leadsto \mathsf{copysign}\left(\left(-0.5 \cdot x\right) \cdot x, x\right) \]
14. 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 2024270 
(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))