Rust f64::asinh

Percentage Accurate: 30.2% → 99.0%

Time: 7.7s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return asinh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 30.2% 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: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 1e-5)
       (copysign (fma (* x x) (/ 0.5 (+ (fabs x) 1.0)) (log1p (fabs x))) x)
       (copysign (log (+ x (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log(((fabs(x) - x) + (-0.5 / x))), x);
	} else if (t_0 <= 1e-5) {
		tmp = copysign(fma((x * x), (0.5 / (fabs(x) + 1.0)), log1p(fabs(x))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 1e-5)
		tmp = copysign(fma(Float64(x * x), Float64(0.5 / Float64(abs(x) + 1.0)), log1p(abs(x))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[With[{TMP1 = Abs[N[(N[(x * x), $MachinePrecision] * N[(0.5 / N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $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[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.8%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000008e-5

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. accelerator-lowering-log1p.f64 N/A

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

      15. fabs-lowering-fabs.f64 100.0

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

   5. Simplified 100.0%

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

   if 1.00000000000000008e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 2: 99.0% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (+ (- (fabs x) x) (/ -0.5 x))) x)
     (if (<= t_0 1e-5)
       (copysign (log1p (fma x (* x 0.5) (fabs x))) x)
       (copysign (log (+ x (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log(((fabs(x) - x) + (-0.5 / x))), x);
	} else if (t_0 <= 1e-5) {
		tmp = copysign(log1p(fma(x, (x * 0.5), fabs(x))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(Float64(abs(x) - x) + Float64(-0.5 / x))), x);
	elseif (t_0 <= 1e-5)
		tmp = copysign(log1p(fma(x, Float64(x * 0.5), abs(x))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision] + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[With[{TMP1 = Abs[N[Log[1 + N[(x * N[(x * 0.5), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.8%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000008e-5

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. fabs-lowering-fabs.f64 7.1

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

   5. Simplified 7.1%

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

      1. +-commutative N/A

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

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

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. sqr-abs N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. fabs-lowering-fabs.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 1.00000000000000008e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.9%

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

Alternative 3: 98.9% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 1e-5)
       (copysign (log1p (fma x (* x 0.5) (fabs x))) x)
       (copysign (log (+ x (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 1e-5) {
		tmp = copysign(log1p(fma(x, (x * 0.5), fabs(x))), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 1e-5)
		tmp = copysign(log1p(fma(x, Float64(x * 0.5), abs(x))), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[With[{TMP1 = Abs[N[Log[1 + N[(x * N[(x * 0.5), $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.2%

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

   3. Taylor expanded in x around -inf

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

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f64 N/A

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

      16. fabs-lowering-fabs.f64 99.4

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

   5. Simplified 99.4%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000008e-5

   1. Initial program 7.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

      4. sqr-abs N/A

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

      5. associate-*r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r* N/A

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

      8. *-lft-identity N/A

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

      9. distribute-rgt-out N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. metadata-eval N/A

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

      16. accelerator-lowering-fma.f64 N/A

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

      17. fabs-lowering-fabs.f64 7.1

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

   5. Simplified 7.1%

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

      1. +-commutative N/A

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

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

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

      5. sqr-abs N/A

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

      6. associate-*l* N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. fabs-lowering-fabs.f64 100.0

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

   7. Applied egg-rr 100.0%

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

   if 1.00000000000000008e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.8%

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

Alternative 4: 98.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -5.0)
     (copysign (log (- (fabs x) x)) x)
     (if (<= t_0 1e-5)
       (copysign (log1p (fabs x)) x)
       (copysign (log (+ x (fabs x))) x)))))

C

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = copysign(log((fabs(x) - x)), x);
	} else if (t_0 <= 1e-5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = Math.copySign(Math.log((Math.abs(x) - x)), x);
	} else if (t_0 <= 1e-5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -5.0:
		tmp = math.copysign(math.log((math.fabs(x) - x)), x)
	elif t_0 <= 1e-5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = copysign(log(Float64(abs(x) - x)), x);
	elseif (t_0 <= 1e-5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 1e-5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 49.2%

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

   3. Taylor expanded in x around -inf

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

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

      4. *-rgt-identity N/A

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

      5. distribute-neg-in N/A

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

      6. mul-1-neg N/A

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

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

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

      8. remove-double-neg N/A

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

      9. sub-neg N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. --lowering--.f64 N/A

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

      16. fabs-lowering-fabs.f64 99.4

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

   5. Simplified 99.4%

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

   if -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000008e-5

   1. Initial program 7.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. fabs-lowering-fabs.f64 98.6

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

   5. Simplified 98.6%

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

   if 1.00000000000000008e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 99.2%

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

Alternative 5: 81.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 10^{-5}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right|\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left|x\right|\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x) 1e-5)
   (copysign (log1p (fabs x)) x)
   (copysign (log (+ x (fabs x))) x)))

C

double code(double x) {
	double tmp;
	if (copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x) <= 1e-5) {
		tmp = copysign(log1p(fabs(x)), x);
	} else {
		tmp = copysign(log((x + fabs(x))), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x) <= 1e-5) {
		tmp = Math.copySign(Math.log1p(Math.abs(x)), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.abs(x))), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x) <= 1e-5:
		tmp = math.copysign(math.log1p(math.fabs(x)), x)
	else:
		tmp = math.copysign(math.log((x + math.fabs(x))), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x) <= 1e-5)
		tmp = copysign(log1p(abs(x)), x);
	else
		tmp = copysign(log(Float64(x + abs(x))), x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], 1e-5], N[With[{TMP1 = Abs[N[Log[1 + N[Abs[x], $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 1.00000000000000008e-5

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

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

      1. accelerator-lowering-log1p.f64 N/A

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

      2. fabs-lowering-fabs.f64 71.3

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

   5. Simplified 71.3%

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

   if 1.00000000000000008e-5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 50.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 100.0%

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

4. Final simplification 78.1%

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

Alternative 6: 12.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot \left(x \cdot 0.5\right)}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.11)
   (copysign (/ (* x (* x 0.5)) (+ (fabs x) 1.0)) x)
   (copysign (log x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.11) {
		tmp = copysign(((x * (x * 0.5)) / (fabs(x) + 1.0)), x);
	} else {
		tmp = copysign(log(x), x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.11) {
		tmp = Math.copySign(((x * (x * 0.5)) / (Math.abs(x) + 1.0)), x);
	} else {
		tmp = Math.copySign(Math.log(x), x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.11:
		tmp = math.copysign(((x * (x * 0.5)) / (math.fabs(x) + 1.0)), x)
	else:
		tmp = math.copysign(math.log(x), x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.11)
		tmp = copysign(Float64(Float64(x * Float64(x * 0.5)) / Float64(abs(x) + 1.0)), x);
	else
		tmp = copysign(log(x), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sign(x) * abs(((x * (x * 0.5)) / (abs(x) + 1.0)));
	else
		tmp = sign(x) * abs(log(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.11], N[With[{TMP1 = Abs[N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 24.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

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

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

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

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

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

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

      14. accelerator-lowering-log1p.f64 N/A

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

      15. fabs-lowering-fabs.f64 61.2

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

   5. Simplified 61.2%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. fabs-lowering-fabs.f64 5.9

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

   8. Simplified 5.9%

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

   if 0.110000000000000001 < x

   1. Initial program 50.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   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. log-lowering-log.f64 31.7

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

   5. Simplified 31.7%

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

4. Final simplification 11.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\mathsf{copysign}\left(\frac{x \cdot \left(x \cdot 0.5\right)}{\left|x\right| + 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.0% 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 30.3%

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

3. Taylor expanded in x around 0

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

   1. accelerator-lowering-log1p.f64 N/A

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

   2. fabs-lowering-fabs.f64 62.1

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

5. Simplified 62.1%

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

Alternative 8: 6.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (copysign (/ (* x (* x 0.5)) (+ (fabs x) 1.0)) x))

C

double code(double x) {
	return copysign(((x * (x * 0.5)) / (fabs(x) + 1.0)), x);
}

Java

public static double code(double x) {
	return Math.copySign(((x * (x * 0.5)) / (Math.abs(x) + 1.0)), x);
}

Python

def code(x):
	return math.copysign(((x * (x * 0.5)) / (math.fabs(x) + 1.0)), x)

Julia

function code(x)
	return copysign(Float64(Float64(x * Float64(x * 0.5)) / Float64(abs(x) + 1.0)), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(((x * (x * 0.5)) / (abs(x) + 1.0)));
end

Wolfram

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

Derivation

1. Initial program 30.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-lft-identity N/A

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

   3. associate-*l/ N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. unpow2 N/A

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

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

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

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

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

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

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

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

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

   14. accelerator-lowering-log1p.f64 N/A

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

   15. fabs-lowering-fabs.f64 48.0

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

5. Simplified 48.0%

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

6. Taylor expanded in x around inf

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

   1. associate-*r/ N/A

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

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

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

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

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

   7. *-commutative N/A

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

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

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

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

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

   10. fabs-lowering-fabs.f64 5.6

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

8. Simplified 5.6%

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

9. Final simplification 5.6%

   \[\leadsto \mathsf{copysign}\left(\frac{x \cdot \left(x \cdot 0.5\right)}{\left|x\right| + 1}, x\right) \]
10. 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 2024198 
(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))