Rust f64::asinh

Percentage Accurate: 30.3% → 99.9%

Time: 2.2s

Alternatives: 3

Speedup: 3.1×

Specification

Math

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

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]

Initial Program: 30.3% accurate, 1.0× speedup

Math

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

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: 99.9% accurate, 1.6× speedup

Math

\[\mathsf{copysign}\left(\sinh^{-1} x, x\right) \]

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[ArcSinh[x], $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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-abs-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f64 99.9%

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

3. Applied rewrites 99.9%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqr-neg-rev N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}\right), x\right) \]

   4. sqrt-prod N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

   6. lower-unsound-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\color{blue}{\sqrt{\mathsf{neg}\left(x\right)}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

   7. lower-neg.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{-x}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

   8. lower-unsound-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{-x} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(x\right)}}\right), x\right) \]

   9. lower-neg.f64 50.4%

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

5. Applied rewrites 50.4%

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

   1. lift-asinh.f64 N/A

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

   2. asinh-def N/A

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

   3. lift-*.f64 N/A

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

   4. fabs-sqr N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-fabs N/A

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

7. Applied rewrites 99.9%

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

   1. lift-copysign.f64 N/A

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

   2. lift-neg.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{neg}\left(\sinh^{-1} x\right)}, x\right) \]

   3. copysign-other-neg N/A

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

   4. lower-copysign.f64 99.9%

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

9. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
10. Add Preprocessing

Alternative 2: 65.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(x):
	t_0 = math.fabs(math.fabs(x))
	tmp = 0
	if math.copysign(math.log((t_0 + math.sqrt(((math.fabs(x) * math.fabs(x)) + 1.0)))), math.fabs(x)) <= 0.5:
		tmp = math.copysign((-1.0 * math.fabs(x)), math.fabs(x))
	else:
		tmp = math.copysign(math.log(t_0), math.fabs(x))
	return math.copysign(1.0, x) * tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	t_0 = abs(abs(x));
	tmp = 0.0;
	if ((sign(abs(x)) * abs(log((t_0 + sqrt(((abs(x) * abs(x)) + 1.0)))))) <= 0.5)
		tmp = sign(abs(x)) * abs((-1.0 * abs(x)));
	else
		tmp = sign(abs(x)) * abs(log(t_0));
	end
	tmp_2 = (sign(x) * abs(1.0)) * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x) < 0.5

   1. Initial program 30.3%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqr-abs-rev N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-fabs.f64 N/A

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

      9. asinh-def-rev N/A

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

      10. lower-asinh.f64 99.9%

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

   3. Applied rewrites 99.9%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqr-neg-rev N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}\right), x\right) \]

      4. sqrt-prod N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

      6. lower-unsound-sqrt.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\color{blue}{\sqrt{\mathsf{neg}\left(x\right)}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{-x}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

      8. lower-unsound-sqrt.f64 N/A

         \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{-x} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(x\right)}}\right), x\right) \]

      9. lower-neg.f64 50.4%

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

   5. Applied rewrites 50.4%

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

      1. lift-asinh.f64 N/A

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

      2. asinh-def N/A

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

      3. lift-*.f64 N/A

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

      4. fabs-sqr N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-neg.f64 N/A

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

      9. neg-fabs N/A

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 52.2%

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

   10. Applied rewrites 52.2%

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

   if 0.5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) #s(literal 1 binary64))))) x)

   1. Initial program 30.3%

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

   2. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fabs.f64 27.4%

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

   4. Applied rewrites 27.4%

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

   5. Taylor expanded in x around 0

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

      1. lower-fabs.f64 18.4%

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

   7. Applied rewrites 18.4%

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

Alternative 3: 52.2% accurate, 3.1× speedup

Math

\[\mathsf{copysign}\left(-1 \cdot x, x\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(-1.0 * x), $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. Step-by-step derivation

   1. lift-log.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. sqr-abs-rev N/A

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

   7. lift-fabs.f64 N/A

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

   8. lift-fabs.f64 N/A

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

   9. asinh-def-rev N/A

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

   10. lower-asinh.f64 99.9%

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

3. Applied rewrites 99.9%

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

   1. lift-fabs.f64 N/A

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

   2. rem-sqrt-square-rev N/A

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

   3. sqr-neg-rev N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)}}\right), x\right) \]

   4. sqrt-prod N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

   5. lower-unsound-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{\mathsf{neg}\left(x\right)} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right)}, x\right) \]

   6. lower-unsound-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\color{blue}{\sqrt{\mathsf{neg}\left(x\right)}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

   7. lower-neg.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{-x}} \cdot \sqrt{\mathsf{neg}\left(x\right)}\right), x\right) \]

   8. lower-unsound-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{-x} \cdot \color{blue}{\sqrt{\mathsf{neg}\left(x\right)}}\right), x\right) \]

   9. lower-neg.f64 50.4%

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

5. Applied rewrites 50.4%

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

   1. lift-asinh.f64 N/A

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

   2. asinh-def N/A

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

   3. lift-*.f64 N/A

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

   4. fabs-sqr N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-fabs N/A

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

7. Applied rewrites 99.9%

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

8. Taylor expanded in x around 0

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

   1. lower-*.f64 52.2%

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

10. Applied rewrites 52.2%

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

Developer Target 1: 100.0% accurate, 0.4× speedup

Math

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

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 2025192 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform c (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))