Rust f64::asinh

Average Accuracy: 29.6% → 98.6%

Time: 4.6s

Precision: binary64

Cost: 65672

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.0)
       (copysign (+ x (* -0.16666666666666666 (pow x 3.0))) x)
       (copysign (log (+ x (- x (/ -0.5 x)))) x)))))

C

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

↓

double code(double x) {
	double t_0 = copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
	double tmp;
	if (t_0 <= -1.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = copysign((x + (-0.16666666666666666 * pow(x, 3.0))), x);
	} else {
		tmp = copysign(log((x + (x - (-0.5 / x)))), x);
	}
	return tmp;
}

Java

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

↓

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 <= -1.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 0.0) {
		tmp = Math.copySign((x + (-0.16666666666666666 * Math.pow(x, 3.0))), x);
	} else {
		tmp = Math.copySign(Math.log((x + (x - (-0.5 / x)))), x);
	}
	return tmp;
}

Python

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

↓

def code(x):
	t_0 = math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)
	tmp = 0
	if t_0 <= -1.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 0.0:
		tmp = math.copysign((x + (-0.16666666666666666 * math.pow(x, 3.0))), x)
	else:
		tmp = math.copysign(math.log((x + (x - (-0.5 / x)))), x)
	return tmp

Julia

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

↓

function code(x)
	t_0 = copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 0.0)
		tmp = copysign(Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0))), x);
	else
		tmp = copysign(log(Float64(x + Float64(x - Float64(-0.5 / x)))), x);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	t_0 = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
	tmp = 0.0;
	if (t_0 <= -1.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 0.0)
		tmp = sign(x) * abs((x + (-0.16666666666666666 * (x ^ 3.0))));
	else
		tmp = sign(x) * abs(log((x + (x - (-0.5 / x)))));
	end
	tmp_2 = tmp;
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]

↓

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, -1.0], N[With[{TMP1 = Abs[(-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision])], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[With[{TMP1 = Abs[N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[(x - N[(-0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Target

Original	29.6%
Target	100.0%
Herbie	98.6%

\[\mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, \frac{1}{\left|x\right|}\right) + \frac{1}{\left|x\right|}}\right), x\right) \]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 50.9%

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

   2. Simplified 100.0%

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

      [Start] 50.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 50.9	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 1.8%

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

      [Start] 100.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      flip-+ [=>] 1.8	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      div-sub [=>] 1.8	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      pow2 [=>] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{x}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      pow2 [<=] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 0.5	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 0.5	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 0.5	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 0.5	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 \cdot 1 + x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.5	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1} + x \cdot x}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 0.0	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   4. Simplified 100.0%

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

      [Start] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 1.8	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      div-sub [<=] 2.7	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      unpow2 [=>] 2.7	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 2.7	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 2.7	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \left(1 + \color{blue}{{x}^{2}}\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [=>] 2.7	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \color{blue}{\left({x}^{2} + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate--r+ [=>] 50.9	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-inverses [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 100.0	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      sub-neg [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{-\color{blue}{\left(x + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      +-commutative [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{-\color{blue}{\left(\left(-\mathsf{hypot}\left(1, x\right)\right) + x\right)}}\right), x\right) \]
      distribute-neg-in [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-\left(-\mathsf{hypot}\left(1, x\right)\right)\right) + \left(-x\right)}}\right), x\right) \]
      remove-double-neg [=>] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)} + \left(-x\right)}\right), x\right) \]
      sub-neg [<=] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   5. Applied egg-rr 100.0%

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

      [Start] 100.0	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right), x\right) \]
      log-div [=>] 100.0	\[ \mathsf{copysign}\left(\color{blue}{\log 1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]
      sub-neg [=>] 100.0	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)}, x\right) \]
      metadata-eval [=>] 100.0	\[ \mathsf{copysign}\left(\color{blue}{0} + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]

   6. Simplified 100.0%

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

      [Start] 100.0	\[ \mathsf{copysign}\left(0 + \left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right), x\right) \]
      +-lft-identity [=>] 100.0	\[ \mathsf{copysign}\left(\color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -1 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 0.0

   1. Initial program 7.0%

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

   2. Simplified 7.0%

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

      [Start] 7.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 7.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 7.0	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Taylor expanded in x around 0 6.8%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 + \left(0.5 \cdot {x}^{2} + \left|x\right|\right)\right)}, x\right) \]

   4. Simplified 6.8%

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

      [Start] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \left(0.5 \cdot {x}^{2} + \left|x\right|\right)\right), x\right) \]
      *-commutative [=>] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{{x}^{2} \cdot 0.5} + \left|x\right|\right)\right), x\right) \]
      unpow2 [=>] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.5 + \left|x\right|\right)\right), x\right) \]
      associate-*l* [=>] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \left(\color{blue}{x \cdot \left(x \cdot 0.5\right)} + \left|x\right|\right)\right), x\right) \]
      fma-def [=>] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \color{blue}{\mathsf{fma}\left(x, x \cdot 0.5, \left|x\right|\right)}\right), x\right) \]
      unpow1 [<=] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \mathsf{fma}\left(x, x \cdot 0.5, \left|\color{blue}{{x}^{1}}\right|\right)\right), x\right) \]
      sqr-pow [=>] 2.6	\[ \mathsf{copysign}\left(\log \left(1 + \mathsf{fma}\left(x, x \cdot 0.5, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|\right)\right), x\right) \]
      fabs-sqr [=>] 2.6	\[ \mathsf{copysign}\left(\log \left(1 + \mathsf{fma}\left(x, x \cdot 0.5, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right)\right), x\right) \]
      sqr-pow [<=] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \mathsf{fma}\left(x, x \cdot 0.5, \color{blue}{{x}^{1}}\right)\right), x\right) \]
      unpow1 [=>] 6.8	\[ \mathsf{copysign}\left(\log \left(1 + \mathsf{fma}\left(x, x \cdot 0.5, \color{blue}{x}\right)\right), x\right) \]

   5. Taylor expanded in x around 0 99.8%

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

   if 0.0 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 50.2%

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

   2. Taylor expanded in x around inf 95.2%

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

   3. Simplified 95.2%

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

      [Start] 95.2	\[ \mathsf{copysign}\left(\log \left(0.5 \cdot \frac{1}{x} + \left(\left|x\right| + x\right)\right), x\right) \]
      +-commutative [=>] 95.2	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + x\right) + 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
      associate-+l+ [=>] 95.2	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \left(x + 0.5 \cdot \frac{1}{x}\right)\right)}, x\right) \]
      rem-square-sqrt [<=] 95.2	\[ \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \left(x + 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      fabs-sqr [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \left(x + 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      rem-square-sqrt [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(\color{blue}{x} + \left(x + 0.5 \cdot \frac{1}{x}\right)\right), x\right) \]
      *-commutative [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x + \color{blue}{\frac{1}{x} \cdot 0.5}\right)\right), x\right) \]
      cancel-sign-sub [<=] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \color{blue}{\left(x - \left(-\frac{1}{x}\right) \cdot 0.5\right)}\right), x\right) \]
      distribute-lft-neg-in [<=] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \color{blue}{\left(-\frac{1}{x} \cdot 0.5\right)}\right)\right), x\right) \]
      *-commutative [<=] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \left(-\color{blue}{0.5 \cdot \frac{1}{x}}\right)\right)\right), x\right) \]
      associate-*r/ [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \left(-\color{blue}{\frac{0.5 \cdot 1}{x}}\right)\right)\right), x\right) \]
      metadata-eval [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \left(-\frac{\color{blue}{0.5}}{x}\right)\right)\right), x\right) \]
      distribute-neg-frac [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \color{blue}{\frac{-0.5}{x}}\right)\right), x\right) \]
      metadata-eval [=>] 95.2	\[ \mathsf{copysign}\left(\log \left(x + \left(x - \frac{\color{blue}{-0.5}}{x}\right)\right), x\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.95:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x - \frac{-0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	82.2%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 6
Accuracy	64.3%
Cost	13124

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

Alternative 7
Accuracy	58.1%
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	51.8%
Cost	6528

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

Reproduce

herbie shell --seed 2023147 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))