Rust f64::asinh

Percentage Accurate: 30.6% → 99.5%

Time: 8.9s

Precision: binary64

Cost: 79048

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

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 -5.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 2e-8)
       (copysign
        (+
         (* -0.16666666666666666 (pow x 3.0))
         (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
        x)
       (copysign (log (+ x (hypot 1.0 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 <= -5.0) {
		tmp = copysign(-log((hypot(1.0, x) - x)), x);
	} else if (t_0 <= 2e-8) {
		tmp = copysign(((-0.16666666666666666 * pow(x, 3.0)) + ((0.075 * pow(x, 5.0)) + (x + (-0.044642857142857144 * pow(x, 7.0))))), x);
	} else {
		tmp = copysign(log((x + hypot(1.0, 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 <= -5.0) {
		tmp = Math.copySign(-Math.log((Math.hypot(1.0, x) - x)), x);
	} else if (t_0 <= 2e-8) {
		tmp = Math.copySign(((-0.16666666666666666 * Math.pow(x, 3.0)) + ((0.075 * Math.pow(x, 5.0)) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))))), x);
	} else {
		tmp = Math.copySign(Math.log((x + Math.hypot(1.0, 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 <= -5.0:
		tmp = math.copysign(-math.log((math.hypot(1.0, x) - x)), x)
	elif t_0 <= 2e-8:
		tmp = math.copysign(((-0.16666666666666666 * math.pow(x, 3.0)) + ((0.075 * math.pow(x, 5.0)) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))), x)
	else:
		tmp = math.copysign(math.log((x + math.hypot(1.0, 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 <= -5.0)
		tmp = copysign(Float64(-log(Float64(hypot(1.0, x) - x))), x);
	elseif (t_0 <= 2e-8)
		tmp = copysign(Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0))))), x);
	else
		tmp = copysign(log(Float64(x + hypot(1.0, 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 <= -5.0)
		tmp = sign(x) * abs(-log((hypot(1.0, x) - x)));
	elseif (t_0 <= 2e-8)
		tmp = sign(x) * abs(((-0.16666666666666666 * (x ^ 3.0)) + ((0.075 * (x ^ 5.0)) + (x + (-0.044642857142857144 * (x ^ 7.0))))));
	else
		tmp = sign(x) * abs(log((x + hypot(1.0, 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, -5.0], 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, 2e-8], N[With[{TMP1 = Abs[N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(x + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], N[With[{TMP1 = Abs[N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]]]

Target

Original	30.6%
Target	99.9%
Herbie	99.5%

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

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) < -5

   1. Initial program 52.4%

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

   2. Simplified 100.0%

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

      [Start] 52.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 52.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      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 3.7%

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

      [Start] 100.0%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      flip-+ [=>] 2.2%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      div-sub [=>] 2.2%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      pow2 [=>] 2.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      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 [<=] 2.2%	\[ \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{x}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      pow2 [<=] 2.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      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 [<=] 0.6%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 0.6%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 0.6%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 0.6%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 \cdot 1 + x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 0.6%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1} + x \cdot x}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      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 [<=] 3.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   4. Simplified 100.0%

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

      [Start] 3.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 3.7%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      div-sub [<=] 5.0%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      unpow2 [=>] 5.0%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 5.0%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 5.0%	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \left(1 + \color{blue}{{x}^{2}}\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [=>] 5.0%	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \color{blue}{\left({x}^{2} + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      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%

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

      [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%

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

      [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 -5 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x) < 2e-8

   1. Initial program 8.2%

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

   2. Simplified 8.2%

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

      [Start] 8.2%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      hypot-1-def [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   3. Applied egg-rr 8.2%

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

      [Start] 8.2%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      flip-+ [=>] 8.2%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      div-sub [=>] 8.1%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right|}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      pow2 [=>] 8.1%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{2}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{{\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|\right)}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 8.1%	\[ \mathsf{copysign}\left(\log \left(\frac{{\color{blue}{x}}^{2}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      pow2 [<=] 8.1%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 7.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 7.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      hypot-udef [=>] 7.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 7.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 \cdot 1 + x \cdot x}}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 7.7%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1} + x \cdot x}{\left|x\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      fabs-sqr [=>] 3.5%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      add-sqr-sqrt [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]

   4. Simplified 8.2%

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

      [Start] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}}}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      div-sub [<=] 8.2%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{{x}^{2} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)}, x\right) \]
      unpow2 [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{x \cdot x} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{{x}^{2}} - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      unpow2 [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \left(1 + \color{blue}{{x}^{2}}\right)}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-commutative [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{{x}^{2} - \color{blue}{\left({x}^{2} + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate--r+ [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\left({x}^{2} - {x}^{2}\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      +-inverses [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      metadata-eval [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right), x\right) \]
      associate-/r* [<=] 8.2%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)}, x\right) \]
      neg-mul-1 [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right), x\right) \]
      sub-neg [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{-\color{blue}{\left(x + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}}\right), x\right) \]
      +-commutative [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{-\color{blue}{\left(\left(-\mathsf{hypot}\left(1, x\right)\right) + x\right)}}\right), x\right) \]
      distribute-neg-in [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\left(-\left(-\mathsf{hypot}\left(1, x\right)\right)\right) + \left(-x\right)}}\right), x\right) \]
      remove-double-neg [=>] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right)} + \left(-x\right)}\right), x\right) \]
      sub-neg [<=] 8.2%	\[ \mathsf{copysign}\left(\log \left(\frac{1}{\color{blue}{\mathsf{hypot}\left(1, x\right) - x}}\right), x\right) \]

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(-0.044642857142857144 \cdot {x}^{7} + x\right)\right)}, x\right) \]

   if 2e-8 < (copysign.f64 (log.f64 (+.f64 (fabs.f64 x) (sqrt.f64 (+.f64 (*.f64 x x) 1)))) x)

   1. Initial program 49.4%

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

   2. Simplified 100.0%

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

      [Start] 49.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
      +-commutative [=>] 49.4%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{1 + x \cdot x}}\right), x\right) \]
      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 100.0%

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

      [Start] 100.0%	\[ \mathsf{copysign}\left(\log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 100.0%	\[ \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      log-prod [=>] 100.0%	\[ \mathsf{copysign}\left(\color{blue}{\log 1 + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      metadata-eval [=>] 100.0%	\[ \mathsf{copysign}\left(\color{blue}{0} + \log \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      *-un-lft-identity [=>] 100.0%	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(1 \cdot \left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]
      *-un-lft-identity [<=] 100.0%	\[ \mathsf{copysign}\left(0 + \log \color{blue}{\left(\left|x\right| + \mathsf{hypot}\left(1, x\right)\right)}, x\right) \]
      add-sqr-sqrt [=>] 100.0%	\[ \mathsf{copysign}\left(0 + \log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      fabs-sqr [=>] 100.0%	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]
      add-sqr-sqrt [<=] 100.0%	\[ \mathsf{copysign}\left(0 + \log \left(\color{blue}{x} + \mathsf{hypot}\left(1, x\right)\right), x\right) \]

   4. Simplified 100.0%

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

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

3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	79048

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -5:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	71880

\[\begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 3
Accuracy	99.6%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{1}{x \cdot -2 - \frac{0.5}{x}}\right), x\right)\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right), x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.4%
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.96:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + \left(x + \frac{0.5}{x}\right)\right), x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.4%
Cost	13576

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

Alternative 6
Accuracy	99.2%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x + -0.16666666666666666 \cdot {x}^{3}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 7
Accuracy	81.8%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(-x\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 8
Accuracy	98.9%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1.26:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x + x\right), x\right)\\ \end{array} \]

Alternative 9
Accuracy	64.1%
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 10
Accuracy	58.1%
Cost	13060

\[\begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(x\right), x\right)\\ \end{array} \]

Alternative 11
Accuracy	51.5%
Cost	6528

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

Reproduce

herbie shell --seed 2023263 
(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))