Hyperbolic arcsine

Average Accuracy: 17.1% → 99.8%

Time: 8.0s

Precision: binary64

Cost: 13320

Math

\[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00082:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00082)
   (- (log (- (hypot 1.0 x) x)))
   (if (<= x 0.0011)
     (+ x (* (* x x) (* x -0.16666666666666666)))
     (log (+ x (hypot 1.0 x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.00082) {
		tmp = -log((hypot(1.0, x) - x));
	} else if (x <= 0.0011) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = log((x + hypot(1.0, x)));
	}
	return tmp;
}

Java

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

↓

public static double code(double x) {
	double tmp;
	if (x <= -0.00082) {
		tmp = -Math.log((Math.hypot(1.0, x) - x));
	} else if (x <= 0.0011) {
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	} else {
		tmp = Math.log((x + Math.hypot(1.0, x)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.00082:
		tmp = -math.log((math.hypot(1.0, x) - x))
	elif x <= 0.0011:
		tmp = x + ((x * x) * (x * -0.16666666666666666))
	else:
		tmp = math.log((x + math.hypot(1.0, x)))
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= -0.00082)
		tmp = Float64(-log(Float64(hypot(1.0, x) - x)));
	elseif (x <= 0.0011)
		tmp = Float64(x + Float64(Float64(x * x) * Float64(x * -0.16666666666666666)));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.00082)
		tmp = -log((hypot(1.0, x) - x));
	elseif (x <= 0.0011)
		tmp = x + ((x * x) * (x * -0.16666666666666666));
	else
		tmp = log((x + hypot(1.0, x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[Log[N[(x + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -0.00082], (-N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), If[LessEqual[x, 0.0011], N[(x + N[(N[(x * x), $MachinePrecision] * N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	17.1%
Target	29.2%
Herbie	99.8%

\[\begin{array}{l} \mathbf{if}\;x < 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if x < -8.1999999999999998e-4

   1. Initial program 2.3%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 2.3%

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

      [Start] 2.3	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 2.3	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 2.3	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Applied egg-rr 2.3%

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

      [Start] 2.3	\[ \log \left(x + \mathsf{hypot}\left(1, x\right)\right) \]
      flip-+ [=>] 2.8	\[ \log \color{blue}{\left(\frac{x \cdot x - \mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      div-sub [=>] 2.2	\[ \log \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\mathsf{hypot}\left(1, x\right) \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      hypot-udef [=>] 2.3	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{\sqrt{1 \cdot 1 + x \cdot x}} \cdot \mathsf{hypot}\left(1, x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      hypot-udef [=>] 2.3	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\sqrt{1 \cdot 1 + x \cdot x} \cdot \color{blue}{\sqrt{1 \cdot 1 + x \cdot x}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      add-sqr-sqrt [<=] 2.3	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1 \cdot 1 + x \cdot x}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 2.3	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{\color{blue}{1} + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   4. Simplified 99.7%

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

      [Start] 2.3	\[ \log \left(\frac{x \cdot x}{x - \mathsf{hypot}\left(1, x\right)} - \frac{1 + x \cdot x}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      div-sub [<=] 3.3	\[ \log \color{blue}{\left(\frac{x \cdot x - \left(1 + x \cdot x\right)}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      +-commutative [=>] 3.3	\[ \log \left(\frac{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate--r+ [=>] 51.0	\[ \log \left(\frac{\color{blue}{\left(x \cdot x - x \cdot x\right) - 1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      +-inverses [=>] 99.7	\[ \log \left(\frac{\color{blue}{0} - 1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      metadata-eval [=>] 99.7	\[ \log \left(\frac{\color{blue}{-1}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]

   5. Applied egg-rr 99.7%

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

      [Start] 99.7	\[ \log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      *-un-lft-identity [=>] 99.7	\[ \log \color{blue}{\left(1 \cdot \frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      *-commutative [=>] 99.7	\[ \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)} \cdot 1\right)} \]
      log-prod [=>] 99.7	\[ \color{blue}{\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right) + \log 1} \]
      metadata-eval [=>] 99.7	\[ \log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right) + \color{blue}{0} \]

   6. Simplified 99.7%

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

      [Start] 99.7	\[ \log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right) + 0 \]
      +-rgt-identity [=>] 99.7	\[ \color{blue}{\log \left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      metadata-eval [<=] 99.7	\[ \log \left(\frac{\color{blue}{\frac{1}{-1}}}{x - \mathsf{hypot}\left(1, x\right)}\right) \]
      associate-/r* [<=] 99.7	\[ \log \color{blue}{\left(\frac{1}{-1 \cdot \left(x - \mathsf{hypot}\left(1, x\right)\right)}\right)} \]
      neg-mul-1 [<=] 99.7	\[ \log \left(\frac{1}{\color{blue}{-\left(x - \mathsf{hypot}\left(1, x\right)\right)}}\right) \]
      log-rec [=>] 99.7	\[ \color{blue}{-\log \left(-\left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
      neg-sub0 [=>] 99.7	\[ -\log \color{blue}{\left(0 - \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)} \]
      sub-neg [=>] 99.7	\[ -\log \left(0 - \color{blue}{\left(x + \left(-\mathsf{hypot}\left(1, x\right)\right)\right)}\right) \]
      +-commutative [<=] 99.7	\[ -\log \left(0 - \color{blue}{\left(\left(-\mathsf{hypot}\left(1, x\right)\right) + x\right)}\right) \]
      associate--r+ [=>] 99.7	\[ -\log \color{blue}{\left(\left(0 - \left(-\mathsf{hypot}\left(1, x\right)\right)\right) - x\right)} \]
      neg-sub0 [<=] 99.7	\[ -\log \left(\color{blue}{\left(-\left(-\mathsf{hypot}\left(1, x\right)\right)\right)} - x\right) \]
      remove-double-neg [=>] 99.7	\[ -\log \left(\color{blue}{\mathsf{hypot}\left(1, x\right)} - x\right) \]

   if -8.1999999999999998e-4 < x < 0.00110000000000000007

   1. Initial program 7.7%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 7.7%

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

      [Start] 7.7	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 7.7	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 7.7	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3} + x} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) - 1\right)} + x \]
      Proof

      [Start] 100.0	\[ -0.16666666666666666 \cdot {x}^{3} + x \]
      expm1-log1p-u [=>] 100.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot {x}^{3}\right)\right)} + x \]
      expm1-udef [=>] 99.7	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(-0.16666666666666666 \cdot {x}^{3}\right)} - 1\right)} + x \]
      log1p-udef [=>] 99.7	\[ \left(e^{\color{blue}{\log \left(1 + -0.16666666666666666 \cdot {x}^{3}\right)}} - 1\right) + x \]
      add-exp-log [<=] 99.7	\[ \left(\color{blue}{\left(1 + -0.16666666666666666 \cdot {x}^{3}\right)} - 1\right) + x \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)} + x \]
      Proof

      [Start] 99.7	\[ \left(\left(1 + -0.16666666666666666 \cdot {x}^{3}\right) - 1\right) + x \]
      add-exp-log [=>] 99.7	\[ \left(\color{blue}{e^{\log \left(1 + -0.16666666666666666 \cdot {x}^{3}\right)}} - 1\right) + x \]
      log1p-udef [<=] 99.7	\[ \left(e^{\color{blue}{\mathsf{log1p}\left(-0.16666666666666666 \cdot {x}^{3}\right)}} - 1\right) + x \]
      expm1-udef [<=] 100.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.16666666666666666 \cdot {x}^{3}\right)\right)} + x \]
      expm1-log1p-u [<=] 100.0	\[ \color{blue}{-0.16666666666666666 \cdot {x}^{3}} + x \]
      *-commutative [=>] 100.0	\[ \color{blue}{{x}^{3} \cdot -0.16666666666666666} + x \]
      unpow3 [=>] 100.0	\[ \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.16666666666666666 + x \]
      associate-*l* [=>] 100.0	\[ \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)} + x \]

   if 0.00110000000000000007 < x

   1. Initial program 48.6%

      \[\log \left(x + \sqrt{x \cdot x + 1}\right) \]

   2. Simplified 99.7%

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

      [Start] 48.6	\[ \log \left(x + \sqrt{x \cdot x + 1}\right) \]
      +-commutative [=>] 48.6	\[ \log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) \]
      hypot-1-def [=>] 99.7	\[ \log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00082:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.0011:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	99.5%
Cost	7240

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

Alternative 3
Accuracy	99.4%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;x \leq -0.95:\\ \;\;\;\;-\log \left(x \cdot -2 - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;-\log \left(x \cdot -2\right)\\ \mathbf{elif}\;x \leq 1.3:\\ \;\;\;\;x + \left(x \cdot x\right) \cdot \left(x \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 6
Accuracy	76.0%
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.3:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Accuracy	51.6%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

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