Hyperbolic arcsine

Average Error: 53.1 → 0.3

Time: 7.0s

Precision: binary64

Cost: 13444

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;1 + \left(-1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.032)
   (+ 1.0 (- -1.0 (log (- (hypot 1.0 x) x))))
   (log1p (+ x (+ (hypot 1.0 x) -1.0)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.032) {
		tmp = 1.0 + (-1.0 - log((hypot(1.0, x) - x)));
	} else {
		tmp = log1p((x + (hypot(1.0, x) + -1.0)));
	}
	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 <= -0.032) {
		tmp = 1.0 + (-1.0 - Math.log((Math.hypot(1.0, x) - x)));
	} else {
		tmp = Math.log1p((x + (Math.hypot(1.0, x) + -1.0)));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -0.032:
		tmp = 1.0 + (-1.0 - math.log((math.hypot(1.0, x) - x)))
	else:
		tmp = math.log1p((x + (math.hypot(1.0, x) + -1.0)))
	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 <= -0.032)
		tmp = Float64(1.0 + Float64(-1.0 - log(Float64(hypot(1.0, x) - x))));
	else
		tmp = log1p(Float64(x + Float64(hypot(1.0, x) + -1.0)));
	end
	return 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, -0.032], N[(1.0 + N[(-1.0 - N[Log[N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(x + N[(N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Target

Original	53.1
Target	45.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -0.032000000000000001

   1. Initial program 62.6

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

   2. Applied egg-rr 62.5

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

   3. Simplified 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)} \]
      Proof
      (/.f64 1 (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (+.f64 1 0)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 1 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x)))) (-.f64 (hypot.f64 1 x) x)): 29 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 1 (*.f64 x x)) (*.f64 x x))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 28 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1)) (*.f64 x x)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 x x 1)) (*.f64 x x)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 (fma.f64 x x 1) (neg.f64 (*.f64 x x)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (*.f64 x x)) (fma.f64 x x 1))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 x x))) (fma.f64 x x 1)) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (-.f64 (hypot.f64 1 x) x)): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite=> sub-neg_binary64 (+.f64 (hypot.f64 1 x) (neg.f64 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 x) (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 x)) (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite<= associate--r-_binary64 (-.f64 0 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (neg.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite<= neg-sub0_binary64 (neg.f64 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (*.f64 x x) (fma.f64 x x 1)))) (neg.f64 (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (*.f64 -1 (-.f64 (*.f64 x x) (fma.f64 x x 1))) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval 1) (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-lft-identity_binary64 (/.f64 (-.f64 (*.f64 x x) (fma.f64 x x 1)) (-.f64 x (hypot.f64 1 x)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 x x) (-.f64 x (hypot.f64 1 x))) (/.f64 (fma.f64 x x 1) (-.f64 x (hypot.f64 1 x))))): 2 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.1

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

   5. Simplified 0.1

      \[\leadsto \color{blue}{-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)} \]
      Proof
      (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-lft-identity_binary64 (+.f64 0 (neg.f64 (log.f64 (-.f64 (hypot.f64 1 x) x))))): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 0.1

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

   if -0.032000000000000001 < x

   1. Initial program 49.9

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

   2. Applied egg-rr 40.2

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

   3. Applied egg-rr 39.3

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)} \]

   4. Simplified 0.4

      \[\leadsto \color{blue}{\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) - 1\right)\right)} \]
      Proof
      (log1p.f64 (+.f64 x (-.f64 (hypot.f64 1 x) 1))): 0 points increase in error, 0 points decrease in error
      (log1p.f64 (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 x (hypot.f64 1 x)) 1))): 184 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.032:\\ \;\;\;\;1 + \left(-1 - \log \left(\mathsf{hypot}\left(1, x\right) - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	13316

\[\begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;\log \left(\frac{1}{\mathsf{hypot}\left(1, x\right) - x}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(x + \left(\mathsf{hypot}\left(1, x\right) + -1\right)\right)\\ \end{array} \]

Alternative 2
Error	29.4
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;-\log \left(\mathsf{hypot}\left(1, x\right) - x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	45.1
Cost	13056

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

Reproduce

herbie shell --seed 2022334 
(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)))))