Hyperbolic arcsine

Average Error: 53.0 → 0.6

Time: 8.6s

Precision: binary64

Cost: 26888

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00800922675203596:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \mathsf{fma}\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -4186642594962.26)
   (log (/ -0.5 x))
   (if (<= x 0.00800922675203596)
     (+
      (* -0.16666666666666666 (pow x 3.0))
      (fma
       (+ (* 0.075 (pow x 5.0)) (* -0.044642857142857144 (pow x 7.0)))
       1.0
       x))
     (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 <= -4186642594962.26) {
		tmp = log((-0.5 / x));
	} else if (x <= 0.00800922675203596) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + fma(((0.075 * pow(x, 5.0)) + (-0.044642857142857144 * pow(x, 7.0))), 1.0, x);
	} else {
		tmp = log((x + 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 <= -4186642594962.26)
		tmp = log(Float64(-0.5 / x));
	elseif (x <= 0.00800922675203596)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + fma(Float64(Float64(0.075 * (x ^ 5.0)) + Float64(-0.044642857142857144 * (x ^ 7.0))), 1.0, x));
	else
		tmp = log(Float64(x + hypot(1.0, x)));
	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, -4186642594962.26], N[Log[N[(-0.5 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 0.00800922675203596], N[(N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.075 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.044642857142857144 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + x), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + N[Sqrt[1.0 ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.0
Target	45.7
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if x < -4186642594962.2598

   1. Initial program 64.0

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

   2. Simplified 64.0

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 28 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around -inf 0.0

      \[\leadsto \log \color{blue}{\left(\frac{-0.5}{x}\right)} \]

   if -4186642594962.2598 < x < 0.00800922675203596078

   1. Initial program 58.5

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

   2. Simplified 58.5

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 28 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in x around 0 1.1

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

   4. Applied egg-rr 1.1

      \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.075, {x}^{5}, -0.044642857142857144 \cdot {x}^{7}\right), 1, x\right)} \]

   5. Taylor expanded in x around 0 1.1

      \[\leadsto -0.16666666666666666 \cdot {x}^{3} + \mathsf{fma}\left(\color{blue}{0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}}, 1, x\right) \]

   if 0.00800922675203596078 < x

   1. Initial program 31.5

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      Proof
      (log.f64 (+.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (Rewrite<= hypot-1-def_binary64 (sqrt.f64 (+.f64 1 (*.f64 x x)))))): 28 points increase in error, 0 points decrease in error
      (log.f64 (+.f64 x (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x x) 1))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00800922675203596:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \mathsf{fma}\left(0.075 \cdot {x}^{5} + -0.044642857142857144 \cdot {x}^{7}, 1, x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	20488

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.00800922675203596:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 2
Error	0.9
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 4.957748253372183 \cdot 10^{-15}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right) + -1\\ \end{array} \]

Alternative 3
Error	0.9
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 4.957748253372183 \cdot 10^{-15}:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 4
Error	0.8
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.04412733013991219:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(x + \frac{0.5}{x}\right)\right)\\ \end{array} \]

Alternative 5
Error	0.9
Cost	7048

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.04412733013991219:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	6856

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{elif}\;x \leq 0.04412733013991219:\\ \;\;\;\;\frac{x \cdot \left(x + 2\right)}{1 + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x \cdot 2\right)\\ \end{array} \]

Alternative 7
Error	15.9
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq -4186642594962.26:\\ \;\;\;\;\log \left(\frac{-0.5}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	30.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022298 
(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)))))