Hyperbolic arcsine

Average Error: 53.2 → 0.3

Time: 6.7s

Precision: binary64

Cost: 20488

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -1.06)
   (log (/ -1.0 (+ (* x 2.0) (* 0.5 (/ 1.0 x)))))
   (if (<= x 1.25)
     (+
      (* -0.16666666666666666 (pow x 3.0))
      (+ (* 0.075 (pow x 5.0)) (+ x (* -0.044642857142857144 (pow x 7.0)))))
     (log (+ x x)))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -1.06) {
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 1.25) {
		tmp = (-0.16666666666666666 * pow(x, 3.0)) + ((0.075 * pow(x, 5.0)) + (x + (-0.044642857142857144 * pow(x, 7.0))));
	} else {
		tmp = log((x + x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log((x + sqrt(((x * x) + 1.0d0))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.06d0)) then
        tmp = log(((-1.0d0) / ((x * 2.0d0) + (0.5d0 * (1.0d0 / x)))))
    else if (x <= 1.25d0) then
        tmp = ((-0.16666666666666666d0) * (x ** 3.0d0)) + ((0.075d0 * (x ** 5.0d0)) + (x + ((-0.044642857142857144d0) * (x ** 7.0d0))))
    else
        tmp = log((x + x))
    end if
    code = tmp
end function

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 <= -1.06) {
		tmp = Math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	} else if (x <= 1.25) {
		tmp = (-0.16666666666666666 * Math.pow(x, 3.0)) + ((0.075 * Math.pow(x, 5.0)) + (x + (-0.044642857142857144 * Math.pow(x, 7.0))));
	} else {
		tmp = Math.log((x + x));
	}
	return tmp;
}

Python

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

↓

def code(x):
	tmp = 0
	if x <= -1.06:
		tmp = math.log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))))
	elif x <= 1.25:
		tmp = (-0.16666666666666666 * math.pow(x, 3.0)) + ((0.075 * math.pow(x, 5.0)) + (x + (-0.044642857142857144 * math.pow(x, 7.0))))
	else:
		tmp = math.log((x + 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 <= -1.06)
		tmp = log(Float64(-1.0 / Float64(Float64(x * 2.0) + Float64(0.5 * Float64(1.0 / x)))));
	elseif (x <= 1.25)
		tmp = Float64(Float64(-0.16666666666666666 * (x ^ 3.0)) + Float64(Float64(0.075 * (x ^ 5.0)) + Float64(x + Float64(-0.044642857142857144 * (x ^ 7.0)))));
	else
		tmp = log(Float64(x + 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 <= -1.06)
		tmp = log((-1.0 / ((x * 2.0) + (0.5 * (1.0 / x)))));
	elseif (x <= 1.25)
		tmp = (-0.16666666666666666 * (x ^ 3.0)) + ((0.075 * (x ^ 5.0)) + (x + (-0.044642857142857144 * (x ^ 7.0))));
	else
		tmp = log((x + 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, -1.06], N[Log[N[(-1.0 / N[(N[(x * 2.0), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.25], 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], N[Log[N[(x + x), $MachinePrecision]], $MachinePrecision]]]

Target

Original	53.2
Target	45.5
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 3 regimes

2. if x < -1.0600000000000001

   1. Initial program 62.9

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

   2. Simplified 62.9

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 27 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. Applied egg-rr 62.8

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

   4. Simplified 0.0

      \[\leadsto \log \color{blue}{\left(\frac{-1}{x - \mathsf{hypot}\left(1, x\right)}\right)} \]
      Proof
      (/.f64 -1 (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= metadata-eval (-.f64 0 1)) (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (Rewrite<= +-inverses_binary64 (-.f64 (*.f64 x x) (*.f64 x x))) 1) (-.f64 x (hypot.f64 1 x))): 41 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 x x) (+.f64 (*.f64 x x) 1))) (-.f64 x (hypot.f64 1 x))): 0 points increase in error, 33 points decrease in error
      (/.f64 (-.f64 (*.f64 x x) (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 x x)))) (-.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 (+.f64 1 (*.f64 x x)) (-.f64 x (hypot.f64 1 x))))): 4 points increase in error, 0 points decrease in error

   5. Taylor expanded in x around -inf 0.3

      \[\leadsto \log \left(\frac{-1}{\color{blue}{2 \cdot x + 0.5 \cdot \frac{1}{x}}}\right) \]

   if -1.0600000000000001 < x < 1.25

   1. Initial program 58.6

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

   2. Simplified 58.6

      \[\leadsto \color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right)} \]
      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)))))): 27 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 0.1

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

   if 1.25 < x

   1. Initial program 32.4

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

   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)))))): 27 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 0.5

      \[\leadsto \log \color{blue}{\left(2 \cdot x\right)} \]

   4. Simplified 0.5

      \[\leadsto \log \color{blue}{\left(x + x\right)} \]
      Proof
      (+.f64 x x): 0 points increase in error, 0 points decrease in error
      (Rewrite=> count-2_binary64 (*.f64 2 x)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.06:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + \left(x + -0.044642857142857144 \cdot {x}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	13576

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.00088:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(1 + \log \left(x + \mathsf{hypot}\left(1, x\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13320

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 0.00094:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(1, x\right)\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	7236

\[\begin{array}{l} \mathbf{if}\;x \leq -0.96:\\ \;\;\;\;\log \left(\frac{-1}{x \cdot 2 + 0.5 \cdot \frac{1}{x}}\right)\\ \mathbf{elif}\;x \leq 1.25:\\ \;\;\;\;x + -0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 4
Error	0.4
Cost	7048

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

Alternative 5
Error	0.6
Cost	6856

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

Alternative 6
Error	15.5
Cost	6724

\[\begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + x\right)\\ \end{array} \]

Alternative 7
Error	30.1
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022332 
(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)))))