Hyperbolic sine

Percentage Accurate: 54.6% → 99.9%

Time: 4.7s

Precision: binary64

Cost: 46729

• Unsound rule application detected in e-graph. Results may not be sound.

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{e^{x} - e^{-x}}{2} \]

↓

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -2 \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 -2.0) (not (<= t_0 5e-8)))
     (/ t_0 2.0)
     (/
      (+
       (* x 2.0)
       (+
        (* 0.3333333333333333 (pow x 3.0))
        (+
         (* 0.0003968253968253968 (pow x 7.0))
         (* 0.016666666666666666 (pow x 5.0)))))
      2.0))))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

↓

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + ((0.3333333333333333 * pow(x, 3.0)) + ((0.0003968253968253968 * pow(x, 7.0)) + (0.016666666666666666 * pow(x, 5.0))))) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(x) - exp(-x)
    if ((t_0 <= (-2.0d0)) .or. (.not. (t_0 <= 5d-8))) then
        tmp = t_0 / 2.0d0
    else
        tmp = ((x * 2.0d0) + ((0.3333333333333333d0 * (x ** 3.0d0)) + ((0.0003968253968253968d0 * (x ** 7.0d0)) + (0.016666666666666666d0 * (x ** 5.0d0))))) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

↓

public static double code(double x) {
	double t_0 = Math.exp(x) - Math.exp(-x);
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 5e-8)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + ((0.3333333333333333 * Math.pow(x, 3.0)) + ((0.0003968253968253968 * Math.pow(x, 7.0)) + (0.016666666666666666 * Math.pow(x, 5.0))))) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

↓

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -2.0) or not (t_0 <= 5e-8):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * 2.0) + ((0.3333333333333333 * math.pow(x, 3.0)) + ((0.0003968253968253968 * math.pow(x, 7.0)) + (0.016666666666666666 * math.pow(x, 5.0))))) / 2.0
	return tmp

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

↓

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= -2.0) || !(t_0 <= 5e-8))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(Float64(0.3333333333333333 * (x ^ 3.0)) + Float64(Float64(0.0003968253968253968 * (x ^ 7.0)) + Float64(0.016666666666666666 * (x ^ 5.0))))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

↓

function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -2.0) || ~((t_0 <= 5e-8)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * 2.0) + ((0.3333333333333333 * (x ^ 3.0)) + ((0.0003968253968253968 * (x ^ 7.0)) + (0.016666666666666666 * (x ^ 5.0))))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2.0], N[Not[LessEqual[t$95$0, 5e-8]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0003968253968253968 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] + N[(0.016666666666666666 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -2 or 4.9999999999999998e-8 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   if -2 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 4.9999999999999998e-8

   1. Initial program 9.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot x + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -2 \lor \neg \left(e^{x} - e^{-x} \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	46729

\[\begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -2 \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + \left(0.0003968253968253968 \cdot {x}^{7} + 0.016666666666666666 \cdot {x}^{5}\right)\right)}{2}\\ \end{array} \]

Alternative 2
Accuracy	99.9%
Cost	40009

\[\begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + \left(0.3333333333333333 \cdot {x}^{3} + 0.016666666666666666 \cdot {x}^{5}\right)}{2}\\ \end{array} \]

Alternative 3
Accuracy	99.9%
Cost	39433

\[\begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -0.005 \lor \neg \left(t_0 \leq 5 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}\\ \end{array} \]

Alternative 4
Accuracy	90.2%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \lor \neg \left(x \leq 5\right):\\ \;\;\;\;\frac{0.016666666666666666 \cdot {x}^{5}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}\\ \end{array} \]

Alternative 5
Accuracy	90.0%
Cost	7040

\[\frac{x \cdot 2 + 0.016666666666666666 \cdot {x}^{5}}{2} \]

Alternative 6
Accuracy	84.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.45\right):\\ \;\;\;\;\frac{x \cdot \left(0.3333333333333333 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]

Alternative 7
Accuracy	84.5%
Cost	704

\[\frac{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2} \]

Alternative 8
Accuracy	51.8%
Cost	320

\[\frac{x \cdot 2}{2} \]

Alternative 9
Accuracy	2.8%
Cost	64

\[-1 \]

Alternative 10
Accuracy	3.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023167 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))