Hyperbolic sine

Average Error: 57.9 → 0.7

Time: 7.6s

Precision: binary64

Cost: 7040

Math

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

↓

\[\frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (/ (+ (* 2.0 x) (* 0.3333333333333333 (pow x 3.0))) 2.0))

C

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

↓

double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * pow(x, 3.0))) / 2.0;
}

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
    code = ((2.0d0 * x) + (0.3333333333333333d0 * (x ** 3.0d0))) / 2.0d0
end function

Java

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

↓

public static double code(double x) {
	return ((2.0 * x) + (0.3333333333333333 * Math.pow(x, 3.0))) / 2.0;
}

Python

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

↓

def code(x):
	return ((2.0 * x) + (0.3333333333333333 * math.pow(x, 3.0))) / 2.0

Julia

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

↓

function code(x)
	return Float64(Float64(Float64(2.0 * x) + Float64(0.3333333333333333 * (x ^ 3.0))) / 2.0)
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((2.0 * x) + (0.3333333333333333 * (x ^ 3.0))) / 2.0;
end

Wolfram

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

↓

code[x_] := N[(N[(N[(2.0 * x), $MachinePrecision] + N[(0.3333333333333333 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 57.9

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

2. Taylor expanded in x around 0 0.7

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]

3. Final simplification 0.7

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}{2} \]

Alternatives

Alternative 1
Error	0.7
Cost	7040

\[\frac{x + \left(x + 0.3333333333333333 \cdot {x}^{3}\right)}{2} \]

Alternative 2
Error	1.1
Cost	320

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

Alternative 3
Error	52.0
Cost	192

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

Reproduce

herbie shell --seed 2023092 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))