Hyperbolic sine

Average Error: 57.9 → 0.0

Time: 4.1s

Precision: binary64

Cost: 6720

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (/ (* 2.0 (sinh x)) 2.0))

C

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

↓

double code(double x) {
	return (2.0 * sinh(x)) / 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 * sinh(x)) / 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 * Math.sinh(x)) / 2.0;
}

Python

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

↓

def code(x):
	return (2.0 * math.sinh(x)) / 2.0

Julia

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

↓

function code(x)
	return Float64(Float64(2.0 * sinh(x)) / 2.0)
end

MATLAB

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

↓

function tmp = code(x)
	tmp = (2.0 * sinh(x)) / 2.0;
end

Wolfram

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

↓

code[x_] := N[(N[(2.0 * N[Sinh[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 57.9

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

2. Taylor expanded in x around -inf 57.9

   \[\leadsto \frac{\color{blue}{e^{x} - e^{-1 \cdot x}}}{2} \]

3. Simplified 0.0

   \[\leadsto \frac{\color{blue}{2 \cdot \sinh x}}{2} \]
   Proof

   [Start] 57.9	\[ \frac{e^{x} - e^{-1 \cdot x}}{2} \]
   mul-1-neg [=>] 57.9	\[ \frac{e^{x} - e^{\color{blue}{-x}}}{2} \]
   *-lft-identity [<=] 57.9	\[ \frac{\color{blue}{1 \cdot e^{x}} - e^{-x}}{2} \]
   *-lft-identity [<=] 57.9	\[ \frac{1 \cdot e^{x} - \color{blue}{1 \cdot e^{-x}}}{2} \]
   distribute-lft-out-- [=>] 57.9	\[ \frac{\color{blue}{1 \cdot \left(e^{x} - e^{-x}\right)}}{2} \]
   metadata-eval [<=] 57.9	\[ \frac{\color{blue}{\frac{2}{2}} \cdot \left(e^{x} - e^{-x}\right)}{2} \]
   associate-/r/ [<=] 57.9	\[ \frac{\color{blue}{\frac{2}{\frac{2}{e^{x} - e^{-x}}}}}{2} \]
   associate-/l* [<=] 57.9	\[ \frac{\color{blue}{\frac{2 \cdot \left(e^{x} - e^{-x}\right)}{2}}}{2} \]
   associate-*r/ [<=] 57.9	\[ \frac{\color{blue}{2 \cdot \frac{e^{x} - e^{-x}}{2}}}{2} \]
   sinh-def [<=] 0.0	\[ \frac{2 \cdot \color{blue}{\sinh x}}{2} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	0.9
Cost	832

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

Alternative 2
Error	1.2
Cost	320

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

Reproduce

herbie shell --seed 2023053 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))